Fine-tuned Parallel Piecewise Sequential Confidence Region and Point Estimation Strategies for the Regression Parameters in a Linear Model: Big Data Context

Abstract

We introduced in our previous paper, fine-tuned parallel piecewise sequential sampling strategies for estimating the mean of a normal population. In this paper, we continue our explorations of sequential experimental designs for statistical inference in the context of big data. We develop fine-tuned parallel piecewise sequential methodologies for estimating the regression parameters in a linear model. Our proposed approaches achieve asymptotic unbiasedness of the stopping variable estimating the optimal fixed-sample size in addition to the operational efficiency with substantial time-savings when it comes to big data as a result of the parallel processing. We present a number of interesting illustrations of the theories and methodologies based on large-scale data analyses from simulated data as well as real data from a health study.

AMS Subject Classification: 62L1O, 62L05, 62L12, 62J05

Keywords

Asymptotic second-order efficiency big data data analysis fine-tuning fixed-size confidence region linear regression minimum risk point estimation parallel piecewise sequential sampling sequential sampling simulations stopping rule

1. INTRODUCTION

Mukhopadhyay and Zhang^{[1, 2, 3]} explored a series of sequential sampling strategies for a wide range of big data inference problems. In Mukhopadhyay and Zhang,^{[2, 3]} we presented a general framework for obtaining asymptotic distribution of stopping times for a variety of parametric and non-parametric inference problems, giving rise to a better understanding, both theoretically and practically, of final sample sizes that may be extremely large.

Mukhopadhyay and Zhang^[1] developed fine-tuned parallel piecewise sequential sampling strategies for estimating the mean of a normal population by exploiting the idea of parallel processing initially introduced by Mukhopadhyay and Sen^[4] With appropriate fine-tuning, our proposed methodology not only achieved operational efficiency in practice but also overcame the (asymptotic) bias of the stopping variable as an estimate for the optimal fixed-sample size.

In this paper, we extend the idea of parallel processing, or distributed computing, to sequential experimental designs for estimation of the regression parameters in a linear model. The linear regression model is one of the most important and useful tools in statistics and machine learning for modeling the relationship between a response variable (or dependent variable) and one or more explanatory variables (or independent variables). We develop appropriate and efficient sampling techniques in the contexts of both (i) fixed-size confidence region (FSCR) estimation and (ii) minimum risk point estimation (MRPE) problems.

Having recorded $n (\geq p + 1)$ observations:

(Y_{i}, x_{i 1}, x_{i 2}, \dots, x_{i p}), i = 1, 2, \dots, n,

we define $y_{n}, X_{n}, β$ and $ε_{n}$ as follows where obviously $y_{n}$ is $n \times 1, X_{n}$ is $n \times p, β$ is $p \times 1$ , and $ε_{n}$ is $n \times 1$ :

y_{n} = (\begin{matrix} Y_{1} \\ Y_{2} \\ ⋮ \\ Y_{n} \end{matrix}), X_{n} = (\begin{matrix} x_{11} & x_{12} & \dots & x_{1 p} \\ x_{21} & x_{22} & \dots & x_{2 p} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ x_{n 1} & x_{n 2} & \dots & x_{n p} \end{matrix}), β = (\begin{matrix} β_{1} \\ β_{2} \\ ⋮ \\ β_{p} \end{matrix}), and ε_{n} = (\begin{matrix} ε_{1} \\ ε_{2} \\ ⋮ \\ ε_{n} \end{matrix}) .

A customary linear regression model is expressed as

\begin{matrix} y_{n} = X_{n} β + ε_{n}, \end{matrix}

(1.1)

where $ε_{n} \sim N_{n} (0_{n \times 1}, σ^{2} I_{n \times n})$ and $r (X_{n}) = p$ (that is, $X_{n}$ has full column rank). Here, $y_{n}, X_{n}, β$ and $ε_{n}$ are, respectively, referred to as the response vector, design matrix, regression parameter vector and error vector. In some situations, one may explicitly want to show an intercept term, in which case one can simply let the first column of the design matrix take the only value 1 ( $β_{1}$ will then be the intercept) and continue to work with the model (1.1).

Consider the least squares estimator (LSE) of β and its distribution:

\begin{matrix} {\hat{β}}_{n} = {(X_{n}^{'} X_{n})}^{- 1} X_{n}^{'} y_{n} \sim N_{p} (β, σ^{2} {(X_{n}^{'} X_{n})}^{- 1 .}) \end{matrix}

(1.2)

The error variance $σ^{2}$ is estimated by the mean squared error (MSE). The MSE and its distribution are, respectively, given by

\begin{matrix} S_{n}^{2} = {(n - p)}^{- 1} {(y_{n} - X_{n} {\hat{β}}_{n})}^{'} (y_{n} - X_{n} {\hat{β}}_{n}) and (n - p) S_{n}^{2} \sim σ^{2} χ_{n - p}^{2} \end{matrix} .

(1.3)

Obviously, ${\hat{β}}_{n}$ and $S_{n}^{2}$ , respectively, estimate β and $σ^{2}$ unbiasedly, and they are the best unbiased estimators (UMVUE).

In Sections 1.1 and 1.2, we give a brief overview of FSCRs, and MRPE, respectively, for estimating the regression parameters in a linear model, based on which subsequent sampling strategies are proposed and appealing asymptotic properties of associated final sample sizes are studied in this paper. In Section 1.3, we provide the outline of this paper.

1.1. A Purely Sequential FSCR Procedure

The problem of constructing an FSCR of β is formulated as follows. Suppose that $1 - α, 0 < α < 1$ , is the preassigned confidence coefficient. Given preassigned $0 < d < \infty$ , we define an ellipsoidal confidence region of β:

\begin{matrix} R_{n} = \{β_{p \times 1} \in ℝ^{p} : n^{- 1} {({\hat{β}}_{n} - β)}^{'} (X_{n}^{'} X_{n}) ({\hat{β}}_{n} - β) \leq d^{2}\} \end{matrix} .

(1.4)

The optimal fixed-sample size is given by $C = χ_{p, α}^{2} σ^{2} / d^{2}$ had $σ^{2}$ been known, where $χ_{p, α}^{2}$ is the 100(1−α)th percentile of a $χ_{p}^{2}$ distribution.

This ellipsoidal confidence region $R_{n}$ has a fixed size in the sense that its volume is a known function of d, and d is preassigned in advance. In other words, by picking d small(er), we can appropriately shrink the volume or the maximal diameter of $R_{n}$ without adversely compromising its associated confidence coefficient.

Mukhopadhyay and Abid^[5] included the following purely sequential estimation strategy in the spirit of Gleser,^[6] Albert,^[7] and Srivastava.^{[8, 9]} Finster^{[10, 11]} broadened the ensuing approaches to generalized linear models. One may additionally review the relevant literature from chapter 12 in Mukhopadhyay and de Silva^[12] combined with Chatterjee,^{[13, 14]} Ghosh and Sen,^[15] and Sinha^[16]

One would begin with initial observations:

(Y_{i}, x_{i 1}, x_{i 2}, \dots, x_{i p}), i = 1, 2, \dots, m,

where $m (\geq p + 1)$ is the pilot sample size. Then, one will proceed with one at-a-time additional observation as needed and terminate sampling according to the stopping rule:

\begin{matrix} N \equiv N_{d} = i n f \{n \geq m : n \geq χ_{p, α}^{2} S_{n}^{2} / d^{2}\}, d > 0 \end{matrix} .

(1.4)

Termination occurs w.p. 1. Upon termination, we estimate β by the fixed-size ellipsoidal confidence region $R_{N}$ from (1.4) based on the full data set:

N, (Y_{i}, x_{i 1}, x_{i 2}, \dots, x_{i p}), i = 1, 2, \dots, N .

1.2. A Purely Sequential MRPE Procedure

Having recorded $n (\geq p + 1)$ observations, we consider the loss in estimating β by its $LSE {\hat{β}}_{n}$ :

L_{n} (β, {\hat{β}}_{n}) = A n^{- 1} {({\hat{β}}_{n} - β)}^{'} (X_{n}^{'} X_{n}) ({\hat{β}}_{n} - β) + c n

with A > 0, c > 0 prespecified. This loss function is composed of both estimation error and cost due to sampling and strikes a balance between the two.

The fixed-sample-size risk function can be expressed as

R_{n} (c) \equiv E_{β, σ} [L_{n} (β, {\hat{β}}_{n})] = A p σ^{2} n^{- 1} + c n .

The optimal fixed-sample size that minimizes the risk is given by $n^{*} = {(A p / c)}^{1 / 2} σ$ had $σ^{2}$ been known.

Mukhopadhyay^[17] introduced the stopping time for the following purely sequential procedure. Finster^{[10, 11]} broadened the ensuing approaches to generalized linear models. One may additionally review the relevant literature from chapter 12 in Mukhopadhyay and de Silva.^[12]

One would start with initial observations:

(Y_{i}, x_{i 1}, x_{i 2}, \dots, x_{i p}), i = 1, 2, \dots, m,

where $m (\geq p + 1)$ is the pilot sample size. Then, one will proceed with one at-a-time additional observation as needed and terminate sampling according to the stopping rule:

\begin{matrix} N \equiv N_{c} = i n f \{n \geq m : n \geq {(A p / c)}^{1 / 2} S_{n}\}, c > 0 \end{matrix},

(1.5)

where $S_{n}^{2}$ , the MSE, was defined in (1.3). Termination occurs w.p. 1. Upon termination, we estimate β by its $LSE {\hat{β}}_{N}$ from (1.2) based on the full data set:

N, (Y_{i}, x_{i 1}, x_{i 2}, \dots, x_{i p}), i = 1, 2, \dots, N .

1.3. Outline of the Paper

In Section 2, we introduce our proposed fine-tuned parallel piecewise sequential procedures for estimating the regression parameters in a linear model. We discuss the FSCR and MRPE problems in Sections 2.1 and 2.2, respectively.

A simulation study is presented in Section 3 to illustrate the advantage of our proposed fine-tuned parallel piecewise sequential procedures. We compare the purely sequential sampling strategy, the parallel piecewise sequential strategy and the fine-tuned parallel piecewise sequential strategy in the context of both FSCR and MRPE problems. In Section 4, we provide additional clarification and our rationale behind our choices of the specific design parameters used in Sections 3.

We provide an interesting illustration using real data from a health study in Section 5. Some concluding thoughts are given in Section 6.

2. Fine-tuned Parallel Piecewise Sequential Approaches

Implementation of a purely sequential sampling strategy can be operationally inconvenient especially in situations where the anticipated (final) sample size is large, because such a procedure collects data one at-a-time until termination.

We explore the idea of parallel processing in the regression set-up in the same way as the fine-tuned parallel piecewise sequential procedures were developed in the context of estimation for the mean of a normal population in Mukhopadhyay and Zhang.^[1] We discuss FSCR and MRPE problems, respectively, in Sections 2.1 and 2.2.

2.1. FSCR Estimation

Suppose k investigators are collecting data from the same population at the same time, but independently of each other. Suppose that $(Y_{i j}, x_{i j 1}, x_{i j 2}, \dots, x_{i j p}), i = 1, \dots, k, j = 1, \dots, n_{i}$ are independent and identically distributed (i.i.d.) observations. Having recorded $(Y_{i j}, x_{i j 1}, x_{i j 2}, \dots, x_{i j p}), j = 1, \dots, n_{i}$ from the $i$ th arm with $n_{i} \geq p + 1$ , let $S_{i n_{i}}^{2}$ be the MSE as in (1.3) for the ith arm, $i = 1, \dots, k$ .

We propose the following stopping time for the fine-tuned parallel piecewise sequential procedure:

\begin{matrix} \begin{array}{r} N \equiv N_{d} = \sum_{i = 1}^{k} N_{i}, where N_{i} \equiv N_{i d} \\ = inf \{n_{i} \geq m : n_{i} + ξ \geq \frac{1}{k} χ_{p, α}^{2} S_{i n_{i}}^{2} / d^{2}\}, d > 0, i = 1, \dots, k, \end{array} \end{matrix}

(2.1)

where $m (\geq p + 1)$ is the pilot sample size, and the fine-tuning parameter is given by

ξ = - \frac{1}{2} - \sum_{n = 1}^{\infty} n^{- 1} E [max \{0, χ_{n}^{2} - 2 n\}] \approx - 1.1828.

Termination occurs w.p. 1. Upon termination, we obtain ${\hat{β}}_{N}$ , the LSE of β as in (1.2), based on all of the observations from the k arms combined. The FSCR for β is then given by

R_{N} = \{β_{p \times 1} \in ℝ^{p} : N^{- 1} {({\hat{β}}_{N} - β)}^{'} (X_{N}^{'} X_{N}) ({\hat{β}}_{N} - β) \leq d^{2}\} .

Theorem 2.1. The stopping variable N from (2.1) enjoys the asymptotic unbiasedness property, that is,

E_{β, σ} [N] = C + o (1) as d \to 0,

if $m \geq p + 3$ .

Proof: We prove Theorem 2.1 based on the nonlinear renewal theory^[18–21] as follows. By observing (1.3), we can rewrite that

S_{i n_{i}}^{2} = {(n_{i} - p)}^{- 1} \sum_{j = 1}^{n_{i} - p} Y_{i j},

where $Y_{i 1}, Y_{i 2}, \dots, Y_{i, n_{i} - p} \overset{i.i.d.}{\sim} σ^{2} χ_{1}^{2}, i = 1, \dots, k$ . Observe that for $i = 1, \dots, k$ , we can express

n_{i} + ξ \geq \frac{1}{k} χ_{p, α}^{2} S_{i n_{i}}^{2} / d^{2} \Leftrightarrow (n_{i} - p) (n_{i} + ξ) \geq {(k d^{2})}^{- 1} χ_{p, α}^{2} σ^{2} \sum_{j = 1}^{n_{i} - p} (Y_{i j} / σ^{2})

\Leftrightarrow (n_{i} - p) (n_{i} + ξ) \geq \frac{1}{k} C \sum_{j = 1}^{n_{i} - p} W_{i j} \Leftrightarrow \sum_{j = 1}^{n_{i} - p} W_{i j} \leq k (n_{i} - p) (n_{i} + ξ) / C,

where $W_{i j} = Y_{i j} / σ^{2}, j = 1, 2, \dots, n_{i} - p$ are i.i.d. $χ_{1}^{2}$ random variables.

Let $n_{i}^{'} = n_{i} - p$ and define

Q_{i} \equiv N_{i} - p

\begin{array}{l} = \inf \{n_{i} \geq m : \sum_{j = 1}^{n_{i} - p} W_{i j} \leq k (n_{i} - p) (n_{i} + ξ) / C\} - p \\ = \inf_{n_{i}^{'}} \{n_{i}^{'} \geq m - p : \sum_{j = 1}^{n_{i}^{'}} W_{i j} \leq k n_{i}^{'} (n_{i}^{'} + p + ξ) / C\} \\ = \inf \{n_{i} \geq m - p : \sum_{j = 1}^{n_{i}} W_{i j} \leq k n_{i}^{2} [1 + (p + ξ) / n_{i}] / C\}, i = 1, \dots, k . \end{array}

(2.2)

In order to apply the nonlinear renewal theory, we follow the notations in Mukhopadhyay and de Silva^{[12, pp. 446–449]} and rewrite (2.2) as follows: For $i = 1, \dots, k$ , let

Q_{i} \equiv Q_{i} (h^{*}) = inf \{n_{i} \geq m_{0} : \sum_{j = 1}^{n_{i}} W_{i j} \leq h^{*} n_{i}^{δ} [1 + L_{0} n_{i}^{- 1} + o (n_{i}^{- 1})]\},

where $m_{0} = m - p, W_{i 1}, W_{i 2}, \dots, W_{i n_{i}} \overset{i.i.d.}{\sim} χ_{1}^{2}, h^{*} = k / C, δ = 2$ , and $L_{0} = p + ξ$ .

It follows that for $i = 1, \dots, k$ ,

θ = E [W_{i 1}] = 1, τ^{2} = E [W_{i 1}^{2}] - θ^{2} = 2, β^{*} = 1 / (δ - 1) = 1,

n_{0}^{*} = {(θ / h^{*})}^{β^{*}} = C / k, q = β^{* 2} τ^{2} / θ^{2} = 2,

κ = β^{*} ν / θ - β^{*} L_{0} - δ β^{* 2} τ^{2} / (2 θ^{2}) = v - p - 2 - ξ

with

\begin{matrix} ν = β^{*} (^{2 θ) - 1} [{(δ - 1)}^{2} θ^{2} + τ^{2}] - \sum_{n_{i} = 1}^{\infty} n_{i}^{- 1} E [\max \{0, (\sum_{j = 1}^{n_{i}} W_{i j}) - n_{i} δ θ\}] \\ = \frac{3}{2} - \sum_{n = 1}^{\infty} n^{- 1} E [\max \{0, χ_{n}^{2} - 2 n\}] \\ = \frac{3}{2} - \sum_{n = 1}^{\infty} n^{- 1} \int_{y = 2 n}^{\infty} (y - 2 n) {[Γ (n / 2)]}^{- 1} 2^{- n / 2} e^{- y / 2} y^{n / 2 - 1} d y \\ \approx 0.8172. \end{matrix}

Additionally, we have $r = \frac{1}{2}$ , since

P (W_{i 1} \leq u) = \int_{0}^{u} 2^{- 1 / 2} {[Γ (1 / 2)]}^{- 1} e^{- w / 2} w^{- 1 / 2} d w

\leq \int_{0}^{u} 2^{- 1 / 2} {[Γ (1 / 2)]}^{- 1} w^{- 1 / 2} d w = {(2 / π)}^{1 / 2} u^{1 / 2} \equiv B u^{r}, i = 1, \dots, k .

By Theorem A.4.2 in Mukhopadhyay and de Silva,^{[12, p. 448]} we claim the following: As $h^{*} \to 0 (d \to 0)$ ,

\begin{matrix} E_{β, σ} [N_{i}] = E_{β, σ} [Q_{i}] + p \\ = \{n_{0}^{*} + [ω κ + ω (ω - 1) q / 2] + o (1)\} + p \\ = \frac{1}{k} C + (ν - 2 - ξ) + o (1), i = 1, \dots, k, \end{matrix}

if $m_{0} > β^{*} / r$ , that is, if $m \geq p + 3$ . Note that here we let $ω = 1$ .

Therefore, as $d \to 0$ ,

\begin{matrix} E_{β, σ} [N] = E_{β, σ} [\sum_{i = 1}^{k} N_{i}] = k E_{β, σ} [N_{i}] \\ = C + k (ν - 2 - ξ) + o (1), \end{matrix}

(2.3)

if $m \geq p + 3$ . By noticing that $ξ = v - 2$ , it immediately follows that $E_{β, σ} [N] = C + o (1)$ as $d \to 0$ , if $m \geq p + 3$ .

We remark that without the fine-tuning parameter ξ in (2.1), the asymptotic bias of the stopping variable N relative to the optimal fixed-sample size C would be given by (by letting ξ = 0 in (2.3)): As $d \to 0$ ,

\begin{matrix} E_{β, σ} [N - C] = k (v - 2) + o (1), if m \geq p + 3, \end{matrix}

(2.4)

which is approximately k(-1.1828); when k (= number of arms or parallel pieces) is large, the undersampling due to the asymptotic bias would be substantial. We also note that the asymptotic bias (2.4) does not depend on p (= number of explanatory variables in the regression model), although $m \geq p + 3$ is required for (2.4) to hold.

2.2. Minimum Risk Point Estimation

Suppose k investigators are collecting data from the same population at the same time, but independently of each other. Suppose that $(Y_{i j}, x_{i j 1}, x_{i j 2}, \dots, x_{i j p}), i = 1, \dots, k, j = 1, \dots, n_{i}$ are i.i.d. observations. Having recorded $(Y_{i j}, x_{i j 1}, x_{i j 2}, \dots, x_{i j p}), j = 1, \dots, n_{i}$ from the ith arm with $n_{i} \geq p + 1$ , let $S_{i n_{i}}^{2}$ be the MSE as in (1.3) for the ith arm, i=1,…k.

We propose the following stopping time for the fine-tuned parallel piecewise sequential procedure:

\begin{matrix} N \equiv N_{c} = \sum_{i = 1}^{k} N_{i} where N_{i} \equiv N_{i c} \\ = \inf \{n_{i} \geq m : n_{i} + ξ_{1} \geq \frac{1}{k} {(A p / c)}^{1 / 2} S_{i n_{i}}\}, c > 0, i = 1, \dots, k, \end{matrix}

(2.5)

where $m (\geq p + 1)$ is the pilot sample size, and the fine-tuning parameter is given by

ξ_{1} = - \frac{1}{2} \sum_{n = 1}^{\infty} n^{- 1} E [max \{0, χ_{n}^{2} - 3 n\}] \approx - 0.1166.

Termination occurs w.p. 1. Upon termination, we estimate β by ${\hat{β}}_{N}$ from (1.2) based on all of the observations from the k arms combined.

Theorem 2.2. The stopping variable N from (2.5) enjoys the asymptotic unbiasedness property, that is,

E_{β, σ} [N] = n^{*} + o (1) as c \to 0,

if $m \geq p + 2$ .

Proof: We prove Theorem 2.2 based on the nonlinear renewal theory^[18–21] as follows. Again by observing (1.3), we can rewrite that

S_{i n_{i}}^{2} = {(n_{i} - p)}^{- 1} \sum_{j = 1}^{n_{i} - p} Y_{i j},

where $Y_{i 1}, Y_{i 2}, \dots, Y_{i, n_{i} - p} \overset{i.i.d.}{\sim} σ^{2} χ_{1}^{2}, i = 1, \dots, k$ . Observe that for $i = 1, \dots, k$ , we can express

{(n_{i} + ξ_{1})}^{2} \geq \frac{1}{k^{2}} A p S_{i n_{i}}^{2} / c \Leftrightarrow (n_{i} - p) {(n_{i} + ξ_{1})}^{2} \geq {(k^{2} c)}^{- 1} A p σ^{2} \sum_{j = 1}^{n_{i} - p} (Y_{i j} / σ^{2})

\Leftrightarrow (n_{i} - p) {(n_{i} + ξ_{1})}^{2} \geq \frac{1}{k^{2}} n^{* 2} \sum_{j = 1}^{n_{i} - p} W_{i j} \Leftrightarrow \sum_{j = 1}^{n_{i} - p} W_{i j} \leq k^{2} (n_{i} - p) {(n_{i} + ξ_{1})}^{2} / n^{* 2},

where $W_{i j} = Y_{i j} / σ^{2}, j = 1, 2, \dots, n_{i} - p$ are i.i.d. $χ_{1}^{2}$ random variables.

Let $n_{i}^{'} = n_{i} - p$ and define: For $i = 1, \dots, k$ ,

\begin{matrix} Q_{i} \equiv N_{i} - p \\ = inf \{n_{i} \geq m : \sum_{j = 1}^{n_{i} - p} W_{i j} \leq k^{2} (n_{i} - p) {(n_{i} + ξ_{1})}^{2} / n^{* 2}\} - p \\ = inf_{n_{i}^{'}} \{n_{i}^{'} \geq m - p : \sum_{j = 1}^{n_{i}^{'}} W_{i j} \leq k^{2} n_{i}^{'} {(n_{i}^{'} + p + ξ_{1})}^{2} / n^{* 2}\}, \end{matrix}

which implies

\begin{matrix} Q_{i} = inf \{n_{i} \geq m - p : \sum_{j = 1}^{n_{i}} W_{i j} \leq k^{2} n_{i}^{3} {[1 + (p + ξ_{1}) / n_{i}]}^{2} / n^{* 2}\} \\ = \inf \{n_{i} \geq m - p : \sum_{j = 1}^{n_{i}} W_{i j} \leq k^{2} n_{i}^{3} [1 + 2 (p + ξ_{1}) / n_{i} + {(p + ξ_{1})}^{2} / n_{i}^{2}] / n^{* 2}\} . \end{matrix}

(2.6)

In order to apply the nonlinear renewal theory, we again follow the notations in Mukhopadhyay and de Silva^{[12, pp. 446-449]} and rewrite (2.6) as follows: For $i = 1, \dots, k$ , let

Q_{i} \equiv Q_{i} (h^{*}) = inf \{n_{i} \geq m_{0} : \sum_{j = 1}^{n_{i}} W_{i j} \leq h^{*} n_{i}^{δ} [1 + L_{0} n_{i}^{- 1} + o (n_{i}^{- 1})]\},

where $m_{0} = m - p, W_{i 1}, W_{i 2}, \dots, W_{i n_{i}} \overset{i.i.d.}{\sim} χ_{1}^{2}, h^{*} = k^{2} / n^{* 2}, δ = 3$ , and $L_{0} = 2 (p + ξ_{1})$ .

It follows that for $i = 1, \dots, k$ ,

θ = E [W_{i 1}] = 1, τ^{2} = E [W_{i 1}^{2}] - θ^{2} = 2, β^{*} = 1 / (δ - 1) = \frac{1}{2},

n_{0}^{*} = {(θ / h^{*})}^{β^{*}} = n^{*} / k, q = β^{* 2} τ^{2} / θ^{2} = \frac{1}{2},

κ = β^{*} v / θ - β^{*} L_{0} - δ β^{* 2} τ^{2} / (2 θ^{2}) = \frac{1}{2} v - p - \frac{3}{4} - ξ_{1}

with

\begin{array}{l} v = β^{*} (^{2 θ) - 1} [{(δ - 1)}^{2} θ^{2} + τ^{2}] - \sum_{n_{i} = 1}^{\infty} n_{i}^{- 1} E [\max \{0, (\sum_{j = 1}^{n_{i}} W_{i j}) - n_{i} δ θ\}] \\ = \frac{3}{2} - \sum_{n = 1}^{\infty} n^{- 1} E [\max \{0, χ_{n}^{2} - 3 n\}] \\ = \frac{3}{2} - \sum_{n = 1}^{\infty} n^{- 1} \int_{y = 3 n}^{\infty} (y - 3 n) [^{Γ (n / 2)] - 1} 2^{- n / 2} e^{- y / 2} y^{n / 2 - 1} d y \\ \approx 1.2669. \end{array}

Additionally, we again have $r = \frac{1}{2}$ , since

\begin{matrix} P (W_{i 1} \leq u) = \int_{0}^{u} 2^{- 1 / 2} {[Γ (1 / 2)]}^{- 1} e^{- w / 2} w^{- 1 / 2} d w \\ \leq \int_{0}^{u} 2^{- 1 / 2} {[Γ (1 / 2)]}^{- 1} w^{- 1 / 2} d w = {(2 / π)}^{1 / 2} u^{1 / 2} \equiv B u^{r}, i = 1, \dots, k . \end{matrix}

By Theorem A.4.2 in Mukhopadhyay and de Silva,^{[12, p. 448]} we claim the following: As $h^{*} \to 0 (c \to 0)$ ,

\begin{matrix} E_{β, σ} [N_{i}] = E_{β, σ} [Q_{i}] + p \\ = \{n_{0}^{*} + [ω κ + ω (ω - 1) q / 2] + o (1)\} + p \\ = \frac{1}{k} n^{*} + (\frac{1}{2} v - \frac{3}{4} - ξ_{1}) + o (1), i = 1, \dots, k, \end{matrix}

if $m_{0} > β^{*} / r$ , that is, if $m \geq p + 2$ . Note that here we let $ω = 1$ .

Therefore, as $c \to 0$ ,

\begin{array}{l} E_{β, σ} [N] = E_{β, σ} [\sum_{i = 1}^{k} N_{i}] = k E_{β, σ} [N_{i}] \\ = n^{*} + k (\frac{1}{2} v - \frac{3}{4} - ξ_{1}) + o (1), \end{array}

(2.7)

if $m \geq p + 2$ . Finally, by noticing that $ξ_{1} = \frac{1}{2} v - \frac{3}{4}$ , it immediately follows that $E_{β, σ} [N] = n^{*} + o (1)$ as $c \to 0$ , if $m \geq p + 2$ .

We remark that without the fine-tuning parameter $ξ_{1}$ in (2.5), the asymptotic bias of the stopping variable N relative to the optimal fixed-sample size $n^{*}$ would be given by (by letting $ξ_{1} = 0$ in (2.7)): As $c \to 0$ ,

\begin{matrix} E_{β, σ} [N - n^{*}] = k (\frac{1}{2} v - \frac{3}{4}) + o (1), if m \geq p + 2, \end{matrix}

(2.8)

which is approximately k(−0.1166); when k (= number of arms or parallel pieces) is large, the undersampling due to the asymptotic bias could be substantial. We also note that the asymptotic bias (2.8) does not depend on p (= number of explanatory variables in the regression model), although $m \geq p + 2$ is required for (2.8) to hold.

3. Summaries from Simulation Studies

First, we summarize performances in the case of a FSCR estimation problem and then we analogously report summaries of performances in the case of a MRPE problem. We accomplish both with the help of large-scale simulations that we had undertaken.

3.1. FSCR Estimation

In this section, we present Tables 1 –6 to compare the purely sequential procedure (1.5), the fine-tuned parallel piecewise sequential procedure (2.1), as well as the parallel piecewise sequential procedure without fine-tuning (by letting ξ = 0 in (2.1)), implemented for FSCR estimation for the regression coefficient

β_{1} = 2

in the linear model

Y_{i} = β_{1} x_{i 1} + ε_{i} with ε_{i} \overset{i.i.d.}{\sim} N (0, 1), i = 1, \dots, n, \dots,

Table 1.

Purely Sequential Procedure for FSCR with $β_{1} = 2, σ^{2} = I, α = 0.05, m = 5, R = 10^{6}$ and ‘Time’ to Completion Shown in Seconds.

C	d	$\bar{n}$	se $(\bar{n})$	$\bar{n} / C$	$\bar{n} - C$	$\hat{p}$	se $(\hat{p})$	Time
30	0.358	26.91	0.010	0.90	-3.09	0.905	0.000	647
50	0.277	47.12	0.013	0.94	-2.88	0.924	0.000	1,187
100	0.196	97.96	0.017	0.98	-2.04	0.940	0.000	2,556
200	0.139	198.45	0.022	0.99	-1.55	0.946	0.000	5,391
500	0.088	498.64	0.033	1.00	-1.36	0.949	0.000	14,076
1,000	0.062	998.76	0.045	1.00	-1.24	0.949	0.000	29,735
2,000	0.044	1,998.81	0.064	1.00	-1.19	0.950	0.000	66,770
5,000	0.028	4,998.79	0.100	1.00	-1.21	0.950	0.000	202,003
10,000	0.020	9,999.06	0.142	1.00	-0.94	0.950	0.000	499,410

Table 2.

Fine-tuned Purely Sequential Procedure for FSCR with $β_{1} = 2, σ^{2} = I, α = 0.05, m = 5, R = 10^{6}$ and 'Time' to Completion Shown in Seconds.

C	d	$\bar{n}$	$s e (\bar{n})$	$\bar{n} / C$	$\bar{n} - C$	$\hat{p}$	$s e (\hat{p})$	Time
30	0.358	28.64	0.010	0.95	-1.36	0.920	0.000	655
50	0.277	48.82	0.012	0.98	-1.18	0.933	0.000	1,172
100	0.196	99.45	0.016	0.99	-0.55	0.945	0.000	2,477
200	0.139	199.74	0.021	1.00	-0.26	0.947	0.000	5,179
500	0.088	499.89	0.032	1.00	-0.11	0.949	0.000	13,455
1,000	0.062	999.96	0.045	1.00	-0.04	0.950	0.000	28,394
2,000	0.044	2,000.07	0.063	1.00	0.07	0.950	0.000	62,182
5,000	0.028	5,000.06	0.100	1.00	0.06	0.949	0.000	192,308
10,000	0.020	10,000.10	0.142	1.00	0.10	0.950	0.000	499,226

Table 3.

Parallel Piecewise Sequential Procedure for FSCR with $β_{1} = 2, σ^{2} = I, α = 0.05, m = 5, k = 2$ , $R = 10^{6}$ and 'Time' to Completion Shown in Seconds.

C	d	$\bar{n}$	$se (\bar{n})$	$\bar{n} / C$	$\bar{n} - C$	$\hat{p}$	$se (\hat{p})$	Time
30	0.358	26.11	0.008	0.87	-3.89	0.916	0.000	290
50	0.277	44.13	0.013	0.88	-5.87	0.920	0.000	536
100	0.196	94.23	0.018	0.94	-5.77	0.936	0.000	1,223
200	0.139	195.98	0.023	0.98	-4.02	0.945	0.000	2,651
500	0.088	497.03	0.034	0.99	-2.97	0.949	0.000	6,996
1,000	0.062	997.34	0.046	1.00	-2.66	0.949	0.000	14,279
2,000	0.044	1,997.56	0.064	1.00	-2.44	0.950	0.000	30,537
5,000	0.028	4,997.56	0.101	1.00	-2.44	0.950	0.000	84,928
10,000	0.020	9,997.83	0.142	1.00	-2.17	0.950	0.000	194,986

Table 4.

Fine-tuned Parallel Piecewise Sequential Procedure for FSCR with $β_{1} = 2, σ^{2} = I, α = 0.05, m = 5$ , $k = 2, R = 10^{6}$ and 'Time' to Completion Shown in Seconds.

C	d	$\bar{n}$	$se (\bar{n})$	$\bar{n} / C$	$\bar{n} - C$	$\hat{p}$	$se (\hat{p})$	Time
30	0.358	28.80	0.008	0.96	-1.20	0.932	0.000	305
50	0.277	47.44	0.012	0.95	-2.56	0.932	0.000	555
100	0.196	97.68	0.017	0.98	-2.32	0.942	0.000	1,220
200	0.139	198.92	0.022	0.99	-1.08	0.947	0.000	2,569
500	0.088	499.62	0.033	1.00	-0.38	0.949	0.000	6,780
1,000	0.062	999.81	0.046	1.00	-0.19	0.950	0.000	13,843
2,000	0.044	1,999.93	0.064	1.00	-0.07	0.950	0.000	29,070
5,000	0.028	5,000.01	0.100	1.00	0.01	0.949	0.000	82,739
10,000	0.020	10,000.22	0.142	1.00	0.22	0.950	0.000	195,910

Table 5.

Parallel Piecewise Sequential Procedure for FSCR with $β_{1} = 2, σ^{2} = I, α = 0.05, m = 5, k = 5$ , $R = 10^{6}$ and 'Time' to Completion Shown in Seconds.

C	d	$\bar{n}$	$se (\bar{n})$	$\bar{n} / C$	$\bar{n} - C$	$\hat{p}$	$se (\hat{p})$	Time
30	0.358	32.81	0.005	1.09	2.81	0.957	0.000	109
50	0.277	45.74	0.009	0.91	-4.26	0.934	0.000	178
100	0.196	87.03	0.017	0.87	-12.97	0.927	0.000	406
200	0.139	184.60	0.026	0.92	-15.40	0.937	0.000	951
500	0.088	489.97	0.037	0.98	-10.03	0.947	0.000	2,655
1,000	0.062	992.17	0.048	0.99	-7.83	0.949	0.000	5,557
2,000	0.044	1,993.27	0.066	1.00	-6.73	0.949	0.000	11,581
5,000	0.028	4,993.70	0.102	1.00	-6.30	0.950	0.000	30,848
10,000	0.020	9,994.16	0.143	1.00	-5.84	0.950	0.000	64,022

Table 6.

Fine-tuned Parallel Piecewise Sequential Procedure for FSCR with $β_{1} = 2, σ^{2} = I, α = 0.05, m = 5$ , $k = 5, R = 10^{6}$ and 'Time' to Completion Shown in Seconds.

C	d	$\bar{n}$	$se (\bar{n})$	$\bar{n} / C$	$\bar{n} - C$	$\hat{p}$	$se (\hat{p})$	Time
30	0.358	36.17	0.005	1.21	6.17	0.966	0.000	116
50	0.277	51.00	0.009	1.02	1.00	0.948	0.000	195
100	0.196	94.80	0.017	0.95	-5.20	0.939	0.000	426
200	0.139	193.35	0.025	0.97	-6.65	0.943	0.000	948
500	0.088	497.25	0.035	0.99	-2.75	0.949	0.000	2,566
1,000	0.062	998.85	0.047	1.00	-1.15	0.949	0.000	5,349
2,000	0.044	1,999.42	0.065	1.00	-0.58	0.950	0.000	10,966
5,000	0.028	4,999.87	0.101	1.00	-0.13	0.950	0.000	29,045
10,000	0.020	10,000.17	0.142	1.00	0.17	0.950	0.000	63,762

where the values of $x_{i 1}, i = 1, \dots, n, \dots$ , are randomly generated from the Uniform(0, 1) distribution. We note that in this case (p = 1) the FSCR for $β \equiv β_{1}$ is equivalent to a fixed-width confidence interval for $β_{1}$ . We consider the preassigned confidence coefficient 1 – α = 0.95, the initial sample size m = 5, the optimal fixed-sample size

C = 30, 50, 100, 200, 500, 1, 000, 2, 000, 5, 000, 10, 000,

and the number of replications R = 10⁶. Two scenarios, k = 2,5, are included for the parallel piecewise sequential procedures (with or without fine-tuning).

The term $E_{β, σ} [N]$ is estimated by the average final sample size $\bar{n}$ with its standard error $se (\bar{n})$ . The observed values of both $\bar{n} / C$ and $\bar{n} - C$ are computed to illustrate the asymptotic first- and second-order efficiencies of the stopping variables in the spirits Theorems 12.2.3(ii) and 12.2.4(i), respectively, from Mukhopadhyay and de Silva.^[12] We also present the observed coverage probability $\hat{p}$ from R = 10⁶ replications with its standard error $se (\hat{p})$ . The last column in each table is the time (in seconds) taken to complete the R = 10⁶ replications in each case (each row in the table). The simulation program is implemented in MATLAB R2017a through a Linux c4.8xlarge (Intel Xeon E5-2666 v3 (Haswell) CPU and 60GB RAM) instance on Amazon Elastic Compute Cloud.

We note that for a parallel piecewise sequential procedure (with or without fine-tuning) the data collection process is completed as soon as the last operator is done sampling and the final estimation can be given immediately after by pooling data from all arms together. Indeed, the observed runtime of our simulation program is consistent with this intuition. The time-savings are substantial using the parallel piecewise sequential strategy and increase as k increases. For k = 2, we find that the parallel piecewise sequential procedure (with or without fine-tuning) for C = 10,000 takes approximately the same time (around 1.9 × 10⁵ seconds) as the purely sequential procedure for C = 5,000, that the former for C = 2,000 takes approximately the same time (around 2.9 × 10⁴ seconds) as the latter for C = 1,000, and so on. The time savings are even more remarkable when k increases to 5 -the parallel piecewise sequential procedure for C = 10,000 now takes approximately the same time (around 6.4 × 10⁴ seconds) as the purely sequential procedure for C = 2,000, that the former for C = 5,000 takes approximately the same time (around 2.9 × 10⁴ seconds) as the latter for C = 1,000, and so on.

As shown in Tables 1, 3 and 5, respectively, and consistent with (2.4), we find that the estimate $\bar{n} - C$ of $E_{β, σ} [N - C]$ from the purely sequential procedure converges to approximately -1.18 as $d \to 0$ , that this estimate from the parallel piecewise sequential procedure (without fine-tuning) with k = 2 arms converges to approximately -2.37 as $d \to 0$ , and that this estimate from the parallel piecewise sequential procedure (without fine-tuning) with $k = 5$ arms converges to approximately -5.91 as $d \to 0$ . Without fine-tuning, the undersampling due to the asymptotic bias (2.4) of the stopping variable relative to the optimal fixed-sample size becomes increasingly substantial as k increases.

On the other hand, according to Tables 2, 4 and 6, the estimate $\bar{n} - C$ of $E_{β, σ} [N - C]$ from the finetuned parallel piecewise sequential procedure (2.1) with either k = 1 (the fine-tuned purely sequential procedure), k = 2, or k = 5 converges to 0 as $d \to 0$ . This is consistent with Theorem 2.1 that the stopping variable N from (2.1) is an asymptotically unbiased estimator of the optimal fixed-sample size $C$ . We note that the point estimate $\bar{n} - C$ of $E_{β, σ} [N - C]$ is quite accurate, as $s e (\bar{n})$ appears to be reasonably small throughout.

Additionally, in all Tables 1 –6, we observe that the estimate $\bar{n} / C$ of $E_{β, σ} [N / C]$ converges to 1 as $d \to 0$ . This is expected as the asymptotic second-order efficiency implies the asymptotic first-order efficiency. We also briefly comment that as $d \to 0$ , the observed coverage probability $\hat{p}$ converges to the preassigned confidence coefficient 1 – α = 0.95 as expected according to the asymptotic consistency property from Mukhopadhyay and de Silva.^{[12, Theorem 12.2.3(iii)]}

3.2. Minimum Risk Point Estimation

In this section, we present Tables 7 –12 to compare the purely sequential procedure (1.6), the fine-tuned parallel piecewise sequential procedure (2.5), as well as the parallel piecewise sequential procedure without fine-tuning (by letting

ξ_{1} = 0

in (2.5)), implemented for MRPE for the regression coefficient

β_{1} = 2

in the linear model

Y_{i} = β_{1} x_{i 1} + ε_{i}

with

ε_{i} \overset{i.i.d.}{\sim} N (0,1), i = 1, \dots, n, \dots

, where the values of

x_{i 1}, i = 1, \dots, n, \dots

, are randomly generated from the Uniform(0, 1) distribution. We consider

A = 50,000, m = 5

, the optimal fixed-sample size

n^{*} = 30,50,100,200,500,1,000,2,000,5,000,10,000, and R = 10^{6} .

Table 7.

Purely Sequential Procedure for MRPE with $β_{1} = 2, σ^{2} = I, A = 50,000, m = 5, R = 10^{6}$ and 'Time' to Completion Shown in Seconds.

$n^{*}$	$c$	$\bar{n}$	$se (\bar{n})$	$\bar{n} / n^{*}$	$\bar{n} - n^{*}$	$\bar{r} / R_{n^{*}}$	$\bar{r} - R_{n^{*}}$	$se (\bar{r})$	Time
30	55.556	29.69	0.004	0.99	-0.31	1.02	55.072	0.320	685
50	20.000	49.81	0.005	1.00	-0.19	1.01	13.599	0.117	1,249
100	5.000	99.86	0.007	1.00	-0.14	1.00	2.784	0.030	2,652
200	1.250	199.87	0.010	1.00	-0.13	1.00	0.658	0.010	5,567
500	0.200	499.89	0.016	1.00	-0.11	1.00	0.101	0.000	14,478
1,000	0.050	999.90	0.022	1.00	-0.10	1.00	0.025	0.000	30,345
2,000	0.013	1,999.85	0.032	1.00	-0.15	1.00	0.006	0.000	66,229
5,000	0.002	4,999.88	0.050	1.00	-0.12	1.00	0.001	0.000	203,195
10,000	0.001	9,999.90	0.071	1.00	-0.10	1.00	0.000	0.000	521,217

Table 8.

Fine-tuned Purely Sequential Procedure for MRPE with $β_{1} = 2, σ^{2} = I, A = 50000, m = 5, R = 10^{6}$ and 'Time' to Completion Shown in Seconds.

$n^{*}$	$c$	$\bar{n}$	$se (\bar{n})$	$\bar{n} / n^{*}$	$\bar{n} - n^{*}$	$\bar{r} / R_{n^{*}}$	$\bar{r} - R_{n^{*}}$	$se (\bar{r})$	Time
30	55.556	29.82	0.004	0.99	-0.18	1.02	52.679	0.304	714
50	20.000	49.93	0.005	1.00	-0.07	1.01	13.412	0.116	1,278
100	5.000	99.96	0.007	1.00	-0.04	1.00	2.785	0.031	2,679
200	1.250	199.98	0.010	1.00	-0.02	1.00	0.654	0.010	5,571
500	0.200	499.98	0.016	1.00	-0.02	1.00	0.101	0.000	14,462
1,000	0.050	1,000.00	0.022	1.00	0.00	1.00	0.025	0.000	30,380
2,000	0.013	2,000.00	0.032	1.00	-0.00	1.00	0.006	0.000	66,259
5,000	0.002	5,000.03	0.050	1.00	0.03	1.00	0.001	0.000	203,224
10,000	0.001	10,000.00	0.071	1.00	0.00	1.00	0.000	0.000	521,230

Table 9.

Parallel Piecewise Sequential Procedure for MRPE with $β_{1} = 2, σ^{2} = 1, A = 50,000, m = 5, k = 2$ , $R = 10^{6}$ and 'Time' to Completion Shown in Seconds.

$n^{*}$	$c$	$\bar{n}$	$se (\bar{n})$	$\bar{n} / n^{*}$	$\bar{n} - n^{*}$	$\bar{r} / R_{n^{*}}$	$\bar{r} - R_{n^{*}}$	$se (\bar{r})$	Time
30	55.556	28.93	0.005	0.96	-1.07	1.02	61.133	0.137	281
50	20.000	49.23	0.006	0.98	-0.77	1.01	17.169	0.046	577
100	5.000	99.61	0.007	1.00	-0.39	1.00	2.967	0.007	1,274
200	1.250	199.70	0.010	1.00	-0.30	1.00	0.669	0.001	2,687
500	0.200	499.73	0.016	1.00	-0.27	1.00	0.102	0.000	7,086
1,000	0.050	999.73	0.022	1.00	-0.27	1.00	0.025	0.000	14,450
2,000	0.013	1,999.74	0.032	1.00	-0.26	1.00	0.006	0.000	30,370
5,000	0.002	4,999.78	0.050	1.00	-0.22	1.00	0.001	0.000	86,167
10,000	0.001	9,999.78	0.071	1.00	-0.22	1.00	0.000	0.000	203,240

Table 10.

Fine-tuned Parallel Piecewise Sequential Procedure for MRPE with $β_{1} = 2, σ^{2} = I, A = 50,000$ , $m = 5, k = 2, R = 10^{6}$ and 'Time' to Completion Shown in Seconds.

$n^{*}$	$c$	$\bar{n}$	$se (\bar{n})$	$\bar{n} / n^{*}$	$\bar{n} - n^{*}$	$\bar{r} / R_{n^{*}}$	$\bar{r} - R_{n^{*}}$	$se (\bar{r})$	Time
30	55.556	29.20	0.005	0.97	-0.80	1.02	57.336	0.128	334
50	20.000	49.50	0.006	0.99	-0.50	1.01	16.436	0.043	599
100	5.000	99.85	0.007	1.00	-0.15	1.00	2.916	0.006	1,333
200	1.250	199.94	0.010	1.00	-0.06	1.00	0.661	0.001	2,802
500	0.200	499.97	0.016	1.00	-0.03	1.00	0.102	0.000	7,364
1,000	0.050	999.99	0.022	1.00	-0.01	1.00	0.025	0.000	15,005
2,000	0.013	2,000.00	0.032	1.00	-0.00	1.00	0.006	0.000	32,208
5,000	0.002	4,999.98	0.050	1.00	-0.02	1.00	0.001	0.000	90,258
10,000	0.001	10,000.13	0.071	1.00	0.13	1.00	0.000	0.000	202,245

Table 11.

Parallel Piecewise Sequential Procedure for MRPE with $β_{1} = 2, σ^{2} = I, A = 50,000, m = 5, k = 5$ , $R = 10^{6}$ and 'Time' to Completion Shown in Seconds.

$n^{*}$	$c$	$\bar{n}$	$se (\bar{n})$	$\bar{n} / n^{*}$	$\bar{n} - n^{*}$	$\bar{r} / R_{n^{*}}$	$\bar{r} - R_{n^{*}}$	$se (\bar{r})$	Time
30	55.556	31.39	0.003	1.05	1.39	1.01	18.121	0.025	63
50	20.000	47.99	0.006	0.96	-2.01	1.01	17.522	0.028	157
100	5.000	97.65	0.009	0.98	-2.35	1.00	4.593	0.009	434
200	1.250	198.81	0.011	0.99	-1.19	1.00	0.772	0.001	999
500	0.200	499.25	0.016	1.00	-0.75	1.00	0.106	0.000	2,682
1,000	0.050	999.32	0.023	1.00	-0.68	1.00	0.026	0.000	5,584
2,000	0.013	1,999.37	0.032	1.00	-0.63	1.00	0.006	0.000	11,455
5,000	0.002	4,999.38	0.050	1.00	-0.62	1.00	0.001	0.000	30,310
10,000	0.001	9,999.48	0.071	1.00	-0.52	1.00	0.000	0.000	66,356

Table 12.

Fine-tuned Parallel Piecewise Sequential Procedure for MRPE with $β_{1} = 2, σ^{2} = I, A = 50,000$ , $m = 5, k = 5, R = 10^{6}$ and 'Time' to Completion Shown in Seconds.

$n^{*}$	$c$	$\bar{n}$	$se (\bar{n})$	$\bar{n} / n^{*}$	$\bar{n} - n^{*}$	$\bar{r} / R_{n^{*}}$	$\bar{r} - R_{n^{*}}$	$se (\bar{r})$	Time
30	55.556	31.79	0.003	1.06	1.79	1.01	20.474	0.028	68
50	20.000	48.60	0.006	0.97	-1.40	1.01	16.098	0.026	162
100	5.000	98.35	0.009	0.98	-1.65	1.00	4.281	0.008	439
200	1.250	199.44	0.011	1.00	-0.56	1.00	0.751	0.001	1,000
500	0.200	499.82	0.016	1.00	-0.18	1.00	0.105	0.000	2,675
1,000	0.050	999.91	0.023	1.00	-0.09	1.00	0.026	0.000	5,580
2,000	0.013	1,999.98	0.032	1.00	-0.02	1.00	0.006	0.000	11,463
5,000	0.002	5,000.04	0.050	1.00	0.04	1.00	0.001	0.000	30,383
10,000	0.001	10,000.05	0.071	1.00	0.05	1.00	0.000	0.000	66,472

Two scenarios, $k = 2,5$ , are included for the parallel piecewise sequential procedures (with or without fine-tuning).

Again, the term $E_{β, σ} [N]$ is estimated by the average final sample size $\bar{n}$ with its standard error $se (\bar{n})$ . The observed values of both $\bar{n} / n^{*}$ and $\bar{n} - n^{*}$ are computed to illustrate the asymptotic first- and secondorder efficiencies of the stopping variables respectively. We also present the estimated risk efficiency $\bar{r} / R_{n^{*}}$ and regret $\bar{r} - R_{n^{*}}$ with the standard error $se (\bar{r})$ of the estimated risk provided as well. One may refer to Mukhopadhyay and de Silva^{[12, Section 12.3.1]} for many related details. The last column in each table is the time (in seconds) taken to complete the R = 10⁶ replications in each case (each row in the table). The simulation program is implemented in MATLAB R2017a through a Linux c4.8xlarge (Intel Xeon E5-2666 v3 (Haswell) CPU and 60GB RAM) instance on Amazon Elastic Compute Cloud.

We again note that for a parallel piecewise sequential procedure (with or without fine-tuning) the data collection process is completed as soon as the last operator is done sampling and the final estimation can be given immediately after by pooling data from all arms together. Indeed, the observed runtime of our simulation program is consistent with this intuition. The time-savings are substantial using the parallel piecewise sequential strategy and increase as k increases. For k = 2, we find that the parallel piecewise sequential procedure (with or without fine-tuning) for $n^{*} = 10,000$ takes approximately the same time (around 2.0 × 10⁵ seconds) as the purely sequential procedure for $n^{*} = 5,000$ , that the former for $n^{*} = 2,000$ takes approximately the same time (around 3.0 × 10⁴ seconds) as the latter for $n^{*} = 1,000$ , and so on. The time-savings are even more remarkable when k increases to 5 -the parallel piecewise sequential procedure for $n^{*} = 10,000$ now takes approximately the same time (around 6.6 × 10⁴ seconds) as the purely sequential procedure for $n^{*} = 2,000$ , that the former for $n^{*} = 5,000$ takes approximately the same time (around 3.0 × 10⁴ seconds) as the latter for $n^{*} = 1,000$ , and so on.

As shown in Tables 7, 9 and 11, respectively, and consistent with (2.8), the estimate $\bar{n} - n^{*}$ of $E_{β, σ} [N - n^{*}]$ from the purely sequential procedure converges to approximately -0.12 as $c \to 0$ , that this estimate from the parallel piecewise sequential procedure (without fine-tuning) with $k = 2 arms$ converges to approximately -0.23 as $c \to 0$ , and that this estimate from the parallel piecewise sequential procedure (without fine-tuning) with $k = 5$ arms converges to approximately -0.58 as $c \to 0$ . Without fine-tuning, the undersampling due to the asymptotic bias (2.8) of the stopping variable relative to the optimal fixed-sample size becomes increasingly noticeable as k increases and can be substantial for large k.

On the other hand, according to Tables 8, 10 and 12, the estimate $\bar{n} - n^{*}$ of $E_{β, σ} [N - n^{*}]$ from the fine-tuned parallel piecewise sequential procedure (2.5) with either $k = 1$ (the fine-tuned purely sequential procedure), $k = 2$ , or $k = 5$ converges to 0 as $c \to 0$ . This is consistent with Theorem 2.2 that the stopping variable N from (2.5) is an asymptotically unbiased estimator of the optimal fixed-sample size $n^{*}$ . We note that the point estimate $\bar{n} - n^{*}$ of $E_{β, σ} [N - n^{*}]$ is quite accurate, as $se (\bar{n})$ appears to be reasonably small throughout.

Additionally, in all Tables 7 –12, we observe that the estimate $\bar{n} / n^{*}$ of $E_{β, σ} [N / n^{*}]$ converges to 1 as $c \to 0$ . This is expected as the asymptotic second-order efficiency implies the asymptotic firstorder efficiency. We also briefly comment that as $c \to 0$ , the estimated risk efficiency $\bar{r} / R_{n^{*}}$ and regret $\bar{r} - R_{n^{*}}$ , respectively, converge to 1 and $c / 2$ , as expected, due to the asymptotic first- and second-order risk efficiencies in the spirits of Theorems 12.3.1(iii) and 12.3.2(i) from Mukhopadhyay and de Silva.^[12]

4. Why Did We Not Include Simulations with More Parameters in the Regression Model?

The simulations (Section 3) were carried out with only one parameter, $β_{1}$ , in the model. Yet, the last rowcolumn entry of Table 8 shows that the time to round up the simulations in a single case shows 521,230 seconds ( $\approx 6$ days). Parallel networking with $k (> 1)$ arms and fine-tuning helped in the reduction of the time requirements towards completion. More specifically, as a benchmark, for optimal fixed-sample size being $n^{*} = 10,000$ , the one-million-rep simulation runtime for the purely sequential procedure was nearly 6 days, while the (fine-tuned) parallel piecewise sequential procedure took about 64,000 seconds, which was approximately 0.75 days.

A reviewer rightfully remarked that now-a-days, given the easy and cheap availability of computing resources, it would be great if we included an illustration under simulations with more parameters in the model. We include a brief discussion of the time complexity by highlighting in Table 13 the total run time to completion in the case of Tables 1 –12.

Table 13.

Total Time in Seconds (as well as in Days) Needed to Individually Compile Successively Tables 1 –12.

FSCR			MRPE
Table	Total Time to Complete a Table		Table	Total Time to Complete a Table
Table	In Secs	In Days	Table	In Secs	In Days
#1: PS	821,775	9.5113	#7: PS	845,617	9.7872
#2: FTPS	805,048	9.3177	#8: FTPS	845,797	9.7893
#3: PP₂S	336,426	3.8938	#9: PP₂S	346,132	4.0062
#4: FTPP₂S	332,991	3.8541	#10: FTPP₂S	352,148	4.0758
#5: PP₅S	116,307	1.3461	#11: PP₅S	118,040	1.3662
#6: FTPP₂S	113,373	1.3122	#12: FTPP₂S	118,242	1.3685
Total	2.5259 × 10⁶	29.235	Total	2.6260 × 10⁶	30.394

In Table 13, we have used the following abbreviations (or legends) for quickly identifying the exact methodology under which each table from Section 3 was constructed:

\begin{matrix} PS: Purely sequential \\ FTPS: Fine-tuned purely sequential \\ {PP}_{k} S : Parallel piecewise sequential with k pieces \\ {FTPP}_{k} S : Fine-tuned parallel piecewise sequential with k pieces \end{matrix}

Understandably, the entries shown in Table 13 correspond to estimation of a single parameter $β_{1}$ in a linear regression model: $Y_{i} = β_{1} x_{i 1} + ε_{i}$ with $ε_{i} \overset{i.i.d.}{\sim} N (0,1)$ . Obviously, those entries are not absolute in any sense, and yet they emphasize the scale of computations we needed under $10^{6}$ replications. We may reiterate that $10^{6}$ replications helped us to grasp any visible impact of second-order asymptotics from some of the rows in Tables 1 –12. Fewer than 10⁶ replications did not help us in this regard. Perhaps we should have used 10¹⁰ replications instead, but that was out of reach.

A personal laptop computer may not be entirely appropriate to run such lengthy jobs since a laptop may seriously overheat after few hours of prolonged number crunching or it may shut itself off or get stuck at some point without warning. Initially, we faced such problems. We ended up using powerful external computing resources to complete the simulations.

In our purely sequential and parallel piecewise sequential estimation strategies, the computation involves repeated evaluation and inversion of matrices at every step of the way as new data arrive. But, matrix inversion is generally an operation with cubic computational complexity so that it has the order $O (p^{3})$ where $p$ is the number of predictors (i.e., explanatory variables). Hence, the computational expense would increase at a cubic rate of p as p increases.

The Wikipedia^[22] (https://en.wikipedia.org/wiki/Computational_complexity_of_mathematical_operations#Matrix_algebra), suggests approximate cubic computational complexity under the regular Gauss-Jordan elimination algorithm for matrix inversions. One may additionally refer to an expert's opinion from MATLAB,^[23] the software we used for the simulations, by going to the link: https://www.mathworks.com/matlabcentral/answers/348944-can-you-give-me-a-mathematicalestimation-about-the-algorithm-complexity-of-matrix-multiplication-an, which corroborates approximate cubic computational complexity for matrix inversion in practice. We have added these links in the list of references.^{[22, 23]}

That said, if we included (i) $β_{1}, β_{2}$ in the model, or (ii) $β_{1}, \dots, β_{5}$ in the model, or (iii) $β_{1}, \dots, β_{10}$ in the model, the computational expense would increase (i) approximately 8 -fold, or (ii) approximately 125 -fold, or (iii) approximately 1,000-fold, respectively. These are rough approximations, as the real computing time may vary depending on the specifics of (i) a design matrix (i.e., whether it is sparse or dense), or (ii) the nature of the computational environment (e.g., hardware, software), or (iii) other relevant factors (e.g., prepared computer coding).

In the real data illustration included in Section 5, we allow five β parameters, but by mimicking real-life applications, we ran only a single rep instead of one million reps. Moreover, the unknown optimal fixed-sample size $n^{*}$ could be reasonably expected to lie in a proximity of 200, making requisite calculations to round up within few seconds.

5. A Real Data Illustration: A Health Study on Body Fat

In this section, we illustrate the fine-tuned parallel piecewise sequential procedures (2.1) and (2.5) using a data set from http://lib.stat.cmu.edu/datasets/bodyfat. This data set was generously supplied by Dr A. Garth Fisher who gave permission to freely distribute the data and use for non-commercial purposes. Indeed, the data site mentions this: 'Classroom use of this data set: This data set can be used to illustrate multiple regression techniques. Accurate measurement of body fat is inconvenient/costly and it is desirable to have easy methods of estimating body fat that are not inconvenient/costly.' One may refer to Behnke and Wilmore.^[24]

For the purpose of this illustration, we treat the data set of size 252 as our population. The response variable of our interest here is the percentage of body fat $(Y)$ for an individual, which has a 'true' mean of 19.15 and a standard deviation of 8.37. The five-number summary is given as follows: $min = 0.00$ , $Q_{1} = 12.47$ , median $= 19.20, Q_{3} = 25.30$ , and $max = 47.5$ . The explanatory variables we consider are age ( $x_{2}$ ), weight $(x_{3})$ , height $(x_{4})$ , and 'abdomen 2 circumference' ( $x_{5}$ , measured 'laterally, at the level of the iliac crests, and anteriorly, at the umbilicus', Behnke and Wilmore,^{[24, pp. 45-48]}). The 'true' regression equation is obtained and (approximately) given by

\begin{array}{l} \begin{matrix} Y_{i} = - 36.705 - 0.006 x_{i 2} - 0.134 x_{i 3} - 0.126 x_{i 4} + 0.960 x_{i 5} + ε_{i}, \end{matrix} \\ with ε_{i} \overset{i.i.d}{~} N (0,19.45), i = 1, \dots, 252. \end{array}

(5.1)

We note that in this case $p = 5$ and $x_{i 1} = 1, i = 1, \dots, 252$ . The (standardized) residual versus fitted value plot (Figure 1a) appears to be a random scatter and the residuals do not contradict a normal distribution as verified by the Shapiro-Wilk $(p$ -value $= .310)$ and Anderson-Darling $(p$ -value $= .506)$ normality tests. Figure 1 b is a histogram of the residuals superimposed with the $N (0,19.45)$ density curve.

Figure 1.

(a) Standardized Residual vs. Fitted Value Plot; (b) Histogram of Residuals Superimposed with N(0,19.45) Density Curve.

A fixed-size 95% confidence region for the regression coefficients with $d = 1, m = 10$ and $k = 2$ is given by $R_{n}$ as in (1.4), where

{\hat{β}}_{n}^{'} = (- 37.651, - 0.015, - 0.156, - 0.132,1.021)

and

X_{n}^{'} X_{n} = 10^{6} \times (\begin{matrix} 0.0002 & 0.0100 & 0.0404 & 0.0158 & 0.0209 \\ 0.0100 & 0.4758 & 1.7802 & 0.6978 & 0.9272 \\ 0.0404 & 1.7802 & 7.4264 & 2.8395 & 3.7959 \\ 0.0158 & 0.6978 & 2.8395 & 1.1142 & 1.4634 \\ 0.0209 & 0.9272 & 3.7959 & 1.4634 & 1.9529 \end{matrix})

with $n_{1} = 106, n_{2} = 120, n = n_{1} + n_{2} = 226, C = 215.35, n - n^{*} \approx 11$ and $S_{n}^{2} = 19.98$ . In fact, we can verify that the 'true' regression coefficients in (5.1) are indeed contained in $R_{n}$ . Additionally, compared to a purely sequential strategy, the parallel processing with two 'arms' helps reduce the length of time required for data collection by about one-half given that the two researchers are equally efficient in collecting data.

Next, let us obtain MRPE for the regression coefficients with $A = 100, c = 0.2, m = 10$ , and $k = 2$ . The point estimate turns out to be ${\hat{β}}_{n}$ where

{\hat{β}}_{n}^{'} = (- 36.027, - 0.014, - 0.150, - 0.146,1.003)

with $n_{1} = 110, n_{2} = 117, n = n_{1} + n_{2} = 227, n^{*} = 220.52, n - n^{*} \approx 6.5$ and $S_{n}^{2} = 20.32$ . It appears that the estimated regression coefficients are fairly close to the 'true' regression coefficients in (5.1). Once again, the length of time required for data collection is reduced by about one-half (compared to a purely sequential strategy) given that the two researchers are equally efficient in collecting data.

6. Some Concluding Thoughts

(a) We first saw the advantages of the fine-tuned parallel piecewise sequential sampling techniques for estimation of the unknown mean of a normal population.^[1] Now, we have substantiated appreciable benefits of similarly improvised methodologies in the estimation of unknown regression parameters in a linear model. It will be a good idea to explore possible extensions to GLM by substantially moving the area forward way beyond Finster's^{[10, 11]} contributions under fine-tuned parallel piecewise sequential sampling.

(b) There are other possible directions that one may follow in the future to make this methodology more broadly applicable in a number of substantial inference problems. Some potential future research areas may include the development of parallel piecewise sequential strategies (with appropriate fine-tuning) for (i) estimating the location of populations with distributions other than a normal distribution such as a heavy-tailed distribution, (ii) two-sample or even multi-sample estimation problems, and (iii) more general multivariate estimation problems, among others.

(c) We have presented the paper in the context of big data. It may not hit a reader right away what the role of stopping times may be in the context of big data especially since big data are generally synonymous with data sets with no cap on their size. Let us briefly provide our perspective with respect to the relationship between stopping times and big data (and data sets in general), in the context of (sequential) experimental designs for statistical inference. We begin by agreeing that when we talk about big data, we generally mean a data set whose size may be extremely large.

From the experimental designs (for statistical inference) perspective, either in the conventional (nonsequential) setting or in the sequential situation (addressed in our paper here), however, an appropriate sample size will need to be determined first in order for one to collect an appropriate amount of data through the experiment. This will then lead to subsequent statistical inference (e.g., estimation or hypothesis testing) eventually carried out to achieve the predetermined level of precision (for estimation problems) or level of significance (for hypothesis testing problems). Such relevance is aptly validated by the recent contributions of Steland and Chang^[25] among other sources.

In the context of sequential experimental designs particularly addressed in our paper, a stopping time (or stopping rule) is a way of determining the most appropriate sample size to achieve the set goal in a statistical inference problem, more specifically, to arrive at a preassigned level of precision (i.e., fixed size) in an FSCR problem or to achieve a minimized risk in a MRPE problem with predetermined 'cost' per data point (unit observation). Generally speaking, if the stopping time (final sample size) is less than the theoretical optimal fixed-sample size, we would be unable to achieve the level of precision required for the FSCR problem or the minimum risk for the MRPE problem. On the other hand, if the stopping time is greater than the optimal fixed-sample size, we would be wasting unnecessary resources (e.g., time and manpower in the traditional sense, data storage and data processing time and cost in the era of big data and cloud computing) by oversampling to achieve the set goal in a statistical inference problem that could have otherwise addressed with less effort, resources, or cost. Therefore, it is critical for one to be able to determine the most appropriate final sample size (stopping time) that is 'just right' by developing a carefully designed stopping rule, on which our paper is focused, in order to most efficiently and effectively perform our hoped statistical inference with the data collected.

(d) A reader may raise issues with our choices of the specific design parameters used in Sections 3 and 4. Those choices made are indeed due to the limitation of computation with the large-scale simulations. It is necessary though to have as many as one million repetitions because only this magnitude (or greater) can produce a reasonably small standard error of the estimate (the difference between final sample size and optimal fixed sample size) when it comes to the last case (row) in each table. However, such limitation under large-scale simulations has nothing to do with the efficiency of our methodology. In practice, instead of incorporating one million or more replications, we would only collect the data points once, and therefore the computation limitation in the sense of the large-scale simulations does not impact application of our methodology in practice.

No simulation study can cover every possible scenario, but a reasonable question may be raised in the sense that we could have added a nonzero intercept and/or an additional predictor. From the perspective of the large-scale simulations, adding either a non-zero intercept or another predictor made the completion of the simulations infeasible given the computation resources we were able to utilize. As a comparison, the time it took to complete the simulations in Mukhopadhyay and Zhang^[1] was substantially less. In fact, we were unable to complete the simulations in the present paper with only the computation resources available locally on the laptop (which we did in our paper^[1]) and had to resort to the commercial and significantly more powerful Amazon AWS Compute Cloud.

As in all simulations, our simulation study is for illustration purposes only, and it is not intended to be comprehensive in the sense of covering every possible scenario. It is our hope that simulations would not replace the solid and comprehensive theoretical proofs and derivations that we have presented in the earlier sections of the paper.

Footnotes

Acknowledgements

The editor, associate editor and the reviewer shared a number of thoughtful comments. The reviewer also raised a number of penetrating issues which led to a new Section 4 in order for us to address such issues precisely. The totality of their comments helped to improve our presentation. We thank them and we remain grateful to them.

Declaration of Conflicting Interests

The authors declare that there is no conflict of interest.

Funding

The authors received no funding from any source to produce this research.

References

Mukhopadhyay

and Zhang

Fine-tuned parallel piecewise sequential confidence interval and point estimation strategies for the mean of a normal population: big data context. In:

Steland

and Tsui

K-L

(eds) Artificial intelligence, big data and sata science in statistics: challenges and solutions in environmetrics, the natural sciences and technology. Invited chapter on Methodologies and Theoretical Studies , Switzerland: Springer Nature, 2022, pp. 51–84.

Mukhopadhyay

and Zhang

EDA on the asymptotic normality of the standardized sequential stopping times, part-I: parametric models. Seq Anal 2018; 37:342–374.

Mukhopadhyay

and Zhang

EDA on the asymptotic normality of the standardized sequential stopping times, part-II: distribution-free models. Seq Anal 2020; 39:367–398.

Mukhopadhyay

and Sen

PK.

Replicated piecewise stopping numbers and sequential analysis. Seq Anal 1993; 12: 179–197.

Mukhopadhyay

and Abid

On fixed-size confidence regions for the regression parameters. Metron 1986; 44: 297–306.

Gleser

LJ.

On the asymptotic theory of fixed-size sequential confidence bounds for linear regression parameters. Ann Math Statist 1965; 36: 463–467. Corrections: 1966; Ann Math Statist 37: 1053–1055.

Albert

Fixed size confidence ellipsoids for linear regression parameters. Ann Math Statist 1966; 37: 16021630.

Srivastava

MS.

On fixed width confidence bounds for regression parameters and mean vector. J Roy Statist Soc, Ser B 1967; 29: 132–140.

Srivastava

MS.

On fixed-width confidence bounds for regression parameters. Ann Math Statist 1971; 42: 1403–1411.

10.

Finster

A frequentist approach to sequential estimation in the general linear model. J Am Statist Assoc 1983; 78: 403–407.

11.

Finster

Estimation in the general linear model when the accuracy is specified before data collection. Ann Statist 1985; 13: 663–675.

12.

Mukhopadhyay

and de Silva

BM.

Sequential methods and their applications . Boca Raton: Chapman & Hall/CRC, 2009.

13.

Chatterjee

SK.

Sequential inference procedures of Stein's type for a class of multivariate regression problems. Ann Math Statist 1962; 33: 1039–1064.

14.

Chatterjee

SK.

Two-stage and multistage procedures. In:

Ghosh

and Sen

(eds) Handbook of sequential analysis . New York: Dekker, 1991, pp. 21–45.

15.

Ghosh

and Sen

PK.

Handbook of sequential analysis . New York: Dekker, 1991.

16.

Sinha

Bimal K.

Multivariate problems. In:

Ghosh

and Sen

(eds) Handbook of sequential analysis . New York: Dekker, 1991, pp. 199–227.

17.

Mukhopadhyay

Sequential estimation of regression parameters in Gauss-Markoff setup. J Ind Statist Assoc 1974; 12: 39–43.

18.

Woodroofe

Second-order approximations for sequential point and interval estimation. Ann Statist 1977; 5: 984–995.

19.

Woodroofe

Nonlinear renewal theory in sequential analysis. CBMS #39 . Philadelphia: SIAM, 1982.

20.

Lai

and Siegmund

A nonlinear renewal theory with applications to sequential analysis I. Ann Statist 1977; 5: 946–954.

21.

Lai

and Siegmund

A nonlinear renewal theory with applications to sequential analysis II. Ann Statist 1979; 7: 60–76.

22.

Wikipedia link. https://en.wikipedia.org/wiki/Computational_complexity_of_mathematical_operations#Matrix_algebra (2020, accessed 4 June 2024).

23.

MATLAB.

An expert's answer from MATLAB. https://www.mathworks.com/matlabcentral/answers/348944can-you-give-me-a-mathematical-estimation-about-the-algorithm-complexity-of-matrix-multiplication-an (2017, accessed 4 June 2024).

24.

Behnke

and Wilmore

JH.

Evaluation and regulation of body build and composition . Englewood Cliffs: Prentice-Hall, 1974.

25.

Steland

and Chang

Y-T.

High-confidence nonparametric fixed-width uncertainty intervals and applications to projected high-dimensional data and common mean estimation. Seq Anal 2021; 40: 97–124.

Fine-tuned Parallel Piecewise Sequential Confidence Region and Point Estimation Strategies for the Regression Parameters in a Linear Model: Big Data Context

Abstract

Keywords

1. INTRODUCTION

2. Fine-tuned Parallel Piecewise Sequential Approaches

2.1. FSCR Estimation

3.1. FSCR Estimation

Table 1.

Purely Sequential Procedure for FSCR with β 1 = 2 , σ 2 = I , α = 0.05 , m = 5 , R = 10 6 and ‘Time’ to Completion Shown in Seconds.

Fine-tuned Purely Sequential Procedure for FSCR with β 1 = 2 , σ 2 = I , α = 0.05 , m = 5 , R = 10 6 and 'Time' to Completion Shown in Seconds.

Parallel Piecewise Sequential Procedure for FSCR with β 1 = 2 , σ 2 = I , α = 0.05 , m = 5 , k = 2 , R = 10 6 and 'Time' to Completion Shown in Seconds.

Fine-tuned Parallel Piecewise Sequential Procedure for FSCR with β 1 = 2 , σ 2 = I , α = 0.05 , m = 5 , k = 2 , R = 10 6 and 'Time' to Completion Shown in Seconds.

Parallel Piecewise Sequential Procedure for FSCR with β 1 = 2 , σ 2 = I , α = 0.05 , m = 5 , k = 5 , R = 10 6 and 'Time' to Completion Shown in Seconds.

Fine-tuned Parallel Piecewise Sequential Procedure for FSCR with β 1 = 2 , σ 2 = I , α = 0.05 , m = 5 , k = 5 , R = 10 6 and 'Time' to Completion Shown in Seconds.

Table 7.

Purely Sequential Procedure for MRPE with β 1 = 2 , σ 2 = I , A = 50,000 , m = 5 , R = 10 6 and 'Time' to Completion Shown in Seconds.

Fine-tuned Purely Sequential Procedure for MRPE with β 1 = 2 , σ 2 = I , A = 50000 , m = 5 , R = 10 6 and 'Time' to Completion Shown in Seconds.

Parallel Piecewise Sequential Procedure for MRPE with β 1 = 2 , σ 2 = 1 , A = 50,000 , m = 5 , k = 2 , R = 10 6 and 'Time' to Completion Shown in Seconds.

Fine-tuned Parallel Piecewise Sequential Procedure for MRPE with β 1 = 2 , σ 2 = I , A = 50,000 , m = 5 , k = 2 , R = 10 6 and 'Time' to Completion Shown in Seconds.

Parallel Piecewise Sequential Procedure for MRPE with β 1 = 2 , σ 2 = I , A = 50,000 , m = 5 , k = 5 , R = 10 6 and 'Time' to Completion Shown in Seconds.

Fine-tuned Parallel Piecewise Sequential Procedure for MRPE with β 1 = 2 , σ 2 = I , A = 50,000 , m = 5 , k = 5 , R = 10 6 and 'Time' to Completion Shown in Seconds.

Table 13.

Total Time in Seconds (as well as in Days) Needed to Individually Compile Successively Tables 1–12.

(a) Standardized Residual vs. Fitted Value Plot; (b) Histogram of Residuals Superimposed with N(0,19.45) Density Curve.

Footnotes

Acknowledgements

Declaration of Conflicting Interests

Funding

References

Purely Sequential Procedure for FSCR with $β_{1} = 2, σ^{2} = I, α = 0.05, m = 5, R = 10^{6}$ and ‘Time’ to Completion Shown in Seconds.

Fine-tuned Purely Sequential Procedure for FSCR with $β_{1} = 2, σ^{2} = I, α = 0.05, m = 5, R = 10^{6}$ and 'Time' to Completion Shown in Seconds.

Parallel Piecewise Sequential Procedure for FSCR with $β_{1} = 2, σ^{2} = I, α = 0.05, m = 5, k = 2$ , $R = 10^{6}$ and 'Time' to Completion Shown in Seconds.

Fine-tuned Parallel Piecewise Sequential Procedure for FSCR with $β_{1} = 2, σ^{2} = I, α = 0.05, m = 5$ , $k = 2, R = 10^{6}$ and 'Time' to Completion Shown in Seconds.

Parallel Piecewise Sequential Procedure for FSCR with $β_{1} = 2, σ^{2} = I, α = 0.05, m = 5, k = 5$ , $R = 10^{6}$ and 'Time' to Completion Shown in Seconds.

Fine-tuned Parallel Piecewise Sequential Procedure for FSCR with $β_{1} = 2, σ^{2} = I, α = 0.05, m = 5$ , $k = 5, R = 10^{6}$ and 'Time' to Completion Shown in Seconds.

Purely Sequential Procedure for MRPE with $β_{1} = 2, σ^{2} = I, A = 50,000, m = 5, R = 10^{6}$ and 'Time' to Completion Shown in Seconds.

Fine-tuned Purely Sequential Procedure for MRPE with $β_{1} = 2, σ^{2} = I, A = 50000, m = 5, R = 10^{6}$ and 'Time' to Completion Shown in Seconds.

Parallel Piecewise Sequential Procedure for MRPE with $β_{1} = 2, σ^{2} = 1, A = 50,000, m = 5, k = 2$ , $R = 10^{6}$ and 'Time' to Completion Shown in Seconds.

Fine-tuned Parallel Piecewise Sequential Procedure for MRPE with $β_{1} = 2, σ^{2} = I, A = 50,000$ , $m = 5, k = 2, R = 10^{6}$ and 'Time' to Completion Shown in Seconds.

Parallel Piecewise Sequential Procedure for MRPE with $β_{1} = 2, σ^{2} = I, A = 50,000, m = 5, k = 5$ , $R = 10^{6}$ and 'Time' to Completion Shown in Seconds.

Fine-tuned Parallel Piecewise Sequential Procedure for MRPE with $β_{1} = 2, σ^{2} = I, A = 50,000$ , $m = 5, k = 5, R = 10^{6}$ and 'Time' to Completion Shown in Seconds.

Total Time in Seconds (as well as in Days) Needed to Individually Compile Successively Tables 1 –12.