On Asymptotic Standard Normality of the Two Sample Pivot

Abstract

The asymptotic solution to the problem of comparing the means of two heteroscedastic populations, based on two random samples from the populations, hinges on the pivot underpinning the construction of the confidence interval and the test statistic being asymptotically standard normal, which is known to happen if the two samples are independent and the ratio of the sample sizes converges to a finite positive number. This restriction on the asymptotic behavior of the ratio of the sample sizes carries the risk of rendering the asymptotic justification of the finite sample approximation invalid. It turns out that neither the restriction on the asymptotic behavior of the ratio of the sample sizes nor the assumption of cross sample independence is necessary for the pivotal convergence in question to take place. If the joint distribution of the standardized sample means converges to a spherically symmetric distribution, then that distribution must be bivariate standard normal (which can happen without the assumption of cross sample independence), and the aforesaid pivotal convergence holds.

AMS Classification: 62E20, 62G20

Keywords

Asymptotic normality Iterated and double limit net and subnet one-point compactification Slutsky's theorem

1. Introduction

The objective of this note is to critically examine the asymptotic solution to the problem of comparing the means of two populations with finite, unequal variances. Let $X_{1, 1}, \dots, X_{1, n_{1}}$ be a random sample from the first population with mean $μ_{1}$ and variance $σ_{1}^{2}$ , and $X_{2, 1}, \dots, X_{2, n_{2}}$ a random sample from the second population with mean $μ_{2}$ and variance $σ_{2}^{2}$ . When all the parameters $μ_{1}$ , $μ_{2}$ , $σ_{1}$ , and $σ_{2}$ are unknown, under the assumption of independence of the two samples, the traditional asymptotic $100 (1 - α)$ % confidence interval for $μ_{1} - μ_{2}$ is

({\overset{̅}{X}}_{1} - {\overset{̅}{X}}_{2} - z_{\frac{α}{2}} \sqrt{\frac{S_{1}^{2}}{n_{1}} + \frac{S_{2}^{2}}{n_{2}}}, {\overset{̅}{X}}_{1} - {\overset{̅}{X}}_{2} + z_{\frac{α}{2}} \sqrt{\frac{S_{1}^{2}}{n_{1}} + \frac{S_{2}^{2}}{n_{2}}}),

(1.1)

where, for $i = 1, 2$ , ${\overset{̅}{X}}_{i} = n_{i}^{- 1} \sum_{j = 1}^{n_{i}} X_{i, j}$ and $S_{i}^{2} = {(n_{i} - 1)}^{- 1} \sum_{j = 1}^{n_{i}} {(X_{i, j} - {\overset{̅}{X}}_{i})}^{2}$ , and $z_{α}$ denotes the $100 (1 - α)$ ^th percentile of the standard normal distribution. The test statistic for testing the null hypothesis $H_{0} : μ_{1} - μ_{2} = D_{0}$ is

\frac{{\overset{̅}{X}}_{1} - {\overset{̅}{X}}_{2} - D_{0}}{\sqrt{\frac{S_{1}^{2}}{n_{1}} + \frac{S_{2}^{2}}{n_{2}}}},

(1.2)

which is calibrated on the standard normal scale for the calculation of the observed level of significance and the rejection regions at various levels of significance.

The simple idea underlying this widely used method is frequently presented in undergraduate (e.g., Wackerly, Mendenhall, and Scheaffer [2008]) and beginning graduate (e.g., Casella and Berger [2002]) textbooks on mathematical statistics. As Casella and Berger (2002, p. 492) explain, the idea is to obtain a point estimator ${\hat{θ}}_{n}$ of the parameter of interest $θ$ with variance $σ_{n}^{2}$ such that $({\hat{θ}}_{n} - θ) / σ_{n}$ is asymptotically standard normal and, if the calculation of $σ_{n}$ involves an unknown parameter other than $θ$ , a consistent estimator ${\hat{σ}}_{n}$ of $σ_{n}$ , so that $({\hat{θ}}_{n} - θ) / {\hat{σ}}_{n}$ is also asymptotically standard normal by Slutsky's theorem (Mukhopadhyay [2000, Theorem 5.3.3]); $({\hat{θ}}_{n} - θ) / {\hat{σ}}_{n}$ can then be used as a pivot for $θ$ , and the tools for inference on the normal mean can be applied. However, Mukhopadhyay (2000) seems to be the only one to explicitly consider the two sample problem from this point of view. Assuming that the two samples are independent and the ratio $n_{1} / n_{2} \to δ \in (0, \infty)$ , he asserts (p. 544) the asymptotic standard normality of the pivot

W_{n_{1}, n_{2}} = \frac{{\overset{̅}{X}}_{1} - {\overset{̅}{X}}_{2} - (μ_{1} - μ_{2})}{\sqrt{\frac{S_{1}^{2}}{n_{1}} + \frac{S_{2}^{2}}{n_{2}}}} = \frac{({\overset{̅}{X}}_{1} - μ_{1}) - ({\overset{̅}{X}}_{2} - μ_{2})}{\sqrt{\frac{S_{1}^{2}}{n_{1}} + \frac{S_{2}^{2}}{n_{2}}}}

(1.3)

underpinning the formulas of the confidence interval in (1.1) and the test statistic in (1.2).

While asymptotically studying a statistical problem using two samples, many authors (see, among others, DasGupta [2008], Pyke and Shorack [1968], Ramdas, Trillos and Cuturi [2017], & van der Vaart and Wellner [1996]) require that $n_{1} / n_{2} \to δ \in (0, \infty)$ as $n_{1}, n_{2} \to \infty$ , though this requirement risks rendering the asymptotic justification of the finite sample approximation invalid. For example, the two-sample problem with $n_{2} > n_{1} (ln (ln n_{1}))$ (think $n_{1} = 50$ and $n_{2} \geq 69$ ) is a fairly common design, at which rate $n_{1} / n_{2} \to 0$ ; the finite sample approximation involved in using the pivot $W_{n_{1}, n_{2}}$ to draw inference on $μ_{1} - μ_{2}$ in such a design may not have a rigorous asymptotic justification.

To investigate if we can derive the asymptotic standard normality of $W_{n_{1}, n_{2}}$ without the restriction that the ratio of the sample sizes converges to a finite positive number, we formulate the question in terms of weak convergence. For a separable metric space $S$ , let $B (S)$ denote the Borel $σ$ -algebra of $S$ and $M (S)$ the set of probability measures on $B (S)$ . Endowed with the topology of weak convergence, $M (S)$ is metrizable as a separable metric space (Parthasarathy [1967, Theorem II.6.2]). Since all the random elements under consideration are Borel measurable, convergence in distribution is equivalent to weak convergence of the induced probability measures (van der Vaart and Wellner [1996, p. 18]). With

F_{n_{1}, n_{2}} \in M (R) denoting the measure induced by W_{n_{1}, n_{2}} on B (R),

(1.4)

the asymptotic standard normality of $W_{n_{1}, n_{2}}$ is equivalent to the convergence of the double sequence $F_{n_{1}, n_{2}}$ to $Φ$ in $M (R)$ , where $Φ$ denotes the standard normal measure on $B (R)$ . Faced with the question of convergence of a double sequence, we examine both the iterated and double limits of $F_{n_{1}, n_{2}}$ .

Before proceeding further, let us spell out without any ambiguity what the assumption of independence of the two samples means in this context. Hereinafter, iid will abbreviate independent and identically distributed. We are making the following three assumptions:

\{X_{1, j} : j \geq 1\} - iid sequence of random variables, mean μ_{1} and variance σ_{1}^{2}

(1.5)

\{X_{2, j} : j \geq 1\} - iid sequence of random variables, mean μ_{2} and variance σ_{2}^{2}

(1.6)

the sequence \{X_{1, j} : j \geq 1\} is independent of the sequence \{X_{2, j} : j \geq 1\} .

(1.7)

The assumption in (1.7) is often stated as the $X_{1, j}$ ’s being independent of the $X_{2, j}$ ’s. Note that the triplet of assumptions (1.5), (1.6), and (1.7) is equivalent to the pair of assumptions:

\{(X_{1, j}, X_{2, j}) : j \geq 1\} - iid sequence of random vectors

(1.8)

for every j \geq 1, X_{1, j} and X_{2, j} are independent .

(1.9)

We are not aware of any result on the iterated limits of $F_{n_{1}, n_{2}}$ . Proposition 1 observes that both iterated limits of $F_{n_{1}, n_{2}}$ equal $Φ$ under only (1.5) and (1.6), that is, without (1.7). As far as the double limit of $F_{n_{1}, n_{2}}$ is concerned, the only published result we are aware of is Mukhopadhyay (2000) cited above. However, by requiring that $n_{1} / n_{2} \to δ \in (0, \infty)$ as $n_{1}, n_{2} \to \infty$ , Mukhopadhyay (2000) does not obtain the double limit of $F_{n_{1}, n_{2}}$ .

We obtain the double limit of $F_{n_{1}, n_{2}}$ by using the fact that a double sequence is a net; see Appendix B for the definition of a net and related details. Let $N$ denote the set of natural numbers; then $N \times N$ is a directed set under the partial ordering $\underline{≻}$ defined by the condition that $(n_{1}, n_{2}) \underline{≻} (m_{1}, m_{2})$ if and only if $(n_{1} \geq m_{1} and n_{2} \geq m_{2})$ . A double sequence $\{x_{n_{1}, n_{2}} : n_{1}, n_{2} \geq 1\}$ taking values in $S$ converges to $x \in S$ as $n_{1}, n_{2} \to \infty$ if and only if the corresponding net $\{x_{α} : α \in N \times N\}$ converges to $x$ . Thus, our objective reduces to obtaining

lim_{α} F_{α} = Φ .

(1.10)

Proposition 2 shows that (1.5) and (1.6) are not sufficient for (1.10). Proposition 3 shows that (1.10) is implied by the convergence of the joint distribution of the standardized sample means to a spherically symmetric distribution, which implies that the limiting spherically symmetric distribution is bivariate standard normal (see Remark 1). Corollary 1 and Remarks 3 and 4 investigate the question of necessity of the convergence of the joint distribution of the standardized sample means to the bivariate standard normal distribution for (1.10), to which Proposition 4 furnishes a partial answer, and Remark 6 outlines the setup wherein a complete answer is obtainable. Detailed statements of these results and their proofs constitute Section 2. The technical results that we draw upon for the proofs of our results are assembled in eight lemmas spread over three Appendices, A, B, and C. Since none of these lemmas contains any original result of significance, we skip their proofs, and refer the reader to the Appendix of Majumdar and Majumdar (2017b).

2. Results and Proofs

In what follows, unless otherwise specified, we will assume that the index $i$ runs from $1$ to $2$ and $\{X_{i, j} : j \geq 1\}$ are as in (1.5) and (1.6).

Proposition 1. With $F_{n_{1}, n_{2}}$ as in (1.4),

lim_{n_{1} \to \infty} lim_{n_{2} \to \infty} F_{n_{1}, n_{2}} = lim_{n_{2} \to \infty} lim_{n_{1} \to \infty} F_{n_{1}, n_{2}} = Φ .

(2.1)

For subsequent use, let us introduce the following notations. For $k \in N$ , let ${\overset{̅}{X}}_{i, k}$ denote $\sum_{j = 1}^{k} X_{i, j} / k$ and $S_{i, k}^{2}$ denote $\sum_{j = 1}^{k} {(X_{i, j} - {\overset{̅}{X}}_{i, k})}^{2} / (k - 1)$ . For $θ \geq 0$ , let $N_{θ} \in M (R)$ denote the centered normal measure with variance $θ$ , so that $N_{1} = Φ$ and $N_{0}$ is the point mass at $0$ . For $ρ \in [- 1, 1]$ , let ${(Φ \times Φ)}_{ρ} \in M (R^{2})$ denote the bivariate normal distribution with means $0$ , variances $1$ , and correlation coefficient $ρ$ . When $ρ = 1$ (respectively, $ρ = - 1$ ), the support of the bivariate normal distribution is the one dimensional subspace represented by the straight line $y = x$ (respectively, $y = - x$ ); obviously, in either of these cases, ${(Φ \times Φ)}_{ρ}$ does not have a density with respect to the two dimensional Lebesgue measure. Since we do not make any use of the Lebesgue density of any member of the family of bivariate normal distributions, we need not accord any special treatment to either of these two cases. In this context, it may be noted that we conceptualize a bivariate normal distribution as an element $Q \in M (R^{2})$ such that every linear functional on $R^{2}$ (endowed with $Q$ ) induces a normal distribution on the line. Thus, ${(Φ \times Φ)}_{0}$ is the product measure $Φ \times Φ$ , the bivariate standard normal distribution. Let $ξ : N \mapsto N \times N$ denote the order preserving and cofinal map given by $ξ (k) = (k, k)$ . Further, let $n_{i} : N \times N \mapsto N$ denote the order preserving and cofinal map that maps $α \in N \times N$ to its $i$ ^th coordinate $n_{i} (α)$ . For $α \in N \times N$ , let

\begin{matrix} [t] & n_{α} = min (n_{1} (α), n_{2} (α)), & m_{α} = \sqrt{n_{1} (α) n_{2} (α)}, & and & e_{α} = n_{1} (α) / n_{2} (α) . \end{matrix}

Proposition 2. Let $\{(X_{1, j}, X_{2, j}) : j \geq 1\}$ be iid, with ${(Φ \times Φ)}_{ρ}$ being the common distribution. Assume $ρ \neq 0$ . Then (1.5) and (1.6) hold, but (1.10) does not.

Definition 1. For $k \in N$ , let $I (R^{k})$ denote the set of isometries on $R^{k}$ (Axler [2015, Defnition 7.37]). An element $Q \in M (R^{k})$ is called spherically symmetric if the Pettis (that is, coordinate wise) integral of the identity function with respect to $Q$ is $0 \in R^{k}$ , and, for every $O \in I (R^{k})$ , $Q \circ O^{- 1} = Q$ , where $Q \circ O^{- 1} \in M (R^{k})$ is the measure induced by $O$ , that is, $Q \circ O^{- 1} (B) = Q (\{(x_{1}, \dots, x_{k}) \in R^{k} : O (x_{1}, \dots, x_{k}) \in B\})$ for any $B \in B (R^{k})$ . Let $S (R^{k})$ denote the set of spherically symmetric elements in $M (R^{k})$ .

Proposition 3. For $α \in N \times N$ , let $Y_{α}^{(i)}$ denote the standardized sample mean

\sqrt{n_{i} (α)} ({\overset{̅}{X}}_{i, n_{i} (α)} - μ_{i}) / σ_{i}

and $H_{α} \in M (R^{2})$ the measure induced by $Y_{α} = (Y_{α}^{(1)}, Y_{α}^{(2)})$ . Then (1.10) is implied by

lim_{α} H_{α} = Q \in S (R^{2}) .

(2.2)

Remark 1. The sufficient condition for (1.10) stated in (2.2) is equivalent to

lim_{α} H_{α} = Φ \times Φ .

(2.3)

Since $Φ \times Φ \in S (R^{2})$ , (2.3) implies (2.2). To show the converse, it suffices to assume (2.2) and show $Q = Φ \times Φ$ , equivalently, $Q \circ ⟨ \cdot, (a_{1}, a_{2}) ⟩^{- 1} = Φ$ for every unit vector $(a_{1}, a_{2})$ . Since given any two unit vectors there exists an isometry mapping one to the other, by the spherical symmetry of $Q$ ,

Q \circ ⟨ \cdot, (a_{1}, a_{2}) ⟩^{- 1} = Q \circ ⟨ \cdot, (1, 0) ⟩^{- 1} .

Since, by the continuous mapping theorem for nets (Lemma C3) and (2.2),

Q \circ ⟨ \cdot, (1, 0) ⟩^{- 1} = lim_{α} H_{α} \circ ⟨ \cdot, (1, 0) ⟩^{- 1},

it suffices to show that ${lim}_{α} H_{α} \circ ⟨ \cdot, (1, 0) ⟩^{- 1} = Φ$ .

For $k \in N$ , let $P_{k}^{(i)} \in M (R)$ denote the measure induced by $\sqrt{k} ({\overset{̅}{X}}_{i, k} - μ_{i}) / σ_{i}$ , which, by (1.5) or (1.6), converges to $Φ$ by the Central Limit Theorem (CLT, hereinafter; Dudley [1989, Theorem 9.5.6]), that is,

lim_{k \in N} P_{k}^{(i)} = Φ in M (R) .

Since $n_{i}$ is order preserving and cofinal, $\{P_{n_{i} (α)}^{(i)} : α \in N \times N\}$ is a subnet of $\{P_{k}^{(i)} : k \in N\}$ ; since convergence in $M (R)$ is metrizable and any subnet of a convergent net has the same limit (Lemma B1),

lim_{α} P_{n_{i} (α)}^{(i)} = lim_{k \in N} P_{k}^{(i)} = Φ .

(2.4)

Since $H_{α} \circ ⟨ \cdot, (1, 0) ⟩^{- 1} = P_{n_{1} (α)}^{(1)}$ , the converse follows from (2.4).

Remark 2. Note that (1.7) implies

H_{α} = P_{n_{1} (α)}^{(1)} \times P_{n_{2} (α)}^{(2)};

(2.5)

since the product of two weakly convergent nets of probability measures converges to the product of the limits (Lemma C4), (2.5) implies (2.3) via (2.4). Thus, the folklore sufficient conditions (1.5), (1.6), and (1.7), equivalently, (1.8) and (1.9), do imply (2.3), and consequently (1.10), without requiring $n_{1} / n_{2}$ to converge to $δ \in (0, \infty)$ .

Remark 1. Does (1.10) imply (2.3)? While we cannot construct a counterexample where (1.10) holds, above and beyond (1.5) and (1.6), but (2.3) does not, because of Corollary 1 we do not believe that the affirmative answer holds.

Corollary 1. Recall the definition of $W_{α}$ from (1.3) and define

U_{α} = \frac{({\overset{̅}{X}}_{1, n_{1} (α)} - μ_{1}) + ({\overset{̅}{X}}_{2, n_{2} (α)} - μ_{2})}{\sqrt{\frac{S_{1, n_{1} (α)}^{2}}{n_{1} (α)} + \frac{S_{2, n_{2} (α)}^{2}}{n_{2} (α)}}};

let $G_{α} \in M (R)$ denote the measure induced by $U_{α}$ on $B (R)$ . Then (2.3) implies

lim_{α} G_{α} = Φ .

(2.6)

Remark 4. If (1.10) were to imply (2.3), by Corollary 1, (1.10) would have to imply (2.6) as well. We see no reason why (1.10), under only (1.5) and (1.6), would imply (2.6) (or vice-versa). However, Proposition 4 does connect (1.10), (2.3), and (2.6).

Proposition 4. If

L = \lim_{α} H_{α}

(2.7)

exists, and (1.10) and (2.6) hold, then (2.3) holds.

Remark 5. By (2.4) and Proposition 9.3.4 of Dudley (1989), $\{P_{k}^{(i)} : k \in N\}$ is uniformly tight. Consequently, $\{P_{n_{i} (α)}^{(i)} : α \in N \times N\}$ , being contained in $\{P_{k}^{(i)} : k \in N\}$ , is uniformly tight as well. By Tychonoff's theorem and Bonferroni's inequality,

\{H_{α} : α \in N \times N\} is uniformly tight .

(2.8)

Can we establish Proposition 4 without assuming (2.7), substituting it by the conclusion drawn in (2.8)? We do not think so, though a counterexample eludes us.

Lemma 1 is a vital cog in the wheel of our investigation of whether (1.10) implies (2.3).

Lemma 1. With ${\hat{F}}_{α} \in M (R)$ denoting the measure induced by (the non-studentized two-sample pivot)

{\hat{W}}_{α} = \frac{({\overset{̅}{X}}_{1, n_{1} (α)} - μ_{1}) - ({\overset{̅}{X}}_{2, n_{2} (α)} - μ_{2})}{\sqrt{\frac{σ_{1}^{2}}{n_{1} (α)} + \frac{σ_{2}^{2}}{n_{2} (α)}}}

on $B (R)$ , (1.10) is equivalent to

lim_{α} {\hat{F}}_{α} = Φ .

(2.9)

Remark 6. Proposition 4 strengthens our belief that (1.10) does not imply (2.3), but we are, as mentioned above, unable to construct a counterexample. One of the major obstacles to constructing such a counterexample is the fact that it is simply impossible to get a handle on the asymptotic distribution of either $\{W_{α} : α \in N \times N\}$ or $\{Y_{α} : α \in N \times N\}$ unless we are willing to assume some specific dependence structure (including independence) for the sequence $\{(X_{1, j}, X_{2, j}) : j \geq 1\}$ . If we assume, above and beyond (1.5) and (1.6), that

\{(X_{1, j}, X_{2, j}) : j \geq 1\} is an independent sequence of random vectors,

(2.10)

then Theorem 1 of Majumdar and Majumdar (2017a) shows that (1.10) implies (2.3) (which renders Proposition 4 moot), by showing that the convergence of the Cesaro means of the sequence of cross-sample correlation coefficients to $0$ is a sufficient condition for (2.3) that turns out to be necessary for (2.9), equivalently, by Lemma 1, (1.10).

The assumption in (2.10), being the assumption in (1.8) with the identically distributed requirement removed, is weaker than (1.8). The aforesaid convergence of Cesaro means assumption is substantially weaker than (1.9). It is easy to see that if we combine the convergence of Cesaro means assumption with (1.5), (1.6), and (2.10), the resulting collection is weaker than the pair of assumptions in (1.8) and (1.9). All we have to do is to consider a pair of dependent but uncorrelated random variables and a sequence of iid copies of the resulting random vector. By Theorem 1 of Majumdar and Majumdar (2017a), (2.3) can hold without (1.7).

We now present the proofs of the results stated previously.

Proof of Proposition 1. The key to the proof is the algebraic representation

W_{n_{1}, n_{2}} = \frac{\sqrt{n_{1}} ({\overset{̅}{X}}_{1} - μ_{1})}{σ_{1}} \times V_{n_{1}, n_{2}}^{(1)} - \frac{\sqrt{n_{2}} ({\overset{̅}{X}}_{2} - μ_{2})}{σ_{2}} \times V_{n_{1}, n_{2}}^{(2)},

(2.11)

where

\begin{matrix} V_{n_{1}, n_{2}}^{(1)} = \frac{σ_{1}}{\sqrt{S_{1}^{2} + \frac{n_{1}}{n_{2}} \times S_{2}^{2}}} \\ V_{n_{1}, n_{2}}^{(2)} = \frac{σ_{2}}{\sqrt{\frac{n_{2}}{n_{1}} \times S_{1}^{2} + S_{2}^{2}}} . \end{matrix}

(2.12)

Now, let us fix $n_{2}$ and let $n_{1} \to \infty$ . Since $S_{1}^{2}$ converges (in probability) to $σ_{1}^{2}$ (Lemma C1), $V_{n_{1}, n_{2}}^{(1)}$ converges to $0$ and $V_{n_{1}, n_{2}}^{(2)}$ converges to $σ_{2} / S_{2}$ . By the CLT and Slutsky's theorem, the first term in RHS(2.11) converges in distribution (and, by Theorem 4.2.9 of Fabian and Hannan [1985], in probability) to $0$ . Since the second term in RHS(2.11) converges in distribution to $\sqrt{n_{2}} ({\overset{̅}{X}}_{2} - μ_{2}) / S_{2}$ , another application of Slutsky's theorem leads to the conclusion that, for fixed $n_{2}$ , as $n_{1} \to \infty$

W_{n_{1}, n_{2}} converges in distribution to - \sqrt{n_{2}} ({\overset{̅}{X}}_{2} - μ_{2}) / S_{2} .

(2.13)

Since $S_{2}^{2}$ converges in probability to $σ_{2}^{2}$ (Lemma C1), by the CLT and Slutsky's theorem, $- \sqrt{n_{2}} ({\overset{̅}{X}}_{2} - μ_{2}) / S_{2}$ converges, as $n_{2} \to \infty$ , in distribution to $Z_{2} \sim Φ$ .

The same argument, with $n_{1}$ and $n_{2}$ interchanged, shows that if we fix $n_{1}$ and let $n_{2} \to \infty$ , $W_{n_{1}, n_{2}}$ converges in distribution to $\sqrt{n_{1}} ({\overset{̅}{X}}_{1} - μ_{1}) / S_{1}$ , which, as $n_{1} \to \infty$ , converges in distribution to $Z_{1} \sim Φ$ . □

Proof of Proposition 2. Clearly, $\{X_{i, j} : j \geq 1\}$ is an iid collection of standard Normal random variables, showing that (1.5) and (1.6) hold, with $μ_{i} = 0$ and $σ_{i} = 1$ . The measure induced by $D_{j} = X_{1, j} - X_{2, j}$ on $B (R)$ is $N_{2 (1 - ρ)}$ for every $j \in N$ , implying that the measure induced by $\sum_{j = 1}^{k} D_{j} / \sqrt{k}$ is $N_{2 (1 - ρ)}$ for every $k \in N$ . Since $S_{i, k}^{2}$ converges in probability to $1$ as $k \to \infty$ (Lemma C1) and $W_{ξ (k)} = (\sum_{j = 1}^{k} D_{j} / \sqrt{k}) {(S_{1, k}^{2} + S_{2, k}^{2})}^{- 1 / 2}$ , by Slutsky's theorem the subnet $\{F_{ξ (k)} : k \in N\}$ of $\{F_{α} : α \in N \times N\}$ converges to $N_{(1 - ρ)}$ , implying, by Lemma B1 and the assumption $ρ \neq 0$ , that (1.10) does not hold. □

Proof of Proposition 3. By Lemma B2, to show (1.10) it suffices to show that given an arbitrary subnet $\{F_{ϕ (β)} : β \in F\}$ of $\{F_{α} : α \in N \times N\}$ , there exists a further subnet $\{F_{ϕ (ϕ (δ))} : δ \in D\}$ such that $lim_{δ} F_{ϕ (ϕ (δ))} = Φ$ . For $α \in N \times N$ , (2.12) implies

V_{α}^{(i)} = \frac{σ_{i}}{\sqrt{e_{α}^{1 - i} S_{1, n_{1} (α)}^{2} + e_{α}^{2 - i} S_{2, n_{2} (α)}^{2}}} .

(2.14)

Let $[0, \infty]$ denote the one-point compactification of $[0, \infty)$ (Dudley [1989, Theorem 2.8.1]). Since every net taking values in a compact set has a convergent subnet (Lemma B3), every subnet $\{e_{ϕ (β)} : β \in F\}$ of $\{e_{α} : α \in N \times N\}$ has a further subnet $\{e_{ϕ (ϕ (δ))} : δ \in D\}$ such that

lim_{δ} e_{ϕ (ϕ (δ))} = κ \in [0, \infty] .

(2.15)

Since convergence in probability on Euclidian spaces is metrizable (Dudley [1989, Theorem 9.2.2]) and $n_{i}$ is order preserving and cofinal, by Lemma C1 and Lemma B1, $S_{i, n_{i} (α)}^{2}$ converges in probability to $σ_{i}^{2}$ ; consequently, by Lemma B1, (2.14), and (2.15), in probability

lim_{δ} V_{ϕ (ϕ (δ))}^{(i)} = a_{i} = \frac{σ_{i}}{\sqrt{κ^{1 - i} σ_{1}^{2} + κ^{2 - i} σ_{2}^{2}}} \in [0, 1] .

(2.16)

Note that $a_{i}$ depends on the subnet $\{e_{ϕ (ϕ (δ))} : δ \in D\}$ through $κ$ ; $κ = 0$ implies $a_{1} = 1$ and $a_{2} = 0$ , $κ = \infty$ implies $a_{1} = 0$ and $a_{2} = 1$ , and, in general

a_{1}^{2} + a_{2}^{2} = 1 .

(2.17)

By (2.2) and Lemma B1, $\{H_{ϕ (ϕ (δ))} : δ \in D\}$ converges to $Q$ . Since $W_{α} = ⟨ Y_{α}, V_{α} ⟩$ by (2.11), where $V_{α} = (V_{α}^{(1)}, - V_{α}^{(2)})$ , by Slutsky's theorem for nets (Lemma C2) and the continuous mapping theorem for nets (Lemma C3), (2.16) implies

lim_{δ} F_{ϕ (ϕ (δ))} = Q \circ ⟨ \cdot, (a_{1}, - a_{2}) ⟩^{- 1} .

(2.18)

Since given any two unit vectors there exists an isometry mapping one to the other, by (2.17) and the spherical symmetry of $Q$ , RHS(2.18) does not depend on the subnet $\{e_{ϕ (ϕ (δ))} : δ \in D\}$ . By Lemma B2, $lim_{α} F_{α} = F$ exists for some $F \in M (R)$ . Since an iterated limit exists and equals the double limit if the latter and the inner limit of the former exists (Lemma A1), by (2.13), $lim_{n_{2} \to \infty} lim_{n_{1} \to \infty} F_{n_{1}, n_{2}} = F$ ; since $F = Φ$ by Proposition 1, (1.10) follows. □

Remark 7. DasGupta (2008, p. 403) considers the Behrens-Fisher problem of comparing the means of two independent heteroscedastic normal populations and the two sample t-statistic that uses the pooled variance

S_{p, n_{1}, n_{2}}^{2} = \frac{(n_{1} - 1) S_{1, n_{1}}^{2} + (n_{2} - 1) S_{2, n_{2}}^{2}}{n_{1} + n_{2} - 2}

for studentization, that is

T_{n_{1}, n_{2}} = \frac{{\overset{̅}{X}}_{1, n_{1}} - {\overset{̅}{X}}_{2, n_{2}} - (μ_{1} - μ_{2})}{\sqrt{S_{p, n_{1}, n_{2}}^{2} (\frac{1}{n_{1}} + \frac{1}{n_{2}})}} = \frac{({\overset{̅}{X}}_{1, n_{1}} - μ_{1}) - ({\overset{̅}{X}}_{2, n_{2}} - μ_{2})}{\sqrt{S_{p, n_{1}, n_{2}}^{2} (\frac{1}{n_{1}} + \frac{1}{n_{2}})}},

as a potential pivot. He observes that if the ratio $n_{1} / n_{2}$ converges to $1$ , that is, the design is asymptotically balanced, then the asymptotic distribution of $T_{n_{1}, n_{2}}$ is standard normal. As outlined below, that observation is a consequence of (1.10), showing that neither the normality of the populations nor their cross-sample independence is necessary for it.

Note that $T_{α} = W_{α} \sqrt{R_{α}}$ , where

R_{α} = \frac{S_{1, n_{1} (α)}^{2} + e_{α} S_{2, n_{2} (α)}^{2}}{S_{p, α}^{2} (1 + e_{α})} .

Given an arbitrary subnet $\{T_{ϕ (β)} : β \in F\}$ of $\{T_{α} : α \in N \times N\}$ , there exists a further subnet $\{T_{ϕ (ϕ (δ))} : δ \in D\}$ such that (2.15) holds. Since $n_{i}$ is order preserving and cofinal, and convergence in probability on Euclidian spaces is metrizable, by Lemma C1 and Lemma B1, in probability

lim_{δ} S_{p, ϕ (ϕ (δ))}^{2} = σ_{1}^{2} (\frac{κ}{κ + 1}) + σ_{2}^{2} (\frac{1}{κ + 1}) = \frac{κ σ_{1}^{2} + σ_{2}^{2}}{κ + 1};

consequently, from Lemma C1 and Lemma B1 again, in probability

lim_{δ} R_{ϕ (ϕ (δ))} = \frac{σ_{1}^{2} + κ σ_{2}^{2}}{κ σ_{1}^{2} + σ_{2}^{2}} = θ (κ) .

Let $K_{α} \in M (R)$ denote the distribution induced by $T_{α}$ . Since $T_{α} = W_{α} \sqrt{R_{α}}$ , by Slutsky's theorem for nets, (1.10) implies

lim_{δ} K_{ϕ (ϕ (δ))} = N_{θ (κ)} .

For asymptotically balanced designs, $κ$ =1 for every subnet $\{e_{ϕ (ϕ (δ))} : δ \in D\}$ , and Dasgupta's observation follows. However, $T_{n_{1}, n_{2}}$ is an inferior choice for a pivot compared to $W_{n_{1}, n_{2}}$ , as asymptotic standard normality of $W_{n_{1}, n_{2}}$ holds for all designs by (1.10), whereas that of $T_{n_{1}, n_{2}}$ holds only for asymptotically balanced designs.

Proof of Corollary 1. The proof of Proposition 3 applies verbatim once we replace $V_{α}$ in that proof by $V_{α}^{*} = (V_{α}^{(1)}, V_{α}^{(2)})$ and note that $U_{α} = ⟨ Y_{α}, V_{α}^{*} ⟩$ . □

Proof of Proposition 4. As observed in Remark 1, $L = Φ \times Φ$ , that is, (2.3) holds, if and only if $L \circ ⟨ \cdot, (a_{1}, a_{2}) ⟩^{- 1} = Φ$ for every unit vector $(a_{1}, a_{2}) \in R^{2}$ .

Corresponding to every $a_{1} \in [0, 1]$ , there exist two unit vectors: $(a_{1}, \sqrt{1 - a_{1}^{2}})$ in the first quadrant and $(a_{1}, - \sqrt{1 - a_{1}^{2}})$ in the fourth quadrant. Since $Φ$ is invariant under the map $x \mapsto - x$ , it suffices to show that for every $a_{1} \in [0, 1]$ ,

L \circ ⟨ \cdot, (a_{1}, \sqrt{1 - a_{1}^{2}}) ⟩^{- 1} = Φ = L \circ ⟨ \cdot, (a_{1}, - \sqrt{1 - a_{1}^{2}}) ⟩^{- 1} .

(2.19)

Given $a_{1} \in [0, 1]$ , let $κ (a_{1}) = {(σ_{1} σ_{2}^{- 1})}^{2} (a_{1}^{- 2} - 1) \in [0, \infty]$ , where $κ (0)$ is interpreted to be $\infty$ . We will show that for every $κ \in [0, \infty]$ , there exists a directed set $F_{κ}$ and an order preserving and cofinal $ϕ_{κ} : F_{κ} \mapsto N \times N$ such that the subnet $\{e_{ϕ_{κ} (β)} : β \in F_{κ}\}$ of $\{e_{α} : α \in N \times N\}$ converges to $κ$ , after using this fact to establish (2.19).

Let $\{e_{ϕ_{κ (a_{1})} (β)} : β \in F\}$ be the subnet of $\{e_{α} : α \in N \times N\}$ that converges to $κ (a_{1})$ . That, as in (2.16), implies, in probability (and hence, in distribution)

\begin{matrix} lim_{β} V_{ϕ_{κ (a_{1})} (β)}^{(1)} = \frac{σ_{1}}{\sqrt{σ_{1}^{2} + κ (a_{1}) σ_{2}^{2}}} = a_{1} \\ lim_{β} V_{ϕ_{κ (a_{1})} (β)}^{(2)} = \frac{σ_{2}}{\sqrt{\frac{σ_{1}^{2}}{κ (a_{1})} + σ_{2}^{2}}} = \sqrt{1 - a_{1}^{2}} . \end{matrix}

By Lemma B1 and (2.7),

lim_{β} H_{ϕ_{κ (a_{1})} (β)} = L;

using $W_{α} = ⟨ Y_{α}, V_{α} ⟩$ and $U_{α} = ⟨ Y_{α}, V_{α}^{*} ⟩$ from the proofs of Proposition 3, and Corollary 1, Lemma C2, and Lemma C3,

\begin{matrix} lim_{β} F_{ϕ_{κ (a_{1})} (β)} = L \circ ⟨ \cdot, (a_{1}, - \sqrt{1 - a_{1}^{2}}) ⟩^{- 1} \\ lim_{β} G_{ϕ_{κ (a_{1})} (β)} = L \circ ⟨ \cdot, (a_{1}, \sqrt{1 - a_{1}^{2}}) ⟩^{- 1}, \end{matrix}

whence (2.19) follows from (2.6) and (1.10) by Lemma B1.

If $κ = \infty$ , let $F_{\infty} = N \times N$ and define $ϕ_{\infty} (α) = (n_{1} (α), J (\sqrt{n_{1} (α)}))$ , where $J$ is the integer part function on $[1, \infty)$ . Clearly, $ϕ_{\infty}$ is order preserving. To show that $ϕ_{\infty}$ is cofinal, given $α \in N \times N$ choose $\hat{α} \in N \times N$ such that $n_{1} (\hat{α}) = max (n_{1} (α), {(n_{2} (α))}^{2})$ ; since $J (\sqrt{n_{1} (\hat{α})}) \geq n_{2} (α)$ , $ϕ_{\infty} (\hat{α}) \underline{≻} α$ . Since $e_{ϕ_{\infty} (α)} = n_{1} (α) / J (\sqrt{n_{1} (α)}) \geq \sqrt{n_{1} (α)}$ , the convergence of $e_{ϕ_{\infty} (α)}$ to $κ = \infty$ follows.

If $κ = 0$ , let $F_{0} = N \times N$ and define $ϕ_{0} (α) = (n_{2} (α), {(n_{2} (α))}^{2})$ . Clearly, $ϕ_{0}$ is order preserving. To show that $ϕ_{0}$ is cofinal, given $α \in N \times N$ choose $\hat{α} \in N \times N$ such that $n_{2} (\hat{α}) = max (n_{1} (α), n_{2} (α))$ ; trivially, ${(n_{2} (\hat{α}))}^{2} \geq n_{2} (α)$ , implying $ϕ_{0} (\hat{α}) \underline{≻} α$ . Since $e_{ϕ_{0} (α)} = n_{2} (α) / {(n_{2} (α))}^{2} = 1 / n_{2} (α)$ , the convergence of $e_{ϕ_{0} (α)}$ to $κ = 0$ follows.

If $κ \in (0, \infty)$ , let $F_{κ} = N$ and define $ϕ_{κ} (r) = (ϕ_{κ}^{'} (r), r + 1)$ , where

ϕ_{κ}^{'} (r) = J (max ((r + 1) (κ - 1 / r), 0)) + 1

is nondecreasing in $r$ , implying that $ϕ_{κ}$ is order preserving. To show that $ϕ_{κ}$ is cofinal, given $α \in N \times N$ choose $\hat{r} \in N$ to equal $max (J (| \overset{*}{r} (κ, α) | + 1), n_{2} (α))$ , where

\overset{*}{r} (κ, α) = \frac{\sqrt{{(κ - 1 - n_{1} (α))}^{2} + 4 κ} - (κ - 1 - n_{1} (α))}{2 κ};

since $κ > 0$ , ${(κ - 1 - n_{1} (α))}^{2} + 4 κ$ is positive and $\overset{*}{r} (κ, α)$ is well-defined. Also, $κ > 0$ implies the quadratic (in $r$ ) $κ r^{2} + (κ - 1 - n_{1} (α)) r - 1$ is convex, so that every $r \in N$ greater than $\overset{*}{r} (κ, α)$ , the bigger root of the quadratic, satisfies the inequality

κ r^{2} + (κ - 1 - n_{1} (α)) r - 1 > 0,

equivalently, the inequality

(r + 1) (κ - 1 / r) > n_{1} (α) .

Since $\hat{r} > | \overset{*}{r} (κ, α) | \geq \overset{*}{r} (κ, α)$ , $ϕ_{κ}^{'} (\hat{r}) > n_{1} (α)$ , implying $ϕ_{κ} (\hat{r}) \underline{≻} α$ . Finally, from the definition of $J$ ,

ϕ_{κ}^{'} (r) - 1 \leq max ((r + 1) (κ - 1 / r), 0) < ϕ_{κ}^{'} (r),

implying

max ((κ - 1 / r), 0) < \frac{ϕ_{κ}^{'} (r)}{r + 1} \leq max ((κ - 1 / r), 0) + \frac{1}{r + 1};

since $κ > 0$ and $ϕ_{κ} (r) = (ϕ_{κ}^{'} (r), r + 1)$ , $lim_{r} e_{ϕ_{κ} (r)} = κ$ . □

Proof of Lemma 1. For $α \in N \times N$ , let

Q_{α} = \frac{\frac{σ_{1}^{2}}{n_{1} (α)} + \frac{σ_{2}^{2}}{n_{2} (α)}}{\frac{S_{1, n_{1} (α)}^{2}}{n_{1} (α)} + \frac{S_{2, n_{2} (α)}^{2}}{n_{2} (α)}} = \frac{σ_{1}^{2} + e_{α} σ_{2}^{2}}{S_{1, n_{1} (α)}^{2} + e_{α} S_{2, n_{2} (α)}^{2}} = \frac{e_{α}^{- 1} σ_{1}^{2} + σ_{2}^{2}}{e_{α}^{- 1} S_{1, n_{1} (α)}^{2} + S_{2, n_{2} (α)}^{2}},

(2.20)

so that

W_{α} = {\hat{W}}_{α} \sqrt{Q_{α}} .

By Lemmas C2 and C3, it suffices to show that $Q_{α}$ converges in distribution to $1$ . Since convergence in distribution is metrizable, by Lemma B2 it suffices to show that given an arbitrary subnet $\{Q_{ϕ (β)} : β \in F\}$ of $\{Q_{α} : α \in N \times N\}$ , there exists a further subnet $\{Q_{ϕ (ϕ (δ))} : δ \in D\}$ such that $Q_{ϕ (ϕ (δ))}$ converges in distribution to 1. Recall from (2.15) the existence of a subnet $\{e_{ϕ (ϕ (δ))} : δ \in D\}$ that converges to $κ$ . Since $\{e_{ϕ (ϕ (δ))} : δ \in D\}$ converges (in distribution) to $κ$ and $S_{i, n_{i} (α)}^{2}$ converges in probability (and hence, in distribution) to $σ_{i}^{2}$ , the assertion follows from (2.19) (for $κ = \infty$ , use the last representation) and Lemmas C2 and C3. □

Remark 8. To what extent can the findings of this note be extended to the $k$ –sample problem? Obviously, there is no unique parameter of interest in the $k$ –sample problem that can be interpreted as the natural extension of $μ_{1} - μ_{2}$ and consequently, there is no natural extension of the pivot $W_{n_{1}, n_{2}}$ to a pivot $W_{n_{1}, n_{2}, \dots, n_{k}}$ . That said, Majumdar and Majumdar (2017a) conjectures in Remark 6 that if the assumption

\{(X_{1, j}, X_{2, j}, \dots, X_{k, j}) : j \geq 1\} an independent sequence of random vectors in R^{k},

which is an extension of (2.10), holds, then the assumption of entry wise convergence of the Cesaro means of the sequence of dispersion matrices of $\{(X_{1, j}, X_{2, j}, \dots, X_{k, j}) : j \geq 1\}$ to the $k \times k$ identity matrix is necessary and sufficient for the convergence of the joint distribution of the standardized sample means from $k$ random samples to the $k$ –variate standard normal distribution. If that conjecture is correct and we are to prove it by extending the tools we have developed for the two sample problem, then we have to formulate appropriate extensions of the findings of this note.

The only approach towards that goal that we can think of involves consideration of the entire collection of contrasts. At this point in time we do not have any idea regarding the feasibility of this approach, but it is definitely worth investigating.

Appendix A. Double and iterated limits

Given a metric space $(S, d)$ , a $S$ –valued double sequence is defined to be a function $x : N \times N \mapsto S$ . We write $x (p, q)$ as $x_{p, q}$ ; recall that $x_{p, q}$ converges to $x \in S$ as $p, q \to \infty$ if, for every $ε > 0$ , there exists $n_{0} (ε) \in N$ such that $p > n_{0}$ and $q > n_{0}$ imply $d (x_{p, q}, x) < ε$ .

Now suppose that for every fixed value of $p$ , $lim_{q \to \infty} x_{p, q} = y_{p}$ exists. Then, $\{y_{p} : p \in N\}$ is a $S$ –valued sequence. If $lim_{p \to \infty} y_{p} = y$ exists, then $y$ is an iterated limit of the double sequence $\{x_{p, q}\}$ and we write $lim_{p \to \infty} lim_{q \to \infty} x_{p, q} = y$ . As illustrated in Section 8.20 of Apostol (1974), the existence of one iterated limit does not imply the existence of the other one (with $S = R$ , let $x (p, q) = {(- 1)}^{p} p {(p + q)}^{- 1}$ ), the existence of both iterated limits does not imply their equality (with $S = R$ , let $x (p, q) = p {(p + q)}^{- 1}$ ), and the equality of the two iterated limits does not imply the existence of the double limit (with $S = R$ , let $x (p, q) = pq {(p^{2} + q^{2})}^{- 1}$ ). However, an iterated limit exists and equals the double limit if the latter and the inner limit of the former exists.

Lemma A1. If the double limit of $\{x_{p, q}\}$ , as $p, q \to \infty$ , exists and is equal to $x$ , then existence of $lim_{q \to \infty} x_{p, q}$ for each fixed $p$ implies $lim_{p \to \infty} lim_{q \to \infty} x_{p, q} = x$ .

Appendix B. Nets and subnets

A set $D$ endowed with a reflexive, anti symmetric, and transitive binary relation $\underline{≻}$ is called a partially ordered set. The pair $(D, \underline{≻})$ is called a directed set if, for each $β, γ \in D$ , there exists $η \in D$ such that $η \underline{≻} β$ and $η \underline{≻} γ$ .

Given a metric space $(S, d)$ and a directed set $(D, \underline{≻})$ , a $S$ –valued net is defined to be a function $x : D \mapsto S$ ; we write the net as $\{x_{β} : β \in D\}$ . Recall that the net $\{x_{β} : β \in D\}$ converges to $x \in S$ if, for every $ε > 0$ , there exists $β_{0} (ε) \in D$ such that $β \underline{≻} β_{0} (ε)$ implies $d (x_{β}, x) < ε$ . It is worth recalling here that a $S$ –valued sequence is a particular $S$ –valued net.

Let $(D, \underline{≻})$ and $(E, ≫)$ be directed sets. Let $ϕ : E \mapsto D$ be order preserving, that is, $i ≫ j \Rightarrow ϕ (i) \underline{≻} ϕ (j)$ , and cofinal, that is, for each $β \in D$ , there exists $γ \in E$ such that $ϕ (γ) \underline{≻} β$ . Then the composite function $y = x \circ ϕ$ , where $x : D \mapsto S$ , defines a net $\{y_{γ} : γ \in E\}$ in $S$ , is called a subnet of $\{x_{β} : β \in D\}$ , and is written as $\{x_{ϕ (γ)} : γ \in E\}$ .

Lemma B1. Let $(D, \underline{≻})$ be a directed set and $\{x_{β} : β \in D\}$ a net taking values in $S$ that converges to $x \in S$ . Then every subnet of $\{x_{β} : β \in D\}$ converges to $x$ .

Lemma B2. Let $(D, \underline{≻})$ be a directed set and $\{x_{β} : β \in D\}$ a net taking values in $S$ . Then $\{x_{β} : β \in D\}$ converges to $x \in S$ if and only if every subnet of $\{x_{β} : β \in D\}$ has a further subnet that converges to $x$ .

Lemma B3. $S$ is compact if and only if every net in $S$ has a convergent subnet.

Appendix C. Miscellaneous results from probability

We have used Lemmas C1, C2, C3, and C4 of this subsection in the note.

Lemma C1. [Consistency of sample standard deviation] As $k \to \infty$ , $S_{i, k}$ converges almost surely (and hence, in probability) to $σ_{i}$ .

Lemma C2. [Slutsky's theorem for nets] Let $D$ and $E$ be metric spaces and $T$ a directed set. Let $\{X_{γ} : γ \in T\}$ and $\{Y_{γ} : γ \in T\}$ be nets of random elements taking values in $D$ and $E$ , respectively, such that $X_{γ} \to X$ weakly and $Y_{γ} \to c$ weakly, where $X$ is a separable random element, that is, there exists a Borel measurable separable subset $M$ of $D$ such that $P (X \in M) = 1$ , and $c$ is a constant. Then, as a net of random elements taking values in $D \times E$ , $(X_{γ}, Y_{γ}) \to (X, c)$ weakly.

Lemma C3. [Continuous mapping theorem for nets] Let $\{X_{γ} : γ \in T\}$ be a net of random elements taking values in $R^{k}$ such that $X_{γ} \to X$ in distribution. Let $g : R^{k} \mapsto R^{m}$ be a continuous function. Then $g (X_{γ}) \to g (X)$ in distribution in $R^{m}$ .

Lemma C4. Let $S_{1}$ and $S_{2}$ be separable metric spaces such that $\{μ_{β} : β \in F\} \subset M (S_{1})$ and $\{ν_{β} : β \in F\} \subset M (S_{2})$ are two nets converging weakly to $μ \in M (S_{1})$ and $ν \in M (S_{2})$ , respectively. Then the net $\{μ_{β} \times ν_{β} : β \in F\} \subset M (S_{1} \times S_{2})$ of product measures converges weakly to $μ \times ν$ .

Footnotes

Declaration of Conflicting Interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

References

Apostol

TM.

Mathematical analysis. Reading, MA: Addison-Wesley; 1974.

Axler

Linear algebra done right

(3rd ed.). New York, NY: Springer; 2015.

Casella

Berger

RL.

Statistical inference

(2nd ed.). Belmont, CA: Brooks/Cole Cengage Learning; 2002.

DasGupta

Asymptotic theory of statistics and probability. New York, NY: Springer; 2008.

Dudley

RM.

Real analysis and probability

(1st ed.). Pacific Grove, CA: Wadsworth and Brooks/Cole; 1989.

Fabian

Hannan

Introduction to probability and mathematical statistics. New York, NY: Wiley; 1985.

Majumdar

Necessary and suffcient condition for asymptotic normality of standardized sample means. 2017a; Retrieved from https://arxiv.org/abs/1710.07275.

Majumdar

On asymptotic standard normality of the two sample pivot. 2017b; Retrieved from https://arxiv.org/abs/1710.08051.

Mukhopadhyay

Probability and statistical inference. New York, NY: Marcel Dekker; 2000.

10.

Parthasarathy

KR.

Probability measures on metric spaces. New York, NY: Academic Press; 1967.

11.

Pyke

Shorack

GR.

Weak convergence of a two-sample empirical process and a new approach to Chernoff-Savage theorems. The Annals of Mathematical Statistics, 39(3):755–771; 1968.

12.

Ramdas

Trillos

Cuturi

On Wasserstein two-sample testing and related families of nonparametric tests. Entropy, 19(2): 47; 2017.

13.

van der Vaart

Wellner

JA.

Weak convergence and empirical processes. New York, NY: Springer; 1996.

14.

Wackerly

Mendenhall

Scheaffer

RL.

Mathematical statistics with applications. Belmont, CA: Brooks/Cole; 2008.