Statistical analysis of competing risks lifetime data from Nadarajaha and Haghighi distribution under type-II censoring

Abstract

Time to failure in the lifetime experiments, common that units or individuals are failure with more than one caused of failure defined by competing risks problem. In this paper, we consider the problem of competing risks model when the failure time of units or individuals have Nadarajaha and Haghighi lifetime distribution [1] based on type-II censoring schemes. The point and interval estimates of model parameters are discussed with maximum likelihood and Bayes methods. In Bayesian approach, the MCMC method is employed. The theoretical results are discussed through real life data analysis and simulation study. Finally, we discussed the numerical results in some brief comments.

Keywords

Nadarajaha and Haghighi distribution competing risks model MCMC classical and bayesian estimation

1 Introduction

Censoring is common phenomenon in life test experiments due to time and cost restrictions in data collection. Ones of the oldest and most widely used type of censoring called type-I and type-II censoring schemes. In both types, censoring scheme does not allow the removal units of the test other than the final termination point. In type-I censoring, number of failure units is random and test time is fixed. But, in type-II censoring scheme failure units is prefixed number and the test time is random. For the possibility of removal units during the test, the two types of censoring scheme is generalized by progressive censoring scheme. The censoring schemes specially, progressive censoring schemes studying in the book of Balakrishnan and Aggarwala [2]. In this article, we are adopted the type-II censoring scheme in discussing the failure time of units under the test when units is failure with more than on case of failure which known with the competing risk problem.

In different area of reliability studying or life testing experiment, the commonly problem is units failure with more than one causes of failure. In these situation, measuring the effect one of the causes respected to the other causes which defined by competing risk problem. This problem is discussed with different author, for exponential lifetime product Cox [3] and for exponential life product under accelerate life tests Balakrishnan and Han [5]. Early, different properties of competing risk model discussed in Crowder [4] discussed under accelerate life test model in Ganguly and Kundu [6], Hanaa Abu-Zinadah and Neveen Ahmed [7]. The problem of the competing risk model is discussed with two different risks under type-II censoring scheme.

Let n independent units are put on a life test, and prior the experiment the integer m is considered, then the first failure T_1;m and its caused of failure ρ₁ are recorded. The second failure T_2;m and its caused of failure ρ₂ are recorded and the experiment is continued until m-th failure T_m;m and its caused ρ_m where ρ_i ∈ {1, 2} , i = 1, 2, . . . , m is recorded. The data (T_1;m, ρ₁)< (T_2;m, ρ₂)< ⋯< (T_m;m, ρ_m) , is called type-II competing risk data.

The joint likelihood function under observed type-II competing risk sample $\underline{t} = {(t_{1; m}, ρ_{1}),$ (t_2;m, ρ₂) , ⋯, (t_m;m, ρ_m)} is presented by

$\begin{matrix} f (\underline{t}) & = \frac{n!}{(n - m)!} {S_{1} (t_{m}) S_{2} (t_{m})}^{(n - m)} \\ \times \prod_{i = 1}^{m} {[f_{1} (t_{i}) S_{2} (t_{i})]}^{I (ρ_{i} = 1)} {[f_{2} (t_{i}) S_{1} (t_{i})]}^{I (ρ_{i} = 2)}, \end{matrix}$ (1) where S (.) and h (.) are the survival and hazard rate functions, respectively and $I (ρ_{i} = j) = {\begin{array}{l} 1, & ρ_{i} = j \\ 0, & ρ_{i} \neq j \end{array}, j = 1, 2,$ (2) for $0 < t_{1} < t_{2} < . . . < t_{m} < \infty .$ Recently work of the competing risk model see, for Burr XII distribution Modhesh and Abd-Elmougod [10], and from bathtub shape lifetime distribution discussed by Algarni et al. [8 , 11]. Also, the analysis of weibull step-stress model under competing risks model discussed by Samanta et al. [12].

Developing estimation procedures of parameters of Nadarajaha and Haghighi lifetime distributions of type-II competing risk data is the main objective of this article. The classical maximum likelihood method as well as Baysian MCMC method are used to compute the point and interval estimates of model parameters. All theoretical results are assessed and compared trough the two, real data set and Monte Carlo simulation.

The paper is organized as follows, The model notations and descriptions discussed in Section 2. The point and approximate intervals under maximum likelihood estimation in Section 3. The MCMC method of Bayesian approach discussed in section 4. In Section 5, theoretical results discussed through real data set. In Section 6, the different methods are compared by conducting Monte Carlo simulations study. Finally, the numerical results discussed through some brief comments in Section 7.

2 Notation and model description

For consideration only two causes of failure some notations are employed to use in this paper

NHD Nadarajaha and Haghighi distribution.

HF Hazard failure rate function.

RF Relability function.

SEL Squard error loss.

MCMC Markov chain Monte Carlo.

MH Metropolis–Hastings.

RMM Radiation male mice.

pdf Probability density function

cdf Cumulative distribution function

HPD Highest credible intervals

GD Gaussian distribution

CI Cofidence interval

ACI Approximate confidence interval

T_i Lifetime of the i-th unit.

T_ij Lifetime of the i-th unit under cause j, j = 1, 2.

F (.) Cumulative distribution function (cdf) of T_i .

f (.) Probability density function (pdf) of T_i .

F_j (.) cdf of T_ij .

f_j (.) pdf of T_ij .

S_j (.) Survival function of $T_{ij} \tilde{.}$

ρ_i Indicator variable denoting the cause of failure of the i-th unit. Ω_j The integer set of failure units under cause j

To simplify the notation we will use henceforth T_i instead of T_i;m, i = 1, 2, ⋯ , m. The model studied in the paper satisfies the following assumptions

Under the time T_ij which the unit number i is failed due to cause j the lifetime of unit is defined by T_i = min {T_i1, T_i2}, i = 1, 2, ⋯ , m.

The lifetime random variable T_ij is distributed with Nadarajaha and Haghighi [1] distribution (NHD) function with shape parameters θ and scale parameters β_j, j = 1, 2 and i = 1, 2, ⋯ , m. Hence, all pdf and cdf of NHD under random variable T_ij, j = 1, 2, for each i = 1, 2, ⋯ , m, are given by

$\begin{matrix} g_{j} (t) = & β_{j} θ (1 + β_{j} t)^{θ - 1} e^{1 - (1 + β_{j} t)^{θ}}, \\ t > 0, θ, β_{j} > 0, \end{matrix}$ (3) $G_{j} (x) = 1 - e^{1 - (1 + β_{j} t)^{θ}},$ (4) where, β_j is a scale parameters and θ is a common shape parameter. Also the corresponding hazard failure rate (HF) and reliability functions (RF) of NHD for some t, respectively are presented by $S_{j} (t) = e^{1 - (1 + β_{j} t)^{θ}}, t > 0,$ (5) $H_{j} (t) = β_{j} θ (1 + β_{j} t)^{θ - 1}, t > 0 .$ (6)

The NHD is introduced by Nadarajaha and Haghighi [1] as an extension of the exponential one to present a distribution equivalent to the Weibull and gamma. The NHD with parameters (β, θ), is agreer with the Weibull, gamma and the exponentiated exponential distributions. The NHD has closed form RF and HF like the Weibull and the exponentiated exponential distributions, also reduced to the exponential distribution for the value of θ = 1 and it has the zero mode. Hence, NHD has increasing, decreasing or constant HF and more properties of NHD is presented by Nadarajaha and Haghighi [1]. Different method of estimation of NHD is discussed by Dey et al. [13] and for recurrence relations under record statistic Kumar and Dey [14]. The NHD has the property that HF is monotonically increasing or decreasing which doesn’t existed in the Weibull, Gamma or exponentiated exponential distributions. Also, when modeling the data that has mode fixed at zero the NHD is suitable than Weibull, Gamma or exponentiated exponential distributions.

3 Maximum likelihood estimates

3.1 The point MLEs

For the competing risk data T_ij from the NHD with parameters (θ, β_j) for j = 1, 2 and i = 1, 2, . . . , m and distribution function given by equations (3) and (4) the joint likelihood function in (1), without normalized constant is reduced to

$\begin{matrix} L (θ, β_{1}, β_{2} | \underline{t}) & = & θ^{m} β_{1}^{m_{1}} β_{2}^{m_{2}} e^{- \sum_{i = 1}^{m} ((1 + β_{1} t_{i})^{θ} + (1 + β_{2} t_{i})^{θ})} \\ \times & e^{(θ - 1) (\sum_{i \in Ω_{1}}^{m_{1}} log (1 + β_{1} t_{i}) + \sum_{i \in Ω_{2}}^{m_{2}} log (1 + β_{2} t_{i}))} \\ \times & e^{- (n - m) ((1 + β_{1} t_{m})^{θ} + (1 + β_{2} t_{m})^{θ})} . \end{matrix}$ (7)

where Ω_j denoted to integer set of index data failed by the case j . Hence, the log-likelihood function written as follows,

$\begin{matrix} ℓ (θ, β_{1}, β_{2} | \underline{t}) = m \log θ + m_{1} \log β_{1} + m_{2} \log β_{2} \\ + (θ - 1) (\sum_{i \in Ω_{1}}^{m_{1}} log (1 + β_{1} t_{i}) + \sum_{i \in Ω_{2}}^{m_{2}} log (1 + β_{2} t_{i})) \\ - \sum_{i = 1}^{m} ((1 + β_{1} t_{i})^{θ} + (1 + β_{2} t_{i})^{θ}) \\ - (n - m) ((1 + β_{1} t_{m})^{θ} + (1 + β_{2} t_{m})^{θ}) . \end{matrix}$ (8) After taking the first partial derivatives of equation (8) respect to β₁ and β₂ and equating to zero, obtaining the following equations

$\begin{matrix} \frac{\partial ℓ (θ, β_{1}, β_{2} | \underline{t})}{\partial β_{1}} = & \frac{m_{1}}{β_{1}} + (θ - 1) \sum_{i \in Ω_{1}}^{m_{1}} \frac{t_{i}}{1 + β_{1} t_{i}} \\ - \sum_{i = 1}^{m} θ t_{i} (1 + β_{1} t_{i})^{θ - 1} \\ - (n - m) θ t_{m} (1 + β_{1} t_{m})^{θ - 1} = 0, \end{matrix}$ (9)

$\begin{matrix} \frac{\partial ℓ (θ, β_{1}, β_{2} | \underline{t})}{\partial β_{2}} = & \frac{m_{2}}{β_{2}} + (θ - 1) \sum_{i \in Ω_{2}}^{m_{2}} \frac{t_{i}}{1 + β_{2} t_{i}} \\ - \sum_{i = 1}^{m} θ t_{i} (1 + β_{2} t_{i})^{θ - 1} \\ - (n - m) θ t_{m} (1 + β_{2} t_{m})^{θ - 1} = 0 . \end{matrix}$ (10) Hence, from (9) and (10) we obtain the ML estimate of β₁ and β₂ from two non-linear equations as follows $β_{1} = \frac{m_{1}}{f_{1} (β_{1}, θ)},$ (11) and $β_{2} = \frac{m_{2}}{f_{2} (β_{2}, θ)},$ (12) where $\begin{matrix} f_{j} (β_{j}, θ) = (n - m) θ t_{m} (1 + β_{j} t_{m})^{θ - 1} + \\ \sum_{i = 1}^{m} θ t_{i} (1 + β_{j} t_{i})^{θ - 1} - (θ - 1) \sum_{i \in Ω_{j}}^{m_{j}} \frac{t_{i}}{1 + β_{j} t_{i}} . \end{matrix}$ (13) The first derivatives of equation (8) is calculated with respect to θ, as follows

$\begin{matrix} \frac{\partial ℓ (θ, β_{1}, β_{2} | \underline{t})}{\partial θ} = \frac{m}{θ} + \sum_{i \in Ω_{1}}^{m_{1}} log (1 + β_{1} t_{i}) \\ + \sum_{i \in Ω_{2}}^{m_{2}} log (1 + β_{2} t_{i}) - \sum_{i = 1}^{m} {(1 + β_{1} t_{i})^{θ} log (1 + β_{1} t_{i}) \\ + (1 + β_{2} t_{i})^{θ} log (1 + β_{2} t_{i})} - (n - m) {(1 + β_{1} t_{m})^{θ} \\ \times log (1 + β_{1} t_{m}) + (1 + β_{2} t_{m})^{θ} log (1 + β_{2} t_{m})} = 0 . \end{matrix}$ (14) Hence the likelihood equation are reduced to three non-linear equations which can be solve with iteration method such as Newton^,s method to obtain the maximum likelihood estimates of parameters θ and β₁ and β₂.

3.2 Approximate confidence intervals

The second partially derivatives of the log-likelihood equation (8) is calculated and presented as follows $\begin{matrix} \frac{\partial^{2} ℓ (θ, β_{1}, β_{2} | \underline{t})}{\partial β_{1}^{2}} = & \frac{- m_{1}}{β_{1}^{2}} - (θ - 1) \sum_{i \in Ω_{1}}^{m_{1}} \frac{t_{i}^{2}}{{(1 + β_{1} t_{i})}^{2}} \\ - \sum_{i = 1}^{m} θ (θ - 1) t_{i}^{2} (1 + β_{1} t_{i})^{θ - 1} \\ - (n - m) θ (θ - 1) t_{m}^{2} (1 + β_{1} t_{m})^{θ - 2}, \end{matrix}$ (15) $\begin{matrix} \frac{\partial^{2} (θ, β_{1}, β_{2} | \underline{t})}{\partial β_{2}^{2}} = & \frac{- m_{2}}{β_{2}^{2}} - (θ - 1) \sum_{i \in Ω_{2}}^{m_{2}} \frac{t_{i}^{2}}{{(1 + β_{2} t_{i})}^{2}} \\ - \sum_{i = 1}^{m} θ (θ - 1) t_{i}^{2} (1 + β_{2} t_{i})^{θ - 1} \\ - (n - m) θ (θ - 1) t_{m}^{2} (1 + β_{2} t_{m})^{θ - 2}, \end{matrix}$ (16) $\begin{matrix} \frac{\partial^{2} ℓ (θ, β_{1}, β_{2} | \underline{t})}{\partial θ^{2}} = & \frac{- m}{θ^{2}} - (n - m) \\ \times {(1 + β_{1} t_{m})^{θ} {log}^{2} (1 + β_{1} t_{m}) \\ + (1 + β_{2} t_{m})^{θ} {log}^{2} (1 + β_{2} t_{m})} . \\ - \sum_{i = 1}^{m} {(1 + β_{1} t_{i})^{θ} log (1 + β_{1} t_{i}) \\ + (1 + β_{2} t_{i})^{θ} log (1 + β_{2} t_{i})} \end{matrix}$ (17) $0 = \frac{\partial^{2} ℓ (θ, β_{1}, β_{2} | \underline{t})}{\partial β_{1} \partial β_{2}} = \frac{\partial^{2} ℓ (θ, β_{1}, β_{2} | \underline{t})}{\partial β_{2} \partial β_{1}}$ (18) $\begin{matrix} \frac{\partial^{2} ℓ (θ, β_{1}, β_{2} | \underline{t})}{\partial β_{1} \partial θ} = & \frac{\partial^{2} ℓ (θ, β_{1}, β_{2} | \underline{t})}{\partial θ \partial β_{1}} = \sum_{i \in Ω_{1}}^{m_{1}} \frac{t_{i}}{1 + β_{1} t_{i}} \\ - \sum_{i = 1}^{m} t_{i} (1 + β_{1} t_{i})^{θ - 1} \\ \times {1 + log (1 + β_{1} t_{i})} \\ - (n - m) t_{m} (1 + β_{1} t_{m})^{θ - 1} \\ \times {1 + log (1 + β_{1} t_{m})} \end{matrix}$ (19) and $\begin{matrix} \frac{\partial^{2} ℓ (θ, β_{1}, β_{2} | \underline{t})}{\partial β_{2} \partial θ} = & \frac{\partial^{2} ℓ (θ, β_{1}, β_{2} | \underline{t})}{\partial θ \partial β_{2}} = \sum_{i \in Ω_{2}}^{m_{2}} \frac{t_{i}}{1 + β_{2} t_{i}} \\ - \sum_{i = 1}^{m} t_{i} (1 + β_{2} t_{i})^{θ - 1} \\ \times {1 + log (1 + β_{2} t_{i})} \\ - (n - m) t_{m} (1 + β_{2} t_{m})^{θ - 1} \\ \times {1 + log (1 + β_{2} t_{m})} . \end{matrix}$ (20) Under taken the minus expectation of equations (15-20) the Fisher information matrix denoted by Ψ (θ, β₁, β₂) is obtained. The asymptotic distribution of ( $\hat{θ},$ ${\hat{β}}_{1},$ ${\hat{β}}_{2})$ is distributed approximately with Gaussian distribution (GD) with mean (θ, β₁, β₂) and Ψ^-1 (θ, β₁, β₂) covariance matrix. The minus expectation, practice cant be obtain, then it is replaced by the estimate $Ψ_{0}^{- 1} (\hat{θ}, {\hat{β}}_{1}, {\hat{β}}_{2})$ . This approximation is described by $(\hat{θ}, {\hat{β}}_{1}, {\hat{β}}_{2}) \to GD ((θ, β_{1}, β_{2}), Ψ_{0}^{- 1} (\hat{θ}, {\hat{β}}_{1}, {\hat{β}}_{2})) .$ The estimate of approximate information matrix $Ψ_{0}^{- 1} (\hat{θ}, {\hat{β}}_{1}, {\hat{β}}_{2})$ is defined by $\begin{matrix} Ψ_{0}^{- 1} (\hat{θ}, {\hat{β}}_{1}, {\hat{β}}_{2}) = & [\begin{matrix} - \frac{\partial^{2} ℓ (θ, β_{1}, β_{2} | \underline{t})}{\partial θ^{2}} - \frac{\partial^{2} ℓ (θ, β_{1}, β_{2} | \underline{t})}{\partial θ \partial β_{1}} \\ - \frac{\partial^{2} ℓ (θ, β_{1}, β_{2} | \underline{t})}{\partial β_{1} \partial θ} - \frac{\partial^{2} ℓ (θ, β_{1}, β_{2} | \underline{t})}{\partial β_{1}^{2}} \\ - \frac{\partial^{2} ℓ (θ, β_{1}, β_{2} | \underline{t})}{\partial β_{2} \partial θ} - \frac{\partial^{2} ℓ (θ, β_{1}, β_{2} | \underline{t})}{\partial β_{2} \partial β_{1}} \end{matrix} \\ {\begin{matrix} - \frac{\partial^{2} ℓ (θ, β_{1}, β_{2} | \underline{t})}{\partial θ \partial β_{2}} \\ - \frac{\partial^{2} ℓ (θ, β_{1}, β_{2} | \underline{t})}{\partial β_{1} \partial β_{2}} \\ - \frac{\partial^{2} ℓ (θ, β_{1}, β_{2} | \underline{t})}{\partial β_{2}^{2}} \end{matrix}]}_{(\hat{θ}, {\hat{β}}_{1}, {\hat{β}}_{2})}^{- 1} . \end{matrix}$ (21) Based on the approximate information matrix $Ψ_{0}^{- 1} (\hat{θ}, {\hat{β}}_{1}, {\hat{β}}_{2}))$ of model parameters (θ, β₁, β₂) and the bivariate Gaussian distribution with mean (θ, β₁, β₂) and covariance matrix $Ψ_{0}^{- 1}$ ( $\hat{θ},$ ${\hat{β}}_{1},$ ${\hat{β}}_{2}$ ). Then, Based on the variances C₁₁, C₂₂ and C₃₃ obtained from The value of the diagonal of the variance covraince matrix in (21) $Ψ_{0}^{- 1} (\hat{θ}, {\hat{β}}_{1}, {\hat{β}}_{2})$ that 100 (1-2α)% approximate confidence intervals for the parameters (θ, β₁, β₂) are given by $\begin{matrix} \hat{θ} \mp z_{α} \sqrt{C_{11}} \\ {\hat{β}}_{1} \mp z_{α} \sqrt{C_{22}} \\ {\hat{β}}_{2} \mp z_{α} \sqrt{C_{33},} \end{matrix}$ (22) respectively, for the standard Gaussian value z_α with tail α .

4 Bayesian MCMC estimation

Under classical estimation discussed in previous section the model parameters are considered as the constant values but in Bayesian approach it is considered as the random variables. So, in this section, the posterior distribution from the model parameters is constructed and the point and the corresponding credible interval is discussed with the help of MCMC method. The prior distribution of parameters (θ, β₁, β₂) is considered to be independent gamma density given by $h_{i}^{*} (φ_{i}) \propto φ_{i}^{a_{i} - 1} e^{- b_{i} φ_{i}}, φ_{i} > 0, a_{i}, b_{i} > 0,$ (23) where (φ_i = (θ, β₁, β₂) , i=1, 2, 3) respectively. Then, the joint prior density of (θ, β₁, β₂) can be expressed as $\begin{matrix} h^{*} (θ, β_{1}, β_{2}) \propto & θ^{a_{1} - 1} β_{1}^{a_{2} - 1} β_{2}^{a_{3} - 1} e^{- b_{1} θ - b_{2} β_{1} - b_{3} β_{2}}, \\ a_{i}, b_{i} > 0 . \end{matrix}$ (24) Multiply the joint prior density (24) with likelihood equation in (8) of (θ, β₁, β₂), the joint posterior density is constructed as follows

$\begin{matrix} h (θ, β_{1}, β_{2} | \underline{t}) = & {Qh}^{*} (θ, β_{1}, β_{2}) \\ \times L (θ, β_{1}, β_{2} | \underline{t}), \end{matrix}$ (25) where

$\begin{matrix} Q & = & 1 / \int_{0}^{\infty} \int_{0}^{\infty} \int_{0}^{\infty} h^{*} (θ, β_{1}, β_{2}) \times \\ \times & L (θ, β_{1}, β_{2} | \underline{t}) d θ d β_{1} d β_{2} . \end{matrix}$ (26) From (8) and (24) the posterior distribution of the model parameters given by

$\begin{matrix} h (θ, β_{1}, β_{2} | \underline{t}) \propto & θ^{m + a_{1} - 1} β_{1}^{m_{1} + a_{2} - 1} β_{2}^{m_{2} + a_{3} - 1} \\ \times e^{- b_{1} θ - b_{2} β_{1} - b_{3} β_{2}} \\ \times e^{(θ - 1) {\sum_{i \in Ω_{1}}^{m_{1}} log (1 + β_{1} t_{i}) + \sum_{i \in Ω_{2}}^{m_{2}} log (1 + β_{2} t_{i})}} \\ \times e^{- \sum_{i = 1}^{m} ((1 + β_{1} t_{i})^{θ} + (1 + β_{2} t_{i})^{θ})} \\ \times e^{- (n - m) {(1 + β_{1} t_{m})^{θ} + (1 + β_{2} t_{m})^{θ}}} \end{matrix}$ (27) The Bayes estimate of the function π (θ, β₁, β₂), is given by $\begin{matrix} {\hat{π}}_{B} (θ, β_{1}, β_{2}) = & Q \int_{0}^{\infty} \int_{0}^{\infty} \int_{0}^{\infty} π (θ, β_{1}, β_{2}) \\ \times h^{*} (θ, β_{1}, β_{2}) L (θ, β_{1}, β_{2} | \underline{t}) \\ \times d θ d β_{1} d β_{2} . \end{matrix}$ (28) This estimate is taken with respect to squared error loss (SEL) function. The equation (28) has a ratio of two integrals generally, need numerical approximation. The important one can be employed in this case is MCMC method which depend on generate samples from the posterior distributions.

Estimations with MCMC method

methodologies of MCMC are used in different branches of science such as in quantum system see Wang et al. [16]. Hence, different type of MCMC approach are available, the important one called Gibbs sampling and more general Metropolis-within-Gibbs samplers. The important technique which employed in this section to approximate the Bayes estimates is called important sample method. The posterior density function defined in (27) can be written as $\begin{matrix} h (θ, β_{1}, β_{2} | \underline{t}) \propto & f_{1} (θ | β_{1}, β_{2} | \underline{t}) f_{2} (β_{1}) f_{3} (β_{2}) \\ \times Z (θ, β_{1}, β_{2}), \end{matrix}$ (29) where $f_{1} (θ | β_{1}, β_{2}, \underline{t})$ distributed as the probability gamma density with the shape parameter m + a₁ and scale parameter $b_{1} - {\sum_{i \in Ω_{1}}^{m_{1}} log (1 + β_{1} t_{i}) + \sum_{i \in Ω_{2}}^{m_{2}} log (1 + β_{2} t_{i})}$ , f₂ (β₁) and f₃ (β₂) is a proper density functions given by $f_{2} (β_{1} | \underline{t}) \propto β_{1}^{m_{1} + a_{2} - 1} e^{- b_{2} β_{1} - \sum_{i \in Ω_{1}}^{m_{1}} log (1 + β_{1} t_{i})},$ (30) and $f_{3} (β_{2} | \underline{t}) \propto β_{2}^{m_{2} + a_{3} - 1} e^{- b_{3} β_{2} - \sum_{i \in Ω_{2}}^{m_{2}} log (1 + β_{2} t_{i})} .$ (31) and

$\begin{matrix} Z (θ, β_{1}, β_{2} | \underline{t}) \propto \\ {(b_{1} - {\sum_{i \in Ω_{1}}^{m_{1}} log (1 + β_{1} t_{i}) + \sum_{i \in Ω_{2}}^{m_{2}} log (1 + β_{2} t_{i})})}^{- (m + a_{1})} \\ \times e^{- \sum_{i = 1}^{m} ((1 + β_{1} t_{i})^{θ} + (1 + β_{2} t_{i})^{θ})} \\ \times e^{- (n - m) {(1 + β_{1} t_{m})^{θ} + (1 + β_{2} t_{m})^{θ}}} \end{matrix}$ (32) $\begin{matrix} Z (θ, β_{1}, β_{2} | \underline{t}) \propto \\ {(b_{1} - {\sum_{i \in Ω_{1}}^{m_{1}} log (1 + β_{1} t_{i}) + \sum_{i \in Ω_{2}}^{m_{2}} log (1 + β_{2} t_{i})})}^{- (m + a_{1})} \\ \times e^{- \sum_{i = 1}^{m} ((1 + β_{1} t_{i})^{θ} + (1 + β_{2} t_{i})^{θ})} \\ \times e^{- (n - m) {(1 + β_{1} t_{m})^{θ} + (1 + β_{2} t_{m})^{θ}}} \end{matrix}$ (32)

Hence, the Bayes estimate in equation (28) is written by $\begin{matrix} {\hat{π}}_{B} (θ, β_{1}, β_{2}) = & Q \int_{0}^{\infty} \int_{0}^{\infty} \int_{0}^{\infty} π (θ, β_{1}, β_{2}) \\ \times f_{1} (θ | β_{1}, β_{2}, \underline{t}) f_{2} (β_{1} | \underline{t}) f_{3} (β_{2} | \underline{t}) \\ \times Z (θ, β_{1}, β_{2} | \underline{t}) d θ d β_{1} d β_{2} . \end{matrix}$ (33) In this case, instead, Gibbs sampling technique to draw MCMC approach, the importance sampling technique is suggested to apply in this approach see Chen and Shao [15] to construct point estimate and corresponding HPD credible intervals, as follows

The plot of the two function $f_{2} (β_{1} | \underline{t})$ and $f_{3} (β_{2} | \underline{t})$ are showed that are similar to Gaussian distribution. Hence, the generate a sample from this distributions, we applied Metropolis–Hastings (MH) method with Gaussian proposal distribution. For generate from gamma distribution $f_{1} (θ | β_{1}, β_{2} | \underline{t})$ is more simple with any standard program. The description of the algorithm are used to generate MCMC sampling described as follows

Let κ = 1 and M =burn-in, to began with initial iteration with maximum likelihood estimates $φ_{i}^{(0)} = (\hat{θ}, {\hat{β}}_{1}, {\hat{β}}_{2})$ .

The value θ^(κ) is generated from gamma density $f_{1} (θ | β_{1}^{(κ - 1)}, β_{2}^{(κ - 1)} | \underline{t}) .$

The two values $β_{1}^{(κ)}$ and $β_{2}^{(κ)}$ are generated from (30) and (31), respectively with (MH) algorithm with Gaussian proposal distribution with mean $β_{1}^{(κ)}$ and $β_{2}^{(κ)}$ and variance obtained from diagonal of variance covariance matrix.

Put κ = 2 and repeat steps 2 and 3 to obtain the data $φ^{(i)} = (θ^{(i)}, β_{1}^{(i)}, β_{2}^{(i)}),$ i = 1, 2, . . . , N.

The Bayes estimate of any function ${\hat{π}}_{B} (θ, β_{1},$ β₂) under a SEL function can be obtained as $\begin{matrix} {\hat{π}}_{B} (θ, β_{1}, β_{2}) = \\ \frac{\sum_{i = M + 1}^{N} π_{B} (φ^{(i)}) Z (φ^{(i)} | \underline{t})}{\sum_{i = M + 1}^{N} Z (φ^{(i)} | \underline{t})} . \end{matrix}$ (34)

Also the posterior variance of ${\hat{π}}_{B} (θ, β_{1}, β_{2})$ cacultated by $\begin{matrix} V ({\hat{π}}_{B} (θ, β_{1}, β_{2})) = \\ \frac{\sum_{i = M + 1}^{N} (π_{B} (φ^{(i)}) - {\hat{π}}_{B})^{2} Z (φ^{(i)} | \underline{t})}{\sum_{i = M + 1}^{N} Z (φ^{(i)} | \underline{t})} . \end{matrix}$ (35) For interval estimation, we adopt the idea of Chen and Shao [15] to obtain the HPD credible intervals of the function π_B (θ, β₁, β₂) as follows

[Step 1.] The MCMC sample of parameters vector $φ_{l}^{(i)} = (θ^{(i)}, β_{1}^{(i)}, β_{2}^{(i)}),$ i = 1, 2, . . . , N - M and l =1, 2, 3 is obtained, then with the help of importance sampling technique, then calculate $Z_{i} = Z (φ_{l}^{(i)}), l = 1, 2, 3 .$

[Step 2.] Sort (Z_l, i = 1, 2, . . . , N - M) to obtain the ordered values, Z₍₁₎ < Z₍₂₎ < . . . < Z_(N-M) .

[Step 3.] Compute the weighted function η_i $η_{i} = \frac{Z (φ_{i}^{(i)})}{\sum_{j = M + 1}^{N} Z (φ_{l}^{(i)})}, l = 1, 2, 3 .$ (36) Then rewrite (η_i, i = M + 1, M + 2, . . . , N) as (η_(i), i = 1, 2, . . . , N - M) so that the ith value η_(i) corresponds to the the value Z_(i).

[Step 4.] Estimate the γ the quantile of the marginal posterior of η by ${\tilde{η}}^{(γ)} = {\begin{matrix} η_{(1)}, if γ = 0 \\ 1 c η_{(i)}, if \sum_{s = 1}^{i - 1} Z_{(i)} < γ < \sum_{s = 1}^{i} Z_{(i)} \end{matrix} .$

[Step 5.] Compute the 100 (1 - α %) credible intervals of θ $(φ^{(L / N)}, φ^{({L + [(1 - α) N]} / N)})$ where L = 1, 2, . . . , N - [(1 - α) N] .

[Step 6.] The 100 (1-α%) HPD interval of is the one, with the smallest interval width among all credible intervals.

5 Real data analysis

In this section, we analyze the real data set presented in Hoel [17] which it is obtained from laboratory experiment for the life of male mice at age of 5-6 weeks received a radiation dose of 300 roentgens. The two causes of failure are chosen, reticulum cell sarcoma and thymic Lymphoma. Our purpose is analysis the two causes of death then, we adopted thymic lymphoma to be cause 1 but the cause 2 is consider be reticulum cell sarcoma. The data of 99 radiation failuremice (RFM) is presented in Table 1. The last data is analyzed with different authors, Sarhan et al. [18], Cramer and Schmiedt [19] and Bakoban and Abd-Elmougod [20]. Hence. the type-II competing risks data under consideration two cause of failure thymic lymphoma and reticulum cell sarcoma and the value of m failure taken to be 35 listed in Table 1 and for computational ease, it divided each sample with 1000. Based on the observed data in Table 2, the MLE and the approximate confidence intervals are presented in Table 3. For Bayesian approach non-informative prior is applied to reflect vague prior information. The results of Bayes point and HPD intervals are also, presented in Table 2. Figs. (1–6) describe the simulation number of the parameters vector generated under MCMC approach as well as the corresponding histogram. Figs. (1–6), show the sequence of 11,000 simulated values observation can be confirmed by examining the autocorrelation function for this sequence. displays convergence diagnostic plots for to assess whether the Markov chains have converged. Inferences should not be made if the Markov chain has not converged. The trace plot shows that the mean of the Markov chain is constant over the graph and is stabilized. The chain is able to traverse the support of the target distribution, and the mixing is good. The trace plot shows that the Markov chain appears to have reached a stationary distribution. The results described by Figs. (1–6) show that the highly correlated of the importance sampling sequences.

Fig.1

Simulation number of θ generated by MCMC method.

Fig.2

Histogram of θ generated MCMC method.

Fig.3

Simulation number of θ generated by MCMC method.

Fig.4

Histogram of θ generated MCMC method.

Fig.5

Simulation number of θ generated by MCMC method.

Fig.6

Histogram of θ generated MCMC method.

Table 1

Hoel [17]

Cause 1: (Thymic Lymphoma)
159	189	191	198	200	207	220	235	245	250	256
261	265	266	280	343	356	383	403	414	428	432
Cause 2: (Reticulum cell sarcoma)
317	318	399	495	525	536	549	552	554	557	558
571	586	594	596	605	612	621	628	631	636	643
647	648	649	661	663	666	670	695	697	700	705
Other causes:
40	42	51	62	163	179	206	222	228	252	249
282	324	333	341	366	385	407	420	431	441	461
462	482	517	517	524	564	567	586	619	620	621
622	647	651	686	761	763

Table 2

Type-II competing risks data for m = 50

i	1	2	3	4	5	6	7	8	9	10	11	12
t _i	0.159	0.189	0.191	0.198	0.200	0.207	0.220	0.235	0.245	0.250	0.256	0.261
ρ _i	1	1	1	1	1	1	1	1	1	1	1	1
i	13	14	15	16	17	18	19	20	21	22	23	24
t _i	0.265	0.266	0.280	0.317	0.318	0.343	0.356	0.383	0.399	0.403	0.414	0.428
ρ _i	1	1	1	2	2	1	1	1	2	1	1	1
i	25	26	27	28	29	30	31	32	33	34	35
t _i	0.432	0.495	0.525	0.536	0.549	0.552	0.554	0.557	0.558	0.571	0.586
ρ _i	1	2	2	2	2	2	2	2	2	2	2

6 Monte Carlo simulation study

The theoretical results developed in this paper are assessed through observed the behavior of the different proposed methods. Hence simulation study are performed under consideration 1000 type-II competing risks samples. The true values of the model parameters to agreed with hybrid parameters values of prior distribution, we proposed the shape and scale hybrid parameters and generate random sample of size 20. Hence the true parameter is choose to be the mean of the random sample $φ_{Ex} ≅ \frac{1}{20} \sum_{i = 1}^{20} φ_{i}$ ,.The simulation study is constructed to test the effect of the change of sample size n and effect sample size m and what the effect of the parameter values change. We used different sample sizes n, different effective sample sizes m and the following two set of parameters φ = (θ, β₁, β₂) ={(1.0, 1.5, 2.5), (3.0, 0.5, 1.0)}. In our studying, we calculate the point MLEs and MCMCs estimates and the two asymptotic confidence intervals and HPD credible intervals. The average and standard errors as well as average confidence lengths and probability coverage are calculated after process is repeated 1000 times. All results are presented Tables 4 to 7.

Table 3
The point and 95% interval estimates of MLEs and Bayes

Parameter (.)_ML 95% ACI Lenth (.)_B - MCMC 95% CI Lenth

θ 4.7413 (0.0077, 15.7138) 15.7061 3.2547 (0.6548, 8.2546) 7.5998

β ₁ 0.3615 (0.0021, 1.2979) 1.2958 0.4521 (0.0432, 1.3014) 1.2582

β ₂ 0.2458 (0.0001, 0.8490) 0.8489 0.3014 (0.0521, 0.7452) 0.6931

Parameter	(.)_ML	95% ACI	Lenth	(.)_B - MCMC	95% CI	Lenth
θ	4.7413	(0.0077, 15.7138)	15.7061	3.2547	(0.6548, 8.2546)	7.5998
β ₁	0.3615	(0.0021, 1.2979)	1.2958	0.4521	(0.0432, 1.3014)	1.2582
β ₂	0.2458	(0.0001, 0.8490)	0.8489	0.3014	(0.0521, 0.7452)	0.6931

Table 4

For (θ, β₁, β₂)= (1.0, 1.5, 2.5), average and mean squared errors (within brackets)

n	m	Metods	θ	β ₁	β ₂
35	15	(.)_MLE	1.432(0.254)	1.842(0.327)	2.879(0.523)
		(.)_BMCMC	0.972(0.124)	1.641(0.209)	2.453(0.475)
	25	(.)_MLE	1.348(0.201)	1.800(0.298)	2.748(0.492)
		(.)_BMCMC	1.325(0.101)	1.641(0.184)	2.421(0.354)
50	25	(.)_MLE	1.325(0.203)	1.745(0.277)	2.702(0.460)
		(.)_BMCMC	1.301(0.122)	1.614(0.163)	2.402(0.335)
	35	(.)_MLE	1.298(0.198)	1.722(0.224)	2.355(0.420)
		(.)_BMCMC	1.145(0.109)	1.600(0.120)	2.672(0.280)
70	40	(.)_MLE	1.230(0.150)	1.707(0.200)	2.442(0.401)
		(.)_BMCMC	1.122(0.101)	1.600(0.120)	2.402(0.307)
	50	(.)_MLE	1.220(0.113)	1.642(0.199)	2.411(0.398)
		(.)_BMCMC	1.120(0.099)	1.551(0.101)	2.462(0.299)

Table 5

For (θ, β₁, β₂)=(1.0, 1.5, 2.5), The average 95% confidence lengths and the corresponding coverage percentages

n	m	Metods	θ	β ₁	β ₂
35	15	(.)_MLE	2.541(0.88)	3.452(0.89)	4.409(0.90)
		(.)_BMCMC	2.019(0.91)	3.112(0.90)	4.178(0.92)
	25	(.)_MLE	2.287(0.90)	3.287(0.91)	4.274(0.91)
		(.)_BMCMC	2.000(0.92)	3.074(0.92)	4.072(0.93)
50	25	(.)_MLE	2.201(0.89)	3.222(0.92)	4.225(0.90)
		(.)_BMCMC	1.991(0.93)	2.974(0.91)	4.014(0.93)
	35	(.)_MLE	2.144(0.91)	3.147(0.91)	4.198(0.93)
		(.)_BMCMC	1.955(0.94)	2.911(0.95)	4.001(0.93)
70	40	(.)_MLE	2.117(0.92)	3.007(0.92)	4.038(0.94)
		(.)_BMCMC	1.907(0.93)	2.789(0.96)	3.881(0.92)
	50	(.)_MLE	2.077(0.93)	2.889(0.92)	4.001(0.93)
		(.)_BMCMC	1.602(0.93)	2.720(0.94)	3.813(0.95)

Table 6

For (θ, β₁, β₂)= (3.0, 0.5, 1.0), average and mean squared errors (within brackets)

n	m	Metods	θ	β ₁	β ₂
35	15	(.)_MLE	4.415(0.854)	0.813(0.102)	1.354(0.321)
		(.)_BMCMC	3.474(0.701)	0.624(0.087)	1.201(0.224)
	25	(.)_MLE	4.013(0.725)	0.689(0.099)	1.238(0.284)
		(.)_BMCMC	3.400(0.624)	0.611(0.081)	1.224(0.207)
50	25	(.)_MLE	4.180(0.714)	0.670(0.093)	1.225(0.260)
		(.)_BMCMC	3.368(0.601)	0.542(0.072)	1.213(0.198)
	35	(.)_MLE	3.998(0.654)	0.549(0.090)	1.211(0.260)
		(.)_BMCMC	3.300(0.564)	0.532(0.065)	1.195(0.172)
70	40	(.)_MLE	3.700(0.612)	0.536(0.082)	1.200(0.178)
		(.)_BMCMC	3.258(0.521)	0.522(0.061)	1.175(0.124)
	50	(.)_MLE	3.623(0.564)	0.530(0.071)	1.220(0.154)
		(.)_BMCMC	3.118(0.421)	0.519(0.054)	1.160(0.111)

Table 7

For (θ, β₁, β₂)=(3.0, 0.5, 1.0), The average 95% confidence lengths and the corresponding coverage percentages

n	m	Metods	θ	β ₁	β ₂
35	15	(.)_MLE	5.522(0.90)	2.411(0.89)	3.265(0.90)
		(.)_BMCMC	5.101(0.91)	2.187(0.90)	3.100(0.92)
	25	(.)_MLE	5.192(0.91)	2.204(0.91)	3.092(0.92)
		(.)_BMCMC	4.879(0.92)	2.001(0.93)	3.009(0.96)
50	25	(.)_MLE	5.197(0.90)	2.199(0.92)	3.120(0.93)
		(.)_BMCMC	4.882(0.93)	2.015(0.93)	3.001(0.94)
	35	(.)_MLE	4.782(0.93)	2.127(0.96)	3.040(0.94)
		(.)_BMCMC	4.542(0.94)	2.000(0.92)	2.901(0.94)
70	40	(.)_MLE	4.562(0.93)	2.007(0.96)	3.019(0.93)
		(.)_BMCMC	4.321(0.92)	1.999(0.95)	2.875(0.94)
	50	(.)_MLE	4.413(0.93)	1.892(0.93)	3.002(0.95)
		(.)_BMCMC	4.129(0.93)	1.788(0.94)	2.820(0.96)

7 Conclusions and some comments

The statistical inference has different applications in more branch of for example in quantum information processing see Batle et al. [21, 22]. This article is dealing with the important problem in life test experiment known with competing risks problem. So, we adopted products has NH lifetime distribution and units is failure with two cause of failure. NHDs with common shape parameter and different scale parameters under type-II censoring scheme is considered. The model parameters are estimated with MLE and Bayes. The inference through real data example also is discussed. We are conducted a simulation study to report the behavior of these estimators. From the results, we observe

From Tables 4 to 7, report that estimation procedure discussed in this article is acceptable.

The results of the point estimates in Table 4 and 6 show that the Bayes estimators perform better than the MLEs in the terms of MSEs.

The results of the interval estimates in Table 5 and 7 show that the Bayes estimators perform better than the MLEs in the terms of mean length and probability coverage.

Increasing the sample proportion $\frac{m}{n}$ , reducing the MSEs and withdraw a length of all estimators

Footnotes

Acknowledgement

This project was funded by the Deanship of Scientific Research (DSR) at King Abdulaziz University, Jeddah, under grant no. (KEP-35-130-39). The authors, therefore, acknowledge with thanks DSR for technical and financial support.

References

Nadarajah and

Haghighi , An extension of the exponential distribution, Statistics 45 (2011), 543.

Balakrishnan and

Aggarwala , Progressive Censoring, Theory, Methods, and Applications User, Boston, 2000.

D.R.

Cox , The analysis of exponentially distributed lifetimes with two types of failures, Journal of the Royal Statistical Society 21 (1959), 411.

M.J.

Crowder , Classical competing risks, Chapman and Hall, London, 2001.

Balakrishnan and

Han , Exact inference for a simple stepstress model with competing risks for failure from exponential distribution under Type-II censoring, Journal of Statistical Planning and Inference 138 (2008), 4172.

Ganguly and

Kundu , Analysis of simple step-stress model in presence of competing risks, Journal of Statistical Computation and Simulation 86 (2016), 1989.

Hanaa and

Sayed-Ahmed , Competing risks model with partially step-stress accelerate life tests in analyses lifetime Chen data under type-II censoring scheme, Open Phys 17 (2019), 8.

M.A.

Sabry ,

H.Z.

Muhammed ,

Nabih and

Shaaban , Parameter estimation for the power generalized weibull distribution based on one- and two-stage ranked set sampling designs, J Stat Appl Prob 8 (2019), 113–128.

Yusuf and

Qureshi , A five parameter statistical distribution with application to real data, J Stat Appl Prob 8 (2019), 11–26.

10.

A.A.

Modhesh and

G.A.

Abd-Elmougod , Analysis of progressive first-failure-censoring in the burr XII model for competing risks data, American Journal of Theoretical and Applied Statistics 4 (2015), 610.

11.

M.M.

Mohie El-Din ,

M.M.

Amein ,

A.M.

Abd El-Raheem ,

H.E.

El-Attar and

E.H.

Hafez , Estimation of the coefficient of variation for lindley distribution based on progressive first failure censored data, J Stat Appl Prob 8 (2019), 83–90.

12.

M.R.

Abonazel , Different estimators for stochastic parameter panel data models with serially correlated errors, J Stat Appl Prob 7 (2018), 423–434.

13.

Dey ,

Zhang ,

Asgharzadeh and

Ghorbannezhad , Comparisons of methods of estimation for the NH distributions, Annals of Data Science 4 (2017), 441.

14.

Kumar and

Dey , Upper record values from extended exponential distribution, Journal of Modern Applied Statistical 17 (2019), 2687.

15.

M.H.

Chen and

Q.M.

Shao , Monte carlo estimation of bayesian credible and HPD intervals, Journal of Computational and Graphical Statistics 8 (1999), 69.

16.

Wang ,

Wu and

Zou , Quantum annealing with markov chain monte carlo simulations and D-wave quantum computers, Statistical Science 31 (2016), 362.

17.

D.G.

Hoel , A representation of mortality data by competing risks, Biometrics 28 (1972), 475.

18.

A.M.

Sarhan ,

D.C.

Hamilton and

Smith , Statistical analysis of competing risks models, Reliability Engineering and System Safety 95 (2010), 953.

19.

Cramer and

A.B.

Schmiedt , Progressively type-II censored competing risks data from Lomax distributions, Computational Statistics and Data Analysis 55 (2011), 1285.

20.

Bakoban and

G.A.

Abd-Elmougod , MCMC in analysis of progressively first failure censored competing risks data for gompertz model, Journal of Computational and Theoretical Nanoscience 13 (2016), 6662.

21.

Batle ,

C.R.

Ooi ,

Farouk ,

Abutalib and

M.S.

Abdalla , Do multipartite correlations speed up adiabatic quantum computation or quantum annealing, Quantum Information Processing 15 (2016), 3081.

22.

Batle ,

Farouk ,

Tarawneh and

Abdalla , Multipartite quantum correlations among atoms in QED cavities, Frontiers of Physics 13 (2018), 130305.