Stress-strength reliability estimation for exponentially distributed system with common minimum guarantee time

This paper continues a study, initiated in [1], of the stochastic independence of $V = X - Y$ and the mean of order $\gamma$, \begin{equation*}\tag{1}M_\gamma\begin{align*} = (1/\gamma) \log \lbrack(e^{\gamma X} + e^{\gamma Y})/2\rbrack\quad\text{for} \gamma \neq 0 \text{and} \gamma \neq \pm \infty\\ = (X + Y)/2\quad\text{for} \gamma = 0\\ = \max (X, Y)\quad\text{for} \gamma = + \infty\\ = \min (X, Y)\quad \text{for} \gamma = - \infty\end{align*}\end{equation*} for independent $X$ and $Y$. The case $\gamma = 0$ is solved by the well known result, if, for independent $X$ and $Y$, the variables $X + Y$ and $X - Y$ are independent, then both $X$ and $Y$ have normal distributions with a common variance. The first proof of this result may be found in Kac [6]. (Although the statement of this result there is much weaker, Kac's proof applies exactly to the statement above.) A corresponding result for the gamma distribution, proved in restricted form by Hogg [5], and proved without restrictions by Lukacs [7], is as follows: if $Z_1$ and $Z_2$ are independent positive random variables for which $Z_1 + Z_2$ and $Z_1/Z_2$ are independent, then both $Z_1$ and $Z_2$ have gamma distributions with a common scale parameter. This result may be used to complete the study of the independence of $M_\gamma$ and $V$ for independent $X$ and $Y$ when $\gamma$ is finite and non-zero, by letting $Z_1 = \exp (\gamma X)$ and $Z_2 = \exp (\gamma Y)$. It is clear, then, from the result of Hogg and Lukacs that both $\exp (\gamma Y)$ and $\exp (\gamma Y)$ must have gamma distributions with a common scale parameter. The resulting distributions of $X$ and $Y$ have been studied in [1]. Therefore, of this study, there remain yet to be completely solved only the cases $\gamma = +\infty$ and $\gamma = - \infty$. These two cases are essentially the same since if $M_{+\infty}$ and $V$ are independent, then so are $\min (-X, -Y)$ and $V$. Thus, the negatives of the distributions for which $M_{+\infty}$ and $V$ are independent will yield independent $M_{-\infty}$ and $V$. We consider only the case $\gamma = -\infty$. This problem is considered in a previous paper, [2], in which the distributions were restricted to be discrete. The main result of that paper is as follows: if $X$ and $Y$ are independent, non-degenerate, discrete random variables, and if $U = \min (X, Y)$ and $V = X - Y$ are independent, then $X$ and $Y$ both have geometric distributions with common location and scale parameters, but possibly different geometric parameters. If the probability mass function of the geometric distribution is written as \begin{equation*}\tag{2}p(x) = (1 - p)p^{(x-\theta)/c}\quad\text{for} x = \theta, \theta + c, \theta + 2c, \cdots\end{equation*} then, the parameter $\theta$ is a location parameter, $c$ is a scale parameter, and $p$ is called the geometric parameter, $c > 0, 0 < p < 1$. In this paper, the distributions of $X$ and $Y$ are restricted to be absolutely continuous. This will result in a characterization of the so-called exponential distribution whose density is \begin{equation*}\tag{3}f(x) = (1/\sigma) \exp (-(x - \theta)/\sigma)\quad\text{if} x > \theta\end{equation*} and zero otherwise. The parameter $\theta$ is a location parameter, and the parameter $\sigma > 0$ is a scale parameter, which is also the standard deviation of the distribution. In these terms the main result of this paper may be stated as follows: if $X$ and $Y$ are independent random variables with absolutely continuous distributions, and if $U = \min (X, Y)$ and $V = X - Y$ are independent, then both $X$ and $Y$ have exponential distributions with a common location parameter but with possibly different scale parameters. This result then gives a characterization of the exponential distribution, since it is easy to show the converse, that if $X$ and $Y$ are independent and if each has an exponential distribution with a common location parameter but with possibly different scale parameters, then $U$ and $V$ are independent. For related results, the reader is referred to papers by Fisz [3] and Rogers [8].

The objective of this study is to examine the properties of Bayes estimators of the shape parameter of the Power Function Distribution (PFD-I), by using two different prior distributions for the parameter θ and different loss functions that were compared with the maximum likelihood estimators. In many practical applications, we may have two different prior information about the prior distribution for the shape parameter of the Power Function Distribution, which influences the parameter estimation. So, we used two different kinds of conjugate priors of shape parameter θ of the Power Function Distribution (PFD-I) to estimate it. The conjugate prior function of the shape parameter θ was considered as a combination of two different prior distributions such as gamma distribution with Erlang distribution and Erlang distribution with exponential distribution and Erlang distribution with non-informative distribution and exponential distribution with the non-informative distribution. We derived Bayes estimators for shape parameter θ of the Power Function Distribution (PFD-I) according to different loss functions such as the squared error loss function (SELF), the weighted error loss function (WSELF) and modified linear exponential (MLINEX) loss function (MLF), with two different double priors. In addition to the classical estimation (maximum likelihood estimation). We used simulation to get the results of this study, for different cases of the shape parameter of the Power Function Distribution used to generate data for different samples sizes.

This article deals with the problem of reliability in a multicomponent stress-strength (MSS) model when both stress and strength variables are from exponentiated Teissier (ET) distributions. The reliability of the system is determined using both classical and Bayesian methods, based on two scenarios where the common scale parameter is unknown or known. In the first scenario, where the common scale parameter is unknown, the maximum likelihood estimation (MLE) and the approximate Bayes estimation are derived. In the second scenario, where the scale parameter is known, the MLE, the uniformly minimum variance unbiased estimator (UMVUE) and the exact Bayes estimation are obtained. In the both scenarios, the asymptotic confidence interval and the highest probability density credible interval are established. Furthermore, two other asymptotic confidence intervals are computed based on the Logit and Arcsin transformations. Monte Carlo simulations are implemented to compare the different proposed methods. Finally, one real example is presented in support of suggested procedures.

Independent samples from different exponential distributions are available in many statistical applications, for example, in queuing or reliability models, where the random variables describe waiting times and lifetimes, respectively. When two samples are observed, inference is then carried out for two location and two scale parameters. In multivariate setups including common location and common scale parameter assumptions, we provide confidence regions for the parameters of interest, which have minimum Lebesgue measure among all those based on the usual pivotal quantities and with the same or higher confidence level; in particular, they improve in terms of area (volume) upon the standard ‘trapezoidal’ confidence regions being constructed by combining independent univariate pivot statistics. The proposed confidence regions do not require any factorization of the overall confidence level, and their calculations need simple Monte Carlo simulations, only. Although focusing on two complete samples, generalizations of the results to more than two and doubly type-II censored samples are possible.

In many real-life scenarios, systems frequently perform badly in difficult operating situations. The multiple failures that take place when systems reach their lower, higher, or extreme functioning states typically receive little attention from researchers. This study uses generalized progressive hybrid censoring to discuss the inference of R=P(X&lt;Y&lt;Z) for a component when it is exposed to two stresses, Y,Z, and it has one strength X that is regarded. We assume that both the stresses and strength variables follow an exponentiated exponential distribution with a common scale parameter. We obtain R’s maximum likelihood estimator and approximate confidence intervals. In addition, the Bayesian estimators for symmetric, such as squared error, and asymmetric loss functions, such as linear exponential, are developed. Credible intervals with the highest posterior densities are established. Monte Carlo simulations are used to evaluate and compare the effectiveness of the many proposed estimators. The process is then precisely described using an analysis of real data.

This paper deals with the estimation of P[Y < X] when X and Y are two independent generalized exponential distributions with different shape parameters but having the same scale parameters. The maximum likelihood estimator and its asymptotic distribution is obtained. The asymptotic distribution is used to construct an asymptotic confidence interval of P[Y < X]. Assuming that the common scale parameter is known, the maximum likelihood estimator, uniformly minimum variance unbiased estimator and Bayes estimator of P[Y < X] are obtained. Different confidence intervals are proposed. Monte Carlo simulations are performed to compare the different proposed methods. Analysis of a simulated data set has also been presented for illustrative purposes.

This paper deals with study and estimation of stress-strength reliability for a system where strength and stress components are connected in series with cold standby redundancy at system level. It is supposed that the random stress and strength both follow Kumaraswamy half-logistic distribution. In this redundant system, we consider that there N subsystems and in each subsystem, there are M statistically independent strength components under the impact of M statistically independent stress components The problem of estimation is solved in two cases. First under the assumption that random stress and strength have common first shape parameter and different second shape parameter and second with the assumption that common shape parameter is known. The stress-strength reliability is estimated using maximum likelihood and Bayesian estimation methods. Also asymptotic and Bayesian intervals for the stress-strength reliability under both the cases are constructed. Monte Carlo simulations will be performed to compare the performance of various methods. Finally a real life data set is analyzed to demonstrate the findings.

In this paper, Bayes estimators of the unknown shape and scale parameters of the Exponentiated Inverse Rayleigh Distribution (EIRD) have been derived using both the frequentist and bayesian methods. The Bayes theorem was adopted to obtain the posterior distribution of the shape and scale parameters of an Exponentiated Inverse Rayleigh Distribution (EIRD) using both conjugate and non-conjugate prior distribution under different loss functions (such as Entropy Loss Function, Linex Loss Function and Scale Invariant Squared Error Loss Function). The posterior distribution derived for both shape and scale parameters are intractable and a Lindley approximation was adopted to obtain the parameters of interest. The loss function were employed to obtain the estimates for both scale and shape parameters with an assumption that the both scale and shape parameters are unknown and independent. Also the Bayes estimate for the simulated datasets and real life datasets were obtained. The Bayes estimates obtained under dierent loss functions are close to the true parameter value of the shape and scale parameters. The estimators are then compared in terms of their Mean Square Error (MSE) using R programming language. We deduce that the MSE reduces as the sample size (n) increases.

Estimation of common location and scale parameters in nonregular cases

This paper considers the problem of estimating the probability P = Pr(X < Y) when X and Y are independent exponential random variables with unequal scale parameters and a common location parameter. Uniformly minimum variance unbiased estimator of P is obtained. The asymptotic distribution of the maximum likelihood estimator is obtained and then the asymptotic equivalence of the two estimators is established. Performance of the two estimators for moderate sample sizes is studied by Monte Carlo simulation. An approximate interval estimator is also obtained. Keywords: stress-strength modelmixed beta distributionuniformly minimum variance unbiased estimatorasymptotic distributionsasymptotic equivalence

In fatigue testing of metals or composite materials under different levels of stress, the underlying distributions are often assumed to be Weibull with a common shape parameter. The problems of estimation and hypothesis tests of the common shape parameter and scale parameters in the Weibull distributions are considered in this paper. The results are: 1) the maximum likelihood equation for estimating the shape parameter, 2) a s-confidence interval and hypothesis test for the shape parameter, 3) s-bias factors and variances of the pooled estimator of shape parameter, and 4) s-confidence interval and hypothesis test for the scale parameters. Examples are given for composite material applications. Results are based on Monte Carlo simulation methods.

From the time of introducing the generalized exponential (GE) distribution, several researches have been done to apply this distribution and its properties in different applications. In this article, it is intended, under asymmetric LINEX loss function, to find Bayes estimation of shape parameter of the GE distribution, when scale parameter is known and also Bayes estimations of shape and scale parameters when both are unknown. Finally, extensive simulations are performed to compare risk efficiencies of the estimators under two squared-error and LINEX loss functions.

Kambo and Awad (1985) defined a test statistic based on doubly censored samples to test the equality of location parameters of K exponential distributions when their common scale parameter is unknown. The power function of the test is derived in this paper and some special cases are studied.

Panel count data often occur in a long-term recurrent event study, where the exact occurrence time of the recurrent events is unknown, but only the occurrence count between any two adjacent observation time points is recorded. Most traditional methods only handle panel count data for a single type of event. In this paper, we propose a Bayesian semiparameteric approach to analyze panel count data for multiple types of events. For each type of recurrent event, the proportional mean model is adopted to model the mean count of the event, where its baseline mean function is approximated by monotone I-splines. The correlation between multiple types of events is modeled by common frailty terms and scale parameters. Unlike many frequentist estimating equation methods, our approach is based on the observed likelihood and makes no assumption on the relationship between the recurrent process and the observation process. Under the Poisson counting process assumption, we develop an efficient Gibbs sampler based on novel data augmentation for the Markov chain Monte Carlo sampling. Simulation studies show good estimation performance of the baseline mean functions and the regression coefficients; meanwhile, the importance of including the scale parameter to flexibly accommodate the correlation between events is also demonstrated. Finally, a skin cancer data example is fully analyzed to illustrate the proposed methods.

This paper proposed a Bayesian method-based structural equation model (SEM) of miners’ work injury for an underground coal mine in India. The environmental and behavioural variables for work injury were identified and causal relationships were developed. For Bayesian modelling, prior distributions of SEM parameters are necessary to develop the model. In this paper, two approaches were adopted to obtain prior distribution for factor loading parameters and structural parameters of SEM. In the first approach, the prior distributions were considered as a fixed distribution function with specific parameter values, whereas, in the second approach, prior distributions of the parameters were generated from experts’ opinions. The posterior distributions of these parameters were obtained by applying Bayesian rule. The Markov Chain Monte Carlo sampling in the form Gibbs sampling was applied for sampling from the posterior distribution. The results revealed that all coefficients of structural and measurement model parameters are statistically significant in experts’ opinion-based priors, whereas, two coefficients are not statistically significant when fixed prior-based distributions are applied. The error statistics reveals that Bayesian structural model provides reasonably good fit of work injury with high coefficient of determination (0.91) and less mean squared error as compared to traditional SEM.