Exact inference and prediction for k-sample exponential case under type-ii censoring

The exact inference and prediction intervals for the K-sample exponential scale parameter under doubly Type-II censored samples are derived using an algorithm of Huffer and Lin [Huffer, F.W. and Lin, C.T., 2001, Computing the joint distribution of general linear combinations of spacings or exponen-tial variates. Statistica Sinica, 11, 1141–1157.]. This approach provides a simple way to determine the exact percentage points of the pivotal quantity based on the best linear unbiased estimator in order to develop exact inference for the scale parameter as well as to construct exact prediction intervals for failure times unobserved in the ith sample. Similarly, exact prediction intervals for failure times of units from a future sample can also be easily obtained.     

On structural predictive distribution with type II progressively censored Weibull data

This note applies the structural method of inference to derive a posterior distribution of Weibull parameters and to obtain predictive probability distributions of a set of future ordered failure times \(Y_{(n_1 )}< Y_{(n_2 )}< ...< Y_{(n_k )} \) from N future observations based on a set of type-II progressively censored sample data from a two-parameter Weibull population. In particular, a predictive distribution of the k^th future failure time is given in an integral form. A brief review of the literature on these topics is also given.     

Discrete time cold standby repairable system: Combinatorial analysis

ABSTRACT

In this article, we obtain exact expression for the distribution of the time to failure of discrete time cold standby repairable system under the classical assumptions that both working time and repair time of components are geometric. Our method is based on alternative representation of lifetime as a waiting time random variable on a binary sequence, and combinatorial arguments. Such an exact expression for the time to failure distribution is new in the literature. Furthermore, we obtain the probability generating function and the first two moments of the lifetime random variable.     

Bayesian estimation and prediction based on generalized Type-II hybrid censored sample

ABSTRACT

In this paper, we consider a general form for the underlying distribution and a general conjugate prior, and develop a general procedure for deriving the maximum likelihood and Bayesian estimators based on an observed generalized Type-II hybrid censored sample. The problems of predicting the future order statistics from the same sample and that from a future sample are also discussed from a Bayesian viewpoint. For the illustration of the developed results, the exponential and Pareto distributions are used as examples. Finally, two numerical examples are presented for illustrating all the inferential procedures developed here.     

The Lorenz Curve for Model Assessment in Exponential Order Statistic Models

A goodness-of-fit technique for random samples from the exponential distribution based on the sample Lorenz curve is adapted for use in the exponential order statistic (EOS) model. In the EOS model, only those observations in a random sample from the exponential distribution of unknown size N that are less than some known stopping time T are observable. The model is known as the Jelinski-Moranda model in software reliability, where it is used to estimate the number of bugs in software during development. Distributional results are derived for the distance between the sample Lorenz curve and the population Lorenz curve so that it can be used as a goodness-of-fit test statistic. Simulations show that the test has good power against several alternative distributions. Simulations also indicate that in some cases, model misspecification leads to poor parameter estimation. A plotting procedure provides a means of graphical assessment of fit.     

Prediction intervals for hypergeometric distributions

Abstract

The problem of constructing prediction intervals (PIs) for a future sample from a hypergeometric distribution is addressed. Simple closed-form approximate PIs based on the Wald approach, the joint sampling approach, and a fiducial approach are proposed and compared in terms of coverage probability and precision. Construction of the proposed PIs are illustrated using an example.     

A weibull process failure mechanism for the economic design of mewma control charts

The design parameters of the multivariate exponentially weighted moving average (MEWMA) control chart may be chosen according to economic and/or statistical considerations. The economic model proposed for the design of the'MEWMA chart assumes a Markovian process failure mechanism following an exponential distribution. We'assess the sensitivity of the resulting economic design for the MEWMA to deviations from this assumption. In particular, the generalization, from an exponential to a Weibull distribution of process failure, is used to study the selection of MEWMA chart parameters given process cost and time information. We conclude that the quality of the resulting design (in terms of expected cost) is not substantially affected by mis-specification of the distribution of process failure.     

Approximate prediction limit for weibull failure based on preliminary test estimator

In this paper we have considered type II censored sample from a two parameter Weibull distribution with the known scale parameter. Using the preliminary test estimator of the unknown shape parameter (3 proposed by Pandey (1983), the paper derives a method of finding the approximate prediction limit for the minimum or, more generally,the j^th smallest of a set of future observations from the Weibull or even extreme-value distribution     

Correcting moments for goodness of fit tests based on two entropy estimates

The sample entropy (Vasicek, 1976) has been most widely used as a nonparametric entropy estimator due to its simplicity, but its underlying distribution function has not been known yet though its moments are required in establishing the entropy-based goodness of test statistic (Soofi et al., 1995). In this paper we derive the nonparametric distribution function of the sample entropy as a piece-wise uniform distribution in the lights of Theil (1980) and Dudwicz and van der Meulen (1987). Then we establish the entropy-based goodness of fit test statistics based on the nonparametric distribution functions of the sample entropy and modified sample entropy (Ebrahimi et al., 1994), and compare their performances for the exponential and normal distributions.     

A moment-based empirical likelihood ratio test for exponentiality using the probability integral transformation

ABSTRACT

A simple and efficient goodness-of-fit test for exponentiality is developed by exploiting the characterization of the exponential distribution using the probability integral transformation. We adopted the empirical likelihood methodology in constructing the test statistic. The proposed test statistic has a chi-square limiting distribution. For small to moderate sample sizes Monte-Carlo simulations revealed that our proposed tests are much more superior under increasing failure rate (IFR) and bathtub decreasing-increasing failure rate (BFR) alternatives. Real data examples were used to demonstrate the robustness and applicability of our proposed tests in practice.     

On the sample characterization criterion for normal distributions

Conventionally, it was shown that the underlying distribution is normal if and only if the sample mean and sample variance from a random sample are independent. This paper focusses on the normal population characterization theorem by showing that, if the joint distribution of a skew normal sample follows certain multivariate skew normal distribution, the sample mean and sample variance are still independent.     

An improper form of Weibull distribution for competing risks analysis with Bayesian approach

ABSTRACT

In survival analysis, individuals may fail due to multiple causes of failure called competing risks setting. Parametric models such as Weibull model are not improper that ignore the assumption of multiple failure times. In this study, a novel extension of Weibull distribution is proposed which is improper and then can incorporate to the competing risks framework. This model includes the original Weibull model before a pre-specified time point and an exponential form for the tail of the time axis. A Bayesian approach is used for parameter estimation. A simulation study is performed to evaluate the proposed model. The conducted simulation study showed identifiability and appropriate convergence of the proposed model. The proposed model and the 3-parameter Gompertz model, another improper parametric distribution, are fitted to the acute lymphoblastic leukemia dataset.     

Exact sample size determination in a weibull test plan when there is time censoring

Engineers who conduct reliability tests need to choose the sample size when designing a test plan. The model parameters and quantiles are the typical quantities of interest. The large-sample procedure relies on the property that the distribution of the t -like quantities is close to the standard normal in large samples. In this paper, we use a new procedure based on both simulation and asymptotic theory to determine the sample size for a test plan. Unlike the complete data case, the t -like quantities are not pivotal quantities in general when data are time censored. However we show that the distribution of the t -like quantities only depend on the expected proportion failing and obtain the distributions by simulation for both complete and time censoring case when data follow Weibull distribution. We find that the large-sample procedure usually underestimates the sample size even when it is said to be 200 or more. The sample size given by the proposed procedure insures the requested nominal accuracy and confidence of the estimation when the test plan results in complete or time censored data. Some useful figures displaying the required sample size for the new procedure are also presented.     

Information approach for a lifetime change-point model based on the exponential-logarithmic distribution

ABSTRACT

The exponential-logarithmic distribution is a distribution which has a decreasing failure function and various applications such as in biological and engineering fields. In this paper, we study a change-point problem of this distribution. A procedure based on Schwarz information criterion is proposed to detect changes in parameters of this distribution. Simulations are conducted to indicate the performance of the proposed procedure under different scenarios. Applications on two real data are provided to illustrate the detection procedure.     

Statistical inference for the accelerated failure time model under two-stage generalized case–cohort design

Abstract

In this article, we propose a two-stage generalized case–cohort design and develop an efficient inference procedure for the data collected with this design. In the first-stage, we observe the failure time, censoring indicator and covariates which are easy or cheap to measure, and in the second-stage, select a subcohort by simple random sampling and a subset of failures in remaining subjects from the first-stage subjects to observe their exposures which are different or expensive to measure. We derive estimators for regression parameters in the accelerated failure time model under the two-stage generalized case–cohort design through the estimated augmented estimating equation and the kernel function method. The resulting estimators are shown to be consistent and asymptotically normal. The finite sample performance of the proposed method is evaluated through the simulation studies. The proposed method is applied to a real data set from the National Wilm’s Tumor Study Group.     

A Statistical Model for Shelf Life Estimation Using Sensory Evaluations Scores

Abstract

This article focuses on the problem of estimating the shelf life of food products by modeling the results coming from sensory evaluations. In such studies, trained panelists are asked to judge food attributes by reference to a scale of numbers (scores varying often from 0 to 6). The usual statistical approach for data analysis is to fit a regression line relating the scores and the time of evaluation. The estimate of the shelf life is obtained by solving the regression equation and replacing the score by a cut-off point (which indicates product “failure”) previously chosen by the food company. The procedure used in these sensory evaluations is such that one never knows the exact “time to failure”. Consequently, data arising from these studies are either right or left censored. We propose a model which incorporates these informations and assumes a Weibull for the underlying distribution of the failure time. Simulation studies were implemented. The approach was used in a real data set coming from sensory evaluations of a dehydrated food product.     

Parameter estimation of Lindley step stress model with independent competing risk under type 1 censoring

Abstract

In literature, Lindley distribution is considered as an alternative to exponential distribution to fit lifetime data. In the present work, a Lindley step-stress model with independent causes of failure is proposed. An algorithm to generate random samples from the proposed model under type 1 censoring scheme is developed. Point and interval estimation of the model parameters is carried out using maximum likelihood method and percentile bootstrap approach. To understand the effectiveness of the resulting estimates, numerical illustration is provided based on simulated and real-life data sets.     

SINGULAR ELLIPTICAL DISTRIBUTION: DENSITY AND APPLICATIONS

ABSTRACT

The paper present an explicit expression for the density of a n-dimensional random vector with a singular Elliptical distribution. Based on this, the densities of the generalized Chi-squared and generalized t distributions are derived, examining the Pearson Type VII distribution and Kotz Type distribution (as specific Elliptical distributions). Finally, the results are applied to the study of the distribution of the residuals of an Elliptical linear model and the distribution of the t-statistic, based on a sample from an Elliptical population.     

Interval estimation for the Pareto distribution based on the progressive Type II censored sample

For the complete sample and the right Type II censored sample, Chen [Joint confidence region for the parameters of Pareto distribution. Metrika 44 (1996), pp. 191–197] proposed the interval estimation of the parameter θ and the joint confidence region of the two parameters of Pareto distribution. This paper proposed two methods to construct the confidence region of the two parameters of the Pareto distribution for the progressive Type II censored sample. A simulation study comparing the performance of the two methods is done and concludes that Method 1 is superior to Method 2 by obtaining a smaller confidence area. The interval estimation of parameter ν is also given under progressive Type II censoring. In addition, the predictive intervals of the future observation and the ratio of the two future consecutive failure times based on the progressive Type II censored sample are also proposed. Finally, one example is given to illustrate all interval estimations in this paper.     

Control chart for monitoring the Weibull shape parameter under two competing risks

ABSTRACT

In this paper, we propose a control chart to monitor the Weibull shape parameter where the observations are censored due to competing risks. We assume that the failure occurs due to two competing risks that are independent and follow Weibull distribution with different shape and scale parameters. The control charts are proposed to monitor one or both of the shape parameters of competing risk distributions and established based on the conditional expected values. The proposed control chart for both shape parameters is used in certain situations and allows to monitor both shape parameters in only one chart. The control limits depend on the sample size, number of failures due to each risk and the desired stable average run length (ARL). We also consider the estimation problem of the target parameters when the Phase I sample is incomplete. We assumed that some of the products that fail during the life testing have a cause of failure that is only known to belong to a certain subset of all possible failures. This case is known as masking. In the presence of masking, the expectation-maximization (EM) algorithm is proposed to estimate the parameters. For both cases, with and without masking, the behaviour of ARLs of charts is studied through the numerical methods. The influence of masking on the performance of proposed charts is also studied through a simulation study. An example illustrates the applicability of the proposed charts.