The odd log-logistic generalized half-normal lifetime distribution: Properties and applications

Cooray and Ananda (2008 Cooray, K., Ananda, M.M.A. (2008). A Generalization of the half-normal distribution with applications to lifetime data. Commun. Stat. - Theory Methods 37:1323–1337.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) pioneered a lifetime model commonly used in reliability studies. Based on this distribution, we propose a new model called the odd log-logistic generalized half-normal distribution for describing fatigue lifetime data. Various of its structural properties are derived. We discuss the method of maximum likelihood to fit the model parameters. For different parameter settings and sample sizes, some simulation studies compare the performance of the new lifetime model. It can be very useful, and its superiority is illustrated by means of a real dataset.     

A double acceptance sampling plan for generalized log-logistic distributions with known shape parameters

A double acceptance sampling plan for the truncated life test is developed assuming that the lifetime of a product follows a generalized log-logistic distribution with known shape parameters. The zero and one failure scheme is mainly considered, where the lot is accepted if no failures are observed from the first sample and it is rejected if two or more failures occur. When there is one failure from the first sample, the second sample is drawn and tested for the same duration as the first sample. The minimum sample sizes of the first and second samples are determined to ensure that the true median life is longer than the given life at the specified consumer’s confidence level. The operating characteristics are analyzed according to various ratios of the true median life to the specified life. The minimum such ratios are also obtained so as to lower the producer’s risk at the specified level. The results are explained with examples.     

The heteroscedastic odd log-logistic generalized gamma regression model for censored data

We propose a four-parameter extended generalized gamma model, which includes as special cases some important distributions and it is very useful for modeling lifetime data. A advantage is that it can represent the error distribution for a new heteroscedastic log-odd log-logistic generalized gamma regression model. The proposed heteroscedastic regression model can be used more effectively in the analysis of survival data since it includes as special models several widely-known regression models. Further, for different parameter settings, sample sizes and censoring percentages, various simulations are performed. Overall, the new regression model is very useful to the analysis of real data.     

Reliability acceptance sampling plans for the Weibull distribution under accelerated Type-I censoring

Type-I censored reliability acceptance sampling plans (RASPs) are developed for the Weibull lifetime distribution with unknown shape and scale parameters such that the producer and consumer risks are satisfied. It is assumed that the life test is conducted at an accelerated condition for which the acceleration factor (AF) is known, and each item is continuously monitored for failure. Sensitivity analyses are also conducted to assess the effect of the uncertainty in the assumed AF on the actual producer and consumer risks, and a method is developed for constructing RASPs that can accommodate the uncertainty in AF.     

The odd log-logistic Lindley Poisson model for lifetime data

We define two new lifetime models called the odd log-logistic Lindley (OLL-L) and odd log-logistic Lindley Poisson (OLL-LP) distributions with various hazard rate shapes such as increasing, decreasing, upside-down bathtub, and bathtub. Various structural properties are derived. Certain characterizations of OLL-L distribution are presented. The maximum likelihood estimators of the unknown parameters are obtained. We propose a flexible cure rate survival model by assuming that the number of competing causes of the event of interest has a Poisson distribution and the time to event has an OLL-L distribution. The applicability of the new models is illustrated by means real datasets.     

Bootstrap confidence intervals of CNpk for inverse Rayleigh and log-logistic distributions

In this article bootstrap confidence intervals of process capability index as suggested by Chen and Pearn [An application of non-normal process capability indices. Qual Reliab Eng Int. 1997;13:355–360] are studied through simulation when the underlying distributions are inverse Rayleigh and log-logistic distributions. The well-known maximum likelihood estimator is used to estimate the parameter. The bootstrap confidence intervals considered in this paper consists of various confidence intervals. A Monte Carlo simulation has been used to investigate the estimated coverage probabilities and average widths of the bootstrap confidence intervals. Application examples on two distributions for process capability indices are provided for practical use.     

Parameter estimation for ihe birnbaum-saunders distribution based on symmetrically censored samples

Ihe Bimbaum-Saunders distribution was derived to model fatigue life. Frequently, it becomes necessary to stop a life testing process before all the test items have failed. Therefore, estimation procedures need to be developed for use when censoring occurs. In this article, we have derived estimators for the parameters of this distribution which may be used for complete samples or Type II symmetrically censored samples A simulation study was also conducted to examine the performance of these estimators.     

Assessing the lifetime performance index with Lomax distribution based on progressive type I interval censored sample

In manufacturing industry, the lifetime performance index CL is applied to evaluate the larger-the-better quality features of products. It can quickly show whether the lifetime performance of products meets the desired level. In this article, first we obtain the maximum likelihood estimator of CL with two unknown parameters in the Lomax distribution on the basis of progressive type I interval censored sample. With the MLE we proposed, some asymptotic confidence intervals of CL are discussed by using the delta method. Furthermore, the MLE of CL is used to establish the hypothesis test procedure under a given lower specification limit L. In addition, we also conduct a hypothesis test procedure when the scale parameter in the Lomax distribution is given. Finally, we illustrate the proposed inspection procedures through a real example. The testing procedure algorithms presented in this paper are efficient and easy to implement.     

On progressively censored generalized inverted exponential distribution

A generalized version of inverted exponential distribution (IED) is considered in this paper. This lifetime distribution is capable of modeling various shapes of failure rates, and hence various shapes of aging criteria. The model can be considered as another useful two-parameter generalization of the IED. Maximum likelihood and Bayes estimates for two parameters of the generalized inverted exponential distribution (GIED) are obtained on the basis of a progressively type-II censored sample. We also showed the existence, uniqueness and finiteness of the maximum likelihood estimates of the parameters of GIED based on progressively type-II censored data. Bayesian estimates are obtained using squared error loss function. These Bayesian estimates are evaluated by applying the Lindley's approximation method and via importance sampling technique. The importance sampling technique is used to compute the Bayes estimates and the associated credible intervals. We further consider the Bayes prediction problem based on the observed samples, and provide the appropriate predictive intervals. Monte Carlo simulations are performed to compare the performances of the proposed methods and a data set has been analyzed for illustrative purposes.     

Pooled estimators for the shape parameter of the three parameter gamma distribution

The aim of the paper is to study the pooled estimator of the shape parameter of the three parameter gamma distribution when k independent samples are available. Sufficient conditions for the existence of the pooled estimator are given and the small as well as the large sample properties are studied. The harmonic mean of the k estimators of the independent samples is proposed in the place of the pooled estimator, in the case in which the latter does not exist.     

Reliability of a soccer player based on the bivariate Rayleigh distribution with right censored and ignorable missing data

In this paper, we study the performance of a soccer player based on analysing an incomplete data set. To achieve this aim, we fit the bivariate Rayleigh distribution to the soccer dataset by the maximum likelihood method. In this way, the missing data and right censoring problems, that usually happen in such studies, are considered. Our aim is to inference about the performance of a soccer player by considering the stress and strength components. The first goal of the player of interest in a match is assumed as the stress component and the second goal of the match is assumed as the strength component. We propose some methods to overcome incomplete data problem and we use these methods to inference about the performance of a soccer player.     

Linear approximate ML estimation in scaled Type I generalized logistic distribution based on Type-II censored samples

The scaled (two-parameter) Type I generalized logistic distribution (GLD) is considered with the known shape parameter. The ML method does not yield an explicit estimator for the scale parameter even in complete samples. In this article, we therefore construct a new linear estimator for scale parameter, based on complete and doubly Type-II censored samples, by making linear approximations to the intractable terms of the likelihood equation using least-squares (LS) method, a new approach of linearization. We call this as linear approximate maximum likelihood estimator (LAMLE). We also construct LAMLE based on Taylor series method of linear approximation and found that this estimator is slightly biased than that based on the LS method. A Monte Carlo simulation is used to investigate the performance of LAMLE and found that it is almost as efficient as MLE, though biased than MLE. We also compare unbiased LAMLE with BLUE based on the exact variances of the estimators and interestingly this new unbiased LAMLE is found just as efficient as the BLUE in both complete and Type-II censored samples. Since MLE is known as asymptotically unbiased, in large samples we compare unbiased LAMLE with MLE and found that this estimator is almost as efficient as MLE. We have also discussed interval estimation of the scale parameter from complete and Type-II censored samples. Finally, we present some numerical examples to illustrate the construction of the new estimators developed here.     

Exact Bayesian variable sampling plans for the exponential distribution with progressive hybrid censoring

From the exact distribution of the maximum likelihood estimator of the average lifetime based on progressive hybrid exponential censored sample, we derive an explicit expression for the Bayes risk of a sampling plan when a quadratic loss function is used. The simulated annealing algorithm is then used to determine the optimal sampling plan. Some optimal Bayes solutions under progressive hybrid and ordinary hybrid censoring schemes are presented to illustrate the effectiveness of the proposed method.     

Maximum likelihood estimation of the bimodal failure rate for censored and tied observations

DIMITROV, RACHEV and YAKOVLEV ( 1985 ) have obtained the isotonic maximum likelihood estimator for the bimodal failure rate function. The authors considered only the complete failure time data. The generalization of this estimator for the case of censored and tied observations is now proposed.     

Analysis of left censored data from the generalized exponential distribution

The generalized exponential distribution proposed by Gupta and Kundu [Gupta, R.D and Kundu, D., 1999, Generalized exponential distributions. Australian and New Zealand Journal of Statistics, 41(2), 173–188.] is an important lifetime distribution in survival analysis. In this paper, we consider the maximum likelihood estimation procedure of the parameters of the generalized exponential distribution when the data are left censored. We obtain the maximum likelihood estimators of the unknown para-meters and the Fisher information matrix. Simulation studies are carried out to observe the performance of the estimators in small sample.     

The exponentiated-log-logistic geometric distribution: Dual activation

ABSTRACT

The log-logistic distribution is commonly used to model lifetime data. We propose a wider distribution, named the exponentiated log-logistic geometric distribution, based on a double activation approach. We obtain the quantile function, ordinary moments, and generating function. The method of maximum likelihood is used to estimate the model parameters. We propose a new extended regression model based on the logarithm of the exponentiated log-logistic geometric distribution. This regression model can be very useful in the analysis of real data and could provide better fits than other special regression models. The potentiality of the new models is illustrated by means of two applications to real lifetime data sets.     

Optimum reliability acceptance sampling plans under progressive type-I interval censoring with random removal using a cost model

A reliability acceptance sampling plan (RASP) is a variable sampling plan, which is used for lot sentencing based on the lifetime of the product under consideration. If a good lot is rejected then there is a loss of sales, whereas if a bad lot is accepted then the post sale cost increases and the brand image of the product is affected. Since cost is an important decision-making factor, adopting an economically optimal RASP is indispensable. This work considers the determination of an asymptotically optimum RASP under progressive type-I interval censoring scheme with random removal (PICR-I). We formulate a decision model for lot sentencing and a cost function is proposed that quantifies the losses. The cost function includes the cost of conducting the life test and warranty cost when the lot is accepted, and the cost of batch disposition when it is rejected. The asymptotically optimal RASP is obtained by minimizing the Bayes risk in a set of decision rules based on the maximum likelihood estimator of the mean lifetime of the items in the lot. For numerical illustration, we consider that lifetimes follow exponential or Weibull distributions.     

Marshall-Olkin extended weibull distribution and its application to censored data

In this paper we show that the Marshall-Olkin extended Weibull distribution can be obtained as a compound distribution with mixing exponential distribution. In addition, we provide simple sufficient conditions for the shape of the hazard rate function of the distribution. Moreover, we extend the considered distribution to accommodate randomly right censored data. Finally, application of the extended distribution to a data set representing the remission times of bladder cancer patients is given and its goodness-of-fit is demonstrated.     

The exact confidence interval for the scale parameter and the mvue of the laplace distribution

The exact distribution of the sample median, and of the maximum likelihood estimator of the scale parameter of the Laplace distribution is derived. Tables of Teans, variances and the distribution functions of the corresponding dislributions are evaluacted. Exact ,solutions to the problem of confidence interval and hypothesrs testing for the scale paramrter are provided. The minimum variance unbiased estimator (MVUE) of the p.d.f. of the Laplace distribution when the location parameter is known is also given.     

Efficient closed-form maximum a posteriori estimators for the gamma distribution

We proposed a new class of maximum a posteriori estimators for the parameters of the Gamma distribution. These estimators have simple closed-form expressions and can be rewritten as a bias-corrected maximum likelihood estimators presented by Ye and Chen [Closed-form estimators for the gamma distribution derived from likelihood equations. Am Statist. 2017;71(2):177–181]. A simulation study was carried out to compare different estimation procedures. Numerical results revels that our new estimation scheme outperforms the existing closed-form estimators and produces extremely efficient estimates for both parameters, even for small sample sizes.