Two popular models Rayleigh and gamma (2) distributions are considered to verify whether one can be an alternative to the other. The cumulative distribution function of gamma (2) is not analytically tractable, whereas for Rayleigh distribution is tractable which motivated for the study. Test statistics based on likelihood ratio is suggested to discriminate between Rayleigh and gamma (2) models. The percentiles and power of the proposed test statistics were also tabulated, and a comparison was made with respect to the power for a given sample and level of significance.

Abstract: A special case of the well known T – X family of distributions proposed by Ayman Alzaatreh et al. (2012) called Pareto – Rayleigh (P – R) distribution is considered. Its scale parameter is estimated using a single order statistic in small samples. Optimal criterion for the choice of single order statistic in a given small sample is worked out. Comparison is made with the corresponding optimal choice of single order statistic with respect to the criterion of asymptotic variance. The results are extended to estimate parametric functions like reliability and hazard rate. Keywords: T-X family, Pareto-Rayleigh distribution, order statistics, optimal estimation, asymptotic variance.

The time to failure of a product is considered as a quality characteristic of following Log-Logistic distribution (β = 3). Control limits are evaluated for the time to failure. Life time data are compared with the control limits to judge the quality performance of the product.

Variable control charts are based on subgroup statistics and variation in the values of the subgroups. In this paper extreme order statistic of the subgroup are considered to develop the control limits to decide upon the in control status of the process. Here, the quality characteristic is assumed to follow Rayleigh (Weibull with shape parameter 2) and gamma with shape parameter 2 distributions are considered. Relevant comparisons are presented with examples.

This paper deals with the Rayleigh distribution as a life time model. Moments of order statistics and an ordered sample are used to define a test statistic for the null hypothesis that the considered random variable has Rayleigh distribution. The percentiles of the test statistic are evaluated and the power of the test is computed. Comparison of Rayleigh distribution verses half-logistic distribution and gamma distribution with shape parameter 2 is presented.

The well known Linear Failure Rate Distribution (LFRD) is considered. A process variate following LFRD is thought of in order to develop control charts for subgroup mean and subgroup range. In view of the limitations on LFRD the theoretical control limits are obtained through some approximations and the resulting control chart limits are worked out. Comparisons with the control limits of similar variable control charts is also presented.

The Burr type X distribution is considered as a life time random variable of a product whose lots are to be decided for acceptance or otherwise on the basis of sample lifetimes drawn from the lot. The sample is divided into various groups in order to develop a group sampling plan in such a way that the life testing experiment is terminated as soon as the first failure in each group is observed. The acceptance criterion based on the theory of order statistics is proposed and is shown to be more economical than a criterion proposed in the earlier similar works.

The two popular life testing models are considered to verify whether one can be an alternative to other. The motivation for this study is as follows. It is well known that the cumulative distribution function of Rayleigh distribution can be analytically inverted where as it is not so with Gamma distribution. Generally analytical inversion of cumulative distribution function would be advantageous in the study of problems of inference. “Whether this advantage can be explored in assessing the discrimination or otherwise of the two models” is studied in this paper.

A new method of generating probability distribution on the bases of given two specified probability models are adopted using the well-known Pareto, Rayleigh distributions. The resulting model is considered as null population and a test statistic is suggested to discriminate the null population between two successive alternative populations Pareto, Rayleigh models. The critical values of the test statistic and the powers are evaluated. A comparative study is presented.

The two parameter Burr type X distribution is considered and its scale parameter is estimated from a censored sample using the classical maximum likelihood method. The estimating equations are modified to get simpler and efficient estimators. Two methods of modification are suggested. The small sample efficiencies are presented.

This paper deals with the log-logistic distribution (β = 3) as a life time model.Moments of order statistics and an ordered sample are used to define a test statistic for the null hypothesis that the considered random variable has log-logistic distribution (β = 3).The percentiles of the test statistic are evaluated, Power's of the test with half-logistic distribution and Rayleigh distribution as alternatives are also evaluated.

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.

A generalization of the Half Logistic Distribution is developed through exponentiation of its survival function and named the Type II Generalized Half Logistic Distribution (GHLD). The distributional characteristics are presented and estimation of its parameters using maximum likelihood and modified maximum likelihood methods is studied with comparisons. Discrimination between Type II GHLD and exponential distribution in pairs is conducted via likelihood ratio criterion.

The Linear Failure Rate Distribution (LFRD) is considered. The graphs of its probability density function are examined for selected parameter combinations. Some of them are similar to the well-known exponential distribution. Incidentally exponential distribution is one of the two component models of the LFRD model. In view of the simpler form of exponential model as applicable in inference, looking at the frequency curves of LFRD, a test statistic is proposed based on ratio of likelihood functions containing the standard forms of the density functions of both LFRD and Exponential to discriminate between LFRD and exponential models. The critical values and the powers of the test statistic are developed.

In this paper, a hybrid group acceptance sampling plan is introduced for a truncated life test if life times of the items follow size biased Lomax model.The minimum number of testers and acceptance number are obtained when the consumer's risk and the test termination time and group size are pre-specified.The operating characteristic values, minimum ratios of the true mean life to the specified mean life for the given producer's risk are also derived.The results are discussed through an example, a comparative study of proposed sampling plan with existing sampling plan are elaborated.

In this article , acceptance sampling plans are developed for the Linear Failure Rate Distribution percentiles when the life test is truncated at a pre-specified time. The minimum sample size necessary to ensure the specified life percentile is obtained under a given consumerś risk. The operating characteristic values of the sampling plans as well as the producerś risk are presented.