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This paper deals with developing sequential procedures for estimating the mean of an inverse Gaussian (IG) distribution when the population coefficient of variation (CV) is known. We consider the minimum risk and bounded risk point estimation problems respectively. Moreover, we make use of a weighted squared-error loss function and aim to control the associated risk functions. Instead of the usual estimator, i.e., the sample mean, Searls J. Amer. Stat. Assoc. 50, 1225–1226 (1964) estimator is utilized for the purpose of estimation. Second-order approximations are also obtained under both estimation set-ups. We establish that Searls’ estimator dominates the usual estimator (sample mean) under proposed sequential sampling procedures. An extensive simulation analysis is carried out to validate the theoretical findings and real data illustrations are also provided to show the practical utility of our proposed sequential stopping strategies.

This paper deals with the classical estimation of the mission time reliability $$R(t)=P(X>t)$$ and stress–strength reliability $$\delta =P(X>Y)$$ for Kumaraswamy-G distributions using progressively first failure censored data. It is assumed that the stress and strength variables follow independent Kumaraswamy-G distributions. The uniformly minimum variance unbiased and maximum likelihood estimators of the parameters, reliability functions R(t) and $$\delta$$ are developed. The asymptotic confidence intervals and hypothesis testing for the parameters and $$\delta$$ are obtained. A simulation study to evaluate the performance of the developed estimators is performed. Finally, a couple of real data examples is analyzed for illustrative purposes.

For the generalized positive exponential family of distributions, the problems of confidence interval estimation, bounded risk point estimation, and minimum risk point estimation of the rth power o...

In this article, the estimation of the stress–strength reliability function δ=P(X>Y) for generalized Maxwell failure distribution is considered. The Maximum-likelihood and Bayesian estimators for δ...

In this article, a study on the stress-strength parameter P=P(X>Y) based on lower record values from one parameter proportional reversed hazard family (PRHF) has been conducted. The classical and B...

In this paper, a two-stage sequential estimation procedure is considered to construct fixed-accuracy confidence intervals of the reliability parameter R under the stress–strength model when the stress and strength are independent exponential random variables with different scale parameters. The exact distribution of the total sample size, explicit formulas for the expected value, and mean squared error of the maximum likelihood estimator of the reliability parameter under the stress–strength model are provided under the two-stage sequential procedure. Performances of the proposed methodology are investigated with the help of simulations. Finally, using three pairs of real datasets, the procedure is clearly illustrated. We used MATLAB in order to implement computations, simulation studies, and real data analysis.

In the first part of this paper, we propose purely sequential and k-stage (k ≥ 3) procedures for estimation of the mean μ of an inverse Gaussian distribution having prescribed ‘proportional closeness’. The problem is constructed in such a manner that the boundedness of the expected loss is equivalent to the estimation of parameter with given ‘proportional closeness’. We obtain the associated second-order approximations for both the procedures. Second part of this paper deals with developing the minimum risk and bounded risk point estimation problems for estimating the mean μ of an inverse Gaussian distribution having unknown scale parameter λ. We propose an useful family of loss functions for both the problems and our aim is to control the associated risk functions. Moreover, we establish the failure of fixed sample size procedures to deal with these problems and hence propose purely sequential and k-stage (k ≥ 3) procedures to estimate the mean μ. We also obtain the second-order approximations associated with our sequential procedures. Further, we provide extensive sets of simulation studies and real data analysis to show the performances of our proposed procedures.

We consider two measures of reliability functions namely R(t)=P(X>t) and P=P(X>Y) for the Moore and Bilikam (1978) family of lifetime distributions which covers fourteen distributions as specific cases. For record data from this family of distributions, preliminary test estimators (PTEs) and preliminary test confidence interval (PTCI) based on uniformly minimum variance unbiased estimator (UMVUE), maximum likelihood estimator (MLE), empirical Bayes estimator (EBE) are obtained for the parameter. The bias and mean square error (MSE) (exact and asymptotic) of the proposed estimators are derived to study their relative efficiency and through simulation studies we establish that PTEs perform better than ordinary UMVUE, MLE and EBE. We also obtain the coverage probability (CP) and the expected length of the PTCI of the parameter and establish that the confidence intervals based on MLE are more precise. An application of the ordinary preliminary test estimator is also considered. To the best of the knowledge of the authors, no PTEs have been derived for R(t) and P based on records and thus we define improved PTEs based on MLE and UMVUE of R(t) and P. A comparative study of different methods of estimation done through simulations establishes that PTEs perform better than ordinary UMVUE and MLE.

In this article, a new three parameter generalized distribution named as Generalized Gamma-Maxwell distribution is introduced. The proposed generalization can be seen as an extension of the...

In this article, we have developed the robust Bayesian inference for the generalized inverted family of distributions (GIFD) under an ϵ-contamination class of prior distributions for the shape para...

The aim of this paper is to develop the estimation procedures for the reliability functions R(t) and P, when the available data have the form of progressively type-II right censored sample....

For the family of distributions considered by Chaturvedi and Alam, the problems of minimum risk point estimation and bounded risk point estimation of the parameter are handled. Consideration is giv...

In this article, we have proposed a three-stage procedure for the estimation of the difference of the means of two multivariate normal populations having unknown and unequal variances. Point as wel...

In the first part of this article, a minimum risk estimation procedure is developed for estimating the mean μ of an inverse Gaussian distribution having an unknown scale parameter λ. A weighted squ...

A very important distribution called Nakagami distribution is taken into consideration. Reliability measures R(t)=Pr(X>t) and P=Pr(X>Y) are considered. Point as well as interval procedures are obtained for estimation of parameters. Uniformly Minimum Variance Unbiased Estimators (U.M.V.U.Es) and Maximum Likelihood Estimators (M.L.Es) are developed for the said parameters. A new technique of obtaining these estimators is introduced. Moment estimators for the parameters of the distribution have been found. Asymptotic confidence intervals of the parameter based on M.L.E and log(M.L.E) are also constructed. Then, testing procedures for various hypotheses are developed. At the end, Monte Carlo simulation is performed for comparing the results obtained. A real data analysis is performed to describe the procedure clearly.

Classical estimators and preliminary test estimators (PTEs) of the powers of model parameter and the two measures of reliability, namely $$R\left( t \right) = P(X > t)$$ and $$P = P(X > Y)$$ of Kumaraswamy-G distribution, are developed under progressive type-II censoring. The preliminary test confidence intervals (PTCIs) are also developed based on uniformly minimum variance unbiased estimators (UMVUEs) and maximum likelihood estimators (MLEs). Merits of PTEs and PTCIs are established. Simulation results are presented, and real-life data set is also analysed.

We consider here the generalized inverted scale family of distributions proposed by Potdar and Shirke [Inference for the parameters of generalized inverted family of distributions. ProbStat Forum. ...

In the current study we develop the robust Bayesian inference for the generalized inverted family of distributions (GIFD) under an e-contamination class of prior distributions for the shape...

This paper presents the extension of the inferences on the stress-strength reliability in more than two states to the system depending on the ratio of the strength and stress values when the stress...

We consider here the general class of distributions proposed by Sankaran and Gupta (2005) by zeroing in on two measures of reliability, R(t) = P(X>t) and P = P(X>Y). Thereafter we develop point estimation and testing procedures for R(t) and ‘P’; and develop uniformly minimum variance unbiased estimators (UMVUES). Then we derive testing procedures for the hypotheses related to different parametric functions. Finally, we compare the results using the Monte Carlo simulation method. Using real data set, we illustrate the procedure clearly.

A family of lifetime distributions is considered which covers many distributions as its special cases. Two measures of reliability are studied, R(t)=P(X > t) and P = P(X > Y). Assuming the availability of some prior information on the parameter of interest, shrinkage estimators are developed for the powers of the scale parameter and reliability functions based on records. These proposed estimators are compared with the maximum likelihood estimators and uniformly minimum variance unbiased estimators of the parametric functions in terms of their relative efficiency. We establish that all the proposed estimators outperform the usual estimators as the true value of parameters approaches their hypothesized value.

The present paper has developed the preliminary test estimators (PTEs) of the model parameter raised to certain power, σp, and the two measures of reliability, namely, the reliability function, R(t ) and the reliability of an item or a system, P of an exponentiated distribution, under Type- II censoring, based on their uniformly minimum variance unbiased estimators (UMVUEs) and maximum likelihood estimators (MLEs). The preliminary test confidence intervals (PTCIs) are also developed for σ, R(t ) and P based on their UMVUEs and MLEs. Further, the paper has derived expression for coverage probability of the PTCI of the model parameter, σ. Merits of the proposed PTEs are also established through analysis of simulated numerical data., Statistica, Vol 79, No 1 (2019)

Two measures of reliability functions, namely R(t)=P(X>t) and P=P(X

A family of lifetime distributions is considered. Two measures of reliability are considered, R(t) = P(X > t) and P = P(X > Y). Point estimation and testing procedures are developed for R(t) and P based on records. Two types of point estimators are developed—uniformly minimum variance unbiased estimators and maximum likelihood estimators. A comparative study of different methods of estimation is done through simulation studies. Testing procedures are developed for the hypothesis related to different parametric functions.

Two measures of reliability are considered, R ( t ) = P ( X > t ) and P = P ( X > Y ) . Estimation and testing procedures are developed for R ( t ) and P under Type II cesoring and a sampling scheme of Bartholomew (1963). Two types of point estimators are considered (i) uniformly minimum variance unbiased estimators (UMVUEs) and (ii) maximum likelihood estimators (MLEs). A new technique of obtaining these estimators is introduced. A comparative study of different methods of estimation is done. Testing procedures are developed for the hypotheses related to different parametric functions.

In this article, characterization based on record values for Kumaraswamy-G distributions is provided. Two measures of reliability are considered, namely R(t) = P(X > t) and P = P(X > Y). Point and interval estimation procedures are developed for unknown parameter(s), R(t) and P, based on records. Two types of point estimators are considered, namely (1) uniformly minimum variance unbiased estimators and (2) maximum likelihood estimators. Testing procedures are also developed for the hypotheses related to various parametric functions. A comparative study of different methods of estimation is done through simulation studies. Real data example is used to illustrate the results.

A generalized inverted scale family of distributions is considered. Two measures of reliability are discussed, namely ρ(t) = P(X > t) and P = P(X > Y ). Point and interval estimation procedures are developed for the parameters, ρ(t) and P based on records. Two types of point estimators are developed - uniformly minimum variance unbiased estimators (UMVUES) and maximum likelihood estimators (MLES). A comparative study of different methods of estimation is done through simulation studies and asymptotic confidence intervals of the parameters based on MLE and log transformed MLE are constructed. Testing procedures are also developed for the parametric functions of the distribution and a real life example has been analysed for illustrative purposes.

SYNOPTIC ABSTRACT: The one-parameter exponential distribution plays an important role in reliability theory. Two measures of reliability for exponential distribution are considered, R(t) = P(X > t) and P = P(X > Y). Sometimes, due to past knowledge or experience, the experimenter may be in a position to make an initial guess on some of the parameters of interest. In such cases, we can provide an improved estimator by incorporating the prior information on the parameters. Preliminary test estimators (PTES) have been developed in the literature for the parameters of various distributions. To the best of the knowledge of the authors, PTES are not available for reliability functions R(t) and P. For record values from exponential distribution, we define PTES based on uniformly minimum variance unbiased estimator (UMVUE), maximum likelihood estimator (MLE), and empirical Bayes estimator (EBE) for the powers of the parameter, R(t) and P. Bias and mean square error (MSE) expressions for the proposed estimators are derived to examine their efficiency. A comparative study of different methods of estimation is done through simulations, and it is established that PTES perform better than ordinary UMVUES, MLES, and EBES.

Athree parameter Burr distribution is considered. Two measures of reliability are discussed. Point and interval estimation procedures are developed for the parameters, and reliability functions under type II and type I censoring. Two types of point estimators namely- uniformly minimum variance unbiased estimators (UMVUES) and maximum likelihood estimators (MLES) are derived. Asymptotic variance-covariance matrix and confidence intervals for MLE’s are obtained. Testing procedures are also developed for various hypotheses., Statistica, Vol 77, No 3 (2017)

Two families of distributions are considered that cover a large number of probability distributions useful in investigations involving reliability studies and survival analyses. The problem of bounded risk point estimation of the parameter and hazard rate function of the two families of distribution is handled. Motivated by Mukhopadhyay and Pepe (2006), Roughani and Mahmoudi (2015), and Mahmoudi and Lalehzari (2017), two-stage procedures are developed based on the maximum likelihood estimator (MLE) as well as uniformly minimum variance unbiased estimator (UMVUE). The estimation problem based on the minimum mean square estimator (MMSE) is also considered. We establish that the MMSE of the parameter and hazard rate provides a smaller risk.

The inverse Weibull distribution has the ability to model failure rates which are quite common in reliability and biological studies. A three parameters generalized inverse Weibull distribution with decreasing and unimodal failure rate is studied. Two measures of reliability are discussed, namely R(t) = P(X > t) and P = P(X > Y). Point and interval estimation procedures are developed for the parameters, R(t) and P based on records. Two point estimators are developed, namely uniformly minimum variance unbiased estimators (UMVUE) and maximum likelihood estimators (MLE). A comparison of different methods of estimation is done through simulations and asymptotic confidence intervals of the parameters based on MLE and log transformed MLE are constructed. Confidence intervals for the MLE and UMVUE of the parametric functions are obtained. Testing procedures are also developed for various hypotheses.

In this paper, Robust Bayesian analysis of the generalized half logistic distribution (GHLD) under an $\epsilon$-contamination class of priors for the shape parameter $\lambda$ is considered. ML-II Bayes estimators of the parameters, reliability function and hazard function are derived under the squared-error loss function (SELF) and linear exponential (LINEX) loss function by considering the Type~II censoring and the sampling scheme of Bartholomew (1963). Both the cases when scale parameter is known and unknown is considered under Type~II censoring and under the sampling scheme of Bartholomew. Simulation study and analysis of a real data set are presented.

Exponentiated distributions are considered. Two measures of reliability are considered, R(t)=P(X>t) and P=P(X>Y). Point estimation and testing procedures are developed for different parametric functions under Type II and Type I censoring. Uniformly minimum variance unbiased estimators (UMVUES) and maximum likelihood estimators (MLES) are derived. A new technique of obtaining these estimators is introduced., Statistica, Vol 77, No 1 (2017)

Problem Statement: The two-parameter exponentiated Rayleigh distribution has been widely used especially in the modelling of life time event data. It provides a statistical model which has a wide variety of application in many areas and the main advantage is its ability in the context of life time event among other distributions. The uniformly minimum variance unbiased and maximum likelihood estimation methods are the way to estimate the parameters of the distribution. In this study we explore and compare the performance of the uniformly minimum variance unbiased and maximum likelihood estimators of the reliability function R(t)=P(X>t) and P=P(X>Y) for the two-parameter exponentiated Rayleigh distribution. Approach: A new technique of obtaining these parametric functions is introduced in which major role is played by the powers of the parameter(s) and the functional forms of the parametric functions to be estimated are not needed. We explore the performance of these estimators numerically under varying conditions. Through the simulation study a comparison are made on the performance of these estimators with respect to the Biasness, Mean Square Error (MSE), 95% confidence length and corresponding coverage percentage. Conclusion: Based on the results of simulation study the UMVUES of R(t) and ‘P’ for the two-parameter exponentiated Rayleigh distribution found to be superior than MLES of R(t) and ‘P’.

A family of exponentiated distributions is proposed. The problems of estimating the reliability function are considered. Uniformly minimum variance unbiased estimators and maximum likelihood estimators are derived. A comparative study of the two methods of estimation is done. Simulation study is preformed.

The problems of estimating the reliability function and P=PrX > Y are considered for the generalized life distributions. Uniformly minimum variance unbiased estimators (UMVUES) of the powers of the parameter involved in the probabilistic model and the probability density function (pdf) at a specified point are derived. The UMVUE of the pdf is utilized to obtain the UMVUE of the reliability function and ‘P’. Our method of obtaining these estimators is quite simple than the traditional approaches. A theoretical method of studying the behaviour of the hazard-rate is provided.

Three-stage and ‘accelerated’ sequential procedures are developed for estimating the mean of a normal population when the population coefficient of variation (CV) is known. In spite of the usual estimator, i.e. the sample mean, Searls' (1964) estimator is utilized for the estimation purpose. It is established that Searls' estimator dominates the sample mean under the two sampling schemes.

SYNOPTIC ABSTRACTSequential procedures are developed for the point estimation of the parameter of the generalized life distributions under (i) squared-error loss function plus linear cost of sampling and (ii) squared-error loss function plus log-cost function. Second-order approximations are obtained for the risk and theoretical justifications to some of the numerical findings of Starr and Woodroofe (1972) is given. ‘Improved’ estimators are proposed and their dominance over the usual estimators is established.

The problem of estimating a linear function of k normal means with unknown variances is considered under an asymmetric loss function such that the associated risk is bounded from above by a known quantity. In the absence of a fixed sample size rule, sequential stopping rules satisfying a general set of assumptions are considered. Two estimators are proposed and second-order asymptotic expansions of their risk functions are derived. It is shown that the usual estimator, namely the linear function of the sample means, is asymptotically inadmissible, being dominated by a shrinkage-type estimator. An example illustrates the use of different multistage sampling schemes and provides asymptotic expansions of the risk functions.

A class of probability density functions is considered, which covers several life-testing models as specific cases. Sequential probability ratio tests are developed for testing simple and composite hypotheses regarding the parameters of the probabilistic model. Expressions for the operating characteristic and the average sample number functions are derived and their behaviour is studied by means of graph-plotting.

The problems of obtaining uniformly minimum variance unbiased estimators of ξ=P(Y>X) and ξ k (where k is a positive integer) when X and Y follow two-parameter exponential distributions considered by Pal et al. [M. Pal, M. Ali Masoom, and J. Woo, Estimation and testing of P(Y>X) in two parameter exponential distributions, Statistics 39 (2005), pp. 415–428] are revisited. Much simpler techniques of obtaining these estimators are provided.

For the model considered by Chaturvedi and Gupta (1995), first of all, we develop a class of 'routine-type' two-stage procedures to construct fixed-width confidence region (which may be interval, ellipsoidal or spherical) for the parameter(s) of interest. The drawbacks of this 'routine-type' two-stage procedure, over other multi-stage procedures, are discussed. In order to remove these drawbacks, along the lines of Takada (1993) and Kumar and Chaturvedi (1993), an 'improved' class of two-stage procedures is proposed. The exact and asymptotic properties of the two classes are studied. By means of examples, we illustrate that many estimation problems fall under our set-up.

The sequential procedure developed by Govindarajulu and Sarkar [Sequential estimation of scale parameter in exponential distributions with unknown location. Utilitas Math. 40 , 161–178 (1991)] for estimating the scale parameter of an exponential distribution, when the location parameter is unknown, is further analyzed. Generalizing the results of Govindarajulu and Sarkar, the ‘asymptotic risk-efficiency’ of the sequential procedure is established for the general loss function. A simple method of obtaining the asymptotic distribution of the stopping time is given. For the case of quadratic loss function and linear cost of sampling, a much simpler proof for obtaining the second-order approximations for the risk is provided.

A family of density functions is considered which contains several life-testing models as specific cases. Uniformly minimum variance unbiased estimators are obtained for the positive and negative powers of the parameter, moments and reliability function. These general results provide the estimators for the specific models.

A sequential procedure is developed in order to construct a confidence interval of “fixed-width and preassigned coverage probability” for the inverse of the coefficient of variation of a normal population. The proposed sequential procedure is proved to be “asymptotically efficient and consistent” in the sense of Chow and Robbins ([1]: Ann. Math. Statist. 36, 457–462 (1965)). Asymptotic distribution of the stopping time is derived.