Ajit Chaturvedi

A note on the estimation of P(Y>X) in two-parameter exponential distributions

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.

Sequential testing procedures for a class of distributions representing various life-testing models

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.

Robust Bayesian analysis of generalized inverted family of distributions

ABSTRACT In the current study we develop the robust Bayesian inference for the generalized inverted family of distributions (GIFD) under an ε-contamination class of prior distributions for the shape parameter α, with different possibilities of known and unknown scale parameter. We used Type II censoring and Bartholomew sampling scheme (1963) for the following derivations under the squared-error loss function (SELF) and linear exponential (LINEX) loss function : ML-II Bayes estimators of the i) parameters; ii) Reliability function and; iii) Hazard function. We also present simulation study and analysis of a real data set.

Estimation and Testing Procedures for the Reliability Functions of a General Class of Distributions

ABSTRACT 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 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.

Estimating the Reliability Function for a Family of Exponentiated Distributions

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.

Second-Order Approximations to Two Classes of Sequential Estimation Procedures

For the model considered by Chaturvedi, Pandey and Gupta (1991), two classes of sequential procedures are developed to construct confidence regions (which may be interval, ellipsoidal or spherical) of ‘pre-assigned width and coverage probability’ for the parameters of interest and for the minimum risk point estimation (taking loss to be quadratic plus linear cost of sampling) of the nuisance parameter. Second-Order approximations are derived for the expected sample size, coverage probability and ‘regret’ associated with the two classes of sequential procedures. A simple and direct method of obtaining the asymptotic distribution of the stopping time is provided. By means of examples, it is illustrated that several estimation problems can be tackled with the help of proposed classes of sequential procedures.

Estimation and testing procedures of the reliability functions of generalized inverted scale family of distributions

ABSTRACT 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. 2013;06:18–28] by zeroing in on two measures of reliability, and , based on Type II censoring and the sampling scheme of Bartholomew [The sampling distribution of an estimate arising in life testing. Technometrics. 1963;5:361–374]. Thereafter, we develop point estimators of powers of α, and ; and develop uniformly minimum variance unbiased (UMVU) estimators and maximum likelihood (ML) estimators. A new technique of obtaining these estimators is introduced. Exact confidence intervals for the parameter α, and are also constructed based on Type II censoring. Then we derive testing procedures for the hypotheses related to different parametric functions. Finally, we compare these results using the Monte Carlo simulation method.

Two-stage procedures for the bounded risk point estimation of the parameter and hazard rate in two families of distributions

ABSTRACT 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.

Preliminary test estimators of the reliability characteristics for the three parameters Burr XII distribution based on records

Some improved estimators and confidence interval of the parametric functions are proposed based on records from three parameters Burr XII distribution. We propose preliminary test estimators (PTES) of the powers of the parameter and reliability functions based on uniformly minimum variance unbiased estimator, maximum likelihood estimator, best invariant estimator and empirical Bayes estimator. We compare the performance of the proposed PTES with the usual estimators by studying their relative efficiencies based on Monte Carlo simulations. We also construct preliminary test confidence interval (PTCI) for the parameter and study its coverage probability and expected length. The results show that the proposed PTES dominate the usual estimators in a wide range of the parametric space. Also it is seen that the proposed PTCI have higher coverage probability while keeping the shorter width in some domain of parametric space. The paper ends up by analysing a real data set.

On the Construction of Preliminary Test Estimators of the Reliability Characteristics for the Exponential Distribution Based on Records

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.