Maximum Likelihood Estimation in the Inverse Weibull Distribution with Type II Censored Data

We consider maximum likelihood estimation for the parameters and certain functions of the parameters in the Inverse Weibull (IW) distribution based on type II censored data. The functions under consideration are the Mean Residual Life (MRL), which is very important in reliability studies, and Tail Value at Risk (TVaR), which is an important measure of risk in actuarial studies. We investigated the performance of the MLE of the parameters and derived functions under various experimental conditions using simulation techniques. The performance criteria are the bias and the mean squared error of the estimators. Recommendations on the use of the MLE in this model are given. We found that the parameter estimators are almost unbiased, while the MRL and TVaR estimators are asymptotically unbiased. Moreover, the mean squared error of all estimators decreased for larger sample sizes and it increased when the censoring proportion is increased for a fixed sample size. The conclusion is that the maximum likelihood method of estimation works well for the parameters and the derived functions of the parameter like the MRL and TVaR. Two examples on a real data set are presented to illustrate the application of the methods used in this paper. The first one is on survival time of pigs while the other is on fire losses.The authors would like to thank the referees for their suggestions and thoughtful comments that resulted in a much-improved version of the paper. This research was supported by a grant from the Office of Research Support at Qatar University, Grant no. QUST-1-CAS-2022-318

Preliminary test estimation in the two parameter exponential distribution based on record values

We considered estimation of the parameters of the exponential distribution based on record data in the presence of prior information. Preliminary test and shrinkage estimators are developed and their properties are investigated using simulation techniques. An example to illustrate the methodology is given. ? Nova Science Publishers, Inc.This work is supported by a grant from Yarmouk University.Scopu

TESTING SYMMETRY USING A TRIMMED LONGEST RUN STATISTIC

A distribution-free test is proposed for the symmetry of a continuous distribution about a specified median. The test is based on a longest run statistic on the upper portion of the sequence of ordered �centred� observations in magnitude. The probability distribution of the longest run statistic is derived, and a computationally simple and accurate approximation of the right-tail probabilities of this statistic is given. This approximation is based on a partial fraction expansion of the corresponding generating function and is derived for use with large samples. The powers of the proposed test, some variations of the test, and other rival tests are investigated under a wide variety of asymmetric alternatives. Simulations indicate that the proposed test is competitive with other tests in terms of power performance.Wiley Online Librar

Interval estimation of quantile difference in the two-parameter exponential distributions

We consider the interval estimation of the difference between the quantiles of two random variables with independent two-parameter exponential distributions based on Type II censored data. We derive asymptotic intervals based on the likelihood function, Bayesian intervals, as well as intervals based on the generalized pivot variable. We include some bootstrap intervals in our comparisons. The performance of the intervals is investigated in terms of their coverage probabilities and expected lengths using simulation techniques. An illustrative example is given.Scopu

Approximating the tail probabilities of the longest run in a sequence of Bernoulli trials

We consider the longest run of either successes or failures in a sequence of (Formula presented.) Bernoulli trials. The exact distribution of this random variable is obtained using probability generating function techniques. We consider approximating the tail areas of this random variable using a partial fraction approximation and a saddlepoint approximation with various continuity corrections. We investigate and compare the performance of the approximations for various values of sequence length and various values of the probability of success

Estimation procedures based on preliminary test in exponential distribution under progressive censoring

This paper considers the preliminary test estimator of the unknown parameter of the exponential distribution based on progressively Type-II censored samples. The mean squared error of the proposed estimator is derived and it is shown that this estimator dominates the corresponding usual estimator in the neighborhood of null hypothesis. The minimax regret estimator and shrinkage estimator are also discussed. A numerical example is given to illustrate the proposed estimators. Nova Science Publishers, Inc.Scopu

Estimation of common location parameter of two exponential populations based on records

Consider the problem of estimating the common location parameter of two exponential populations using record data when the scale parameters are unknown. We derive the maximum likelihood estimator (MLE), the modified maximum likelihood estimator (MMLE) and the uniformly minimum variance unbiased estimator (UMVUE) of the common location parameter. Further, we derive a general result for inadmissibility of an equivariant estimator under the scaled-squared error loss function. Using this result, we conclude that the MLE and the UMVUE are inadmissible and better estimators are provided. A simulation study is conducted for comparing the performances of various competing estimators. - 2018 Taylor & Francis Group, LL

ESTIMATION OF THE LOCATION-SCALE PARAMETERS BASED ON RANKED SET SAMPLING COMBINED WITH ADDITIONAL BINOMIAL INFORMATION

In this paper, we propose a modified version of ranked set sample which allows for the incorporation of more information in the inference procedure at almost no cost. This procedure is studied for the location-scale family using a likelihood framework. Our results show that the modified procedure produce more accurate inferences than the original ranked set sampling scheme. © 2023 Pakistan Journal of Statistics

Likelihood and Bayesian Inference in the Lomax Distribution under Progressive Censoring

The Lomax distribution has been used as a statistical model in several fields, especially for business failure data and reliability engineering. Accurate parameter estimation is very important because it is the base for most inferences from this model. In this paper, we shall study this problem in detail. We developed several points and interval estimators for the parameters of this model assuming the data are type II progressively censored. Specifically, we derive the maximum likelihood estimator and the associated Wald interval. Bayesian point and interval estimators were considered. Since they can't be obtained in a closed form, we used a Markov chains Monte Carlo technique, the so called the Metropolis – Hastings algorithm to obtain approximate Bayes estimators and credible intervals. The asymptotic approximation of Lindley to the Bayes estimator is obtained for the present problem. Moreover, we obtained the least squares and the weighted least squares estimators for the parameters of the Lomax model. Simulation techniques were used to investigate and compare the performance of the various estimators and intervals developed in this paper. We found that the Lindley's approximation to the Bayes estimator has the least mean squared error among all estimators and that the Bayes interval obtained using the Metropolis – Hastings to have better overall performance than the Wald intervals in terms of coverage probabilities and expected interval lengths. Therefore, Bayesian techniques are recommended for inference in this model. An example of real data on total rain volume is given to illustrate the application of the methods developed in this paper