Estimation of P(Y < X) for Weibull Distribution Under Progressive Type-II Censoring

Based on progressively Type II censored samples, we consider the estimation of R = P(Y < X) when X and Y are two independent Weibull distributions with different shape parameters, but having the same scale parameter. The maximum likelihood estimator, approximate maximum likelihood estimator, and Bayes estimator of R are obtained. Based on the asymptotic distribution of R, the confidence interval of R are obtained. Two bootstrap confidence intervals are also proposed. Analysis of a real data set is given for illustrative purposes. Monte Carlo simulations are also performed to compare the different proposed methods.     

Inference of stress-strength for the Type-II generalized logistic distribution under progressively Type-II censored samples

This article studies the estimation of the reliability R = P[Y < X] when X and Y come from two independent generalized logistic distributions of Type-II with different parameters, based on progressively Type-II censored samples. When the common scale parameter is unknown, the maximum likelihood estimator and its asymptotic distribution are proposed. The asymptotic distribution is used to construct an asymptotic confidence interval of R. Bayes estimator of R and the corresponding credible interval using the Gibbs sampling technique have been proposed too. Assuming that the common scale parameter is known, the maximum likelihood estimator, uniformly minimum variance unbiased estimator, Bayes estimation, and confidence interval of R are extracted. Monte Carlo simulations are performed to compare the different proposed methods. Analysis of a real dataset is given for illustrative purposes. Finally, methods are extended for proportional hazard rate models.     

SOME ASYMPTOTIC PROPERTIES OF THE SIMPLE LOGLINEAR MODEL

Consistency and asymptotic normality of the maximum likelihood estimator of β in the loglinear model E(y_i) = e^α+βXi, where y_i are independent Poisson observations, 1 iaan, are proved under conditions which are near necessary and sufficient. The asymptotic distribution of the deviance test for β=β₀ is shown to be chi-squared with 1 degree of freedom under the same conditions, and a second order correction to the deviance is derived. The exponential model for censored survival data is also treated by the same methods.     

Inferences on Stress-Strength Reliability from Lindley Distributions

This article deals with the estimation of the stress-strength parameter R = P(Y < X) when X and Y are independent Lindley random variables with different shape parameters. The uniformly minimum variance unbiased estimator has explicit expression, however, its exact or asymptotic distribution is very difficult to obtain. The maximum likelihood estimator of the unknown parameter can also be obtained in explicit form. We obtain the asymptotic distribution of the maximum likelihood estimator and it can be used to construct confidence interval of R. Different parametric bootstrap confidence intervals are also proposed. Bayes estimator and the associated credible interval based on independent gamma priors on the unknown parameters are obtained using Monte Carlo methods. Different methods are compared using simulations and one data analysis has been performed for illustrative purposes.     

Inference on P(Y < X) in Bivariate Rayleigh Distribution

This paper deals with the estimation of reliability R = P(Y < X) when X is a random strength of a component subjected to a random stress Y, and (X, Y) follows a bivariate Rayleigh distribution. The maximum likelihood estimator of R and its asymptotic distribution are obtained. An asymptotic confidence interval of R is constructed using the asymptotic distribution. Also, two confidence intervals are proposed based on Bootstrap method and a computational approach. Testing of the reliability based on asymptotic distribution of R is discussed. Simulation study to investigate performance of the confidence intervals and tests has been carried out. Also, a numerical example is given to illustrate the proposed approaches.     

Stress-Strength Reliability of a Two-Parameter Bathtub-shaped Lifetime Distribution Based on Progressively Censored Samples

Based on progressively Type-II censored samples, this article deals with inference for the stress-strength reliability R = P(Y < X) when X and Y are two independent two-parameter bathtub-shape lifetime distributions with different scale parameters, but having the same shape parameter. Different methods for estimating the reliability are applied. The maximum likelihood estimate of R is derived. Also, its asymptotic distribution is used to construct an asymptotic confidence interval for R. Assuming that the shape parameter is known, the maximum likelihood estimator of R is obtained. Based on the exact distribution of the maximum likelihood estimator of R an exact confidence interval of that has been obtained. The uniformly minimum variance unbiased estimator are calculated for R. Bayes estimate of R and the associated credible interval are also got under the assumption of independent gamma priors. Monte Carlo simulations are performed to compare the performances of the proposed estimators. One data analysis has been performed for illustrative purpose. Finally, we will generalize this distribution to the proportional hazard family with two parameters and derive various estimators in this family.     

Adaptive Warped Kernel Estimators

In this work, we develop a method of adaptive non‐parametric estimation, based on ‘warped’ kernels. The aim is to estimate a real‐valued function s from a sample of random couples (X,Y). We deal with transformed data (Φ(X),Y), with Φ a one‐to‐one function, to build a collection of kernel estimators. The data‐driven bandwidth selection is performed with a method inspired by Goldenshluger and Lepski (Ann. Statist., 39, 2011, 1608). The method permits to handle various problems such as additive and multiplicative regression, conditional density estimation, hazard rate estimation based on randomly right‐censored data, and cumulative distribution function estimation from current‐status data. The interest is threefold. First, the squared‐bias/variance trade‐off is automatically realized. Next, non‐asymptotic risk bounds are derived. Lastly, the estimator is easily computed, thanks to its simple expression: a short simulation study is presented.     

Weak and strong uniform consistency rates of kernel density estimates for randomly censored data

Let X₁,., X_n, be i.i.d. random variables with distribution function F, and let Y₁,.,.,Y_n be i.i.d. with distribution function G. For i = 1, 2,.,., n set δ_i, = 1 if X_i ≤ Y_i, and 0 otherwise, and X_i, = min{X_i, K_i}. A kernel-type density estimate of f, the density function of F w.r.t. Lebesgue measure on the Borel o-field, based on the censored data (δ_i, X_i), i = 1,.,.,n, is considered. Weak and strong uniform consistency properties over the whole real line are studied. Rates of convergence results are established under higher-order differentiability assumption on f. A procedure for relaxing such assumptions is also proposed.     

Inferential analysis for the reliability parameter based on the three-parameter Lindley distribution

ABSTRACT

In this article, we consider the estimation of R = P(Y < X), when Y and X are two independent three-parameter Lindley (LI) random variables. On the basis of two independent samples, the modified maximum likelihood estimator along its asymptotic behavior and conditional likelihood-based estimator are used to estimate R. We also propose sample-based estimate of R and the associated credible interval based on importance sampling procedure. A real life data set involving the times to breakdown of an insulating fluid is presented and analyzed for illustrative purposes.     

Improved estimators for the selected location parameters

_i , i = 1, 2, ..., k be k independent exponential populations with different unknown location parameters θ _i , i = 1, 2, ..., k and common known scale parameter σ. Let Y _i denote the smallest observation based on a random sample of size n from the i-th population. Suppose a subset of the given k population is selected using the subset selection procedure according to which the population π _i is selected iff Y _i ≥Y ₍₁₎−d, where Y ₍₁₎ is the largest of the Y _i 's and d is some suitable constant. The estimation of the location parameters associated with the selected populations is considered for the squared error loss. It is observed that the natural estimator dominates the unbiased estimator. It is also shown that the natural estimator itself is inadmissible and a class of improved estimators that dominate the natural estimator is obtained. The improved estimators are consistent and their risks are shown to be O(kn ⁻²). As a special case, we obtain the coresponding results for the estimation of θ₍₁₎, the parameter associated with Y ₍₁₎. Received: January 6, 1998; revised version: July 11, 2000     

Inference of R = P[X < Y] for skew normal distribution

This article studies the estimation of R = P[X < Y] when X and Y are two independent skew normal distribution with different parameters. When the scale parameter is unknown, the maximum likelihood estimator of R is proposed. The maximum likelihood estimator, uniformly minimum variance unbiased estimator, Bayes estimation, and confidence interval of R are obtained when the common scale parameter is known. In the general case, the maximum likelihood estimator of R is also discussed. To compare the different proposed methods, Monte Carlo simulations are performed. At last, the analysis of a real dataset has been presented for illustrative purposes too.     

Estimation of P(Y < X) for progressively first-failure-censored generalized inverted exponential distribution

In this article, we consider the problem of estimation of the stress–strength parameter δ?=?P(Y?<?X) based on progressively first-failure-censored samples, when X and Y both follow two-parameter generalized inverted exponential distribution with different and unknown shape and scale parameters. The maximum likelihood estimator of δ and its asymptotic confidence interval based on observed Fisher information are constructed. Two parametric bootstrap boot-p and boot-t confidence intervals are proposed. We also apply Markov Chain Monte Carlo techniques to carry out Bayes estimation procedures. Bayes estimate under squared error loss function and the HPD credible interval of δ are obtained using informative and non-informative priors. A Monte Carlo simulation study is carried out for comparing the proposed methods of estimation. Finally, the methods developed are illustrated with a couple of real data examples.     

On estimation of R=P(Y<X) for exponential distribution under progressive type-II censoring

This paper deals with the estimation of the stress–strength parameter R=P(Y<X), when X and Y are independent exponential random variables, and the data obtained from both distributions are progressively type-II censored. The uniformly minimum variance unbiased estimator and the maximum-likelihood estimator (MLE) are obtained for the stress–strength parameter. Based on the exact distribution of the MLE of R, an exact confidence interval of R has been obtained. Bayes estimate of R and the associated credible interval are also obtained under the assumption of independent inverse gamma priors. An extensive computer simulation is used to compare the performances of the proposed estimators. One data analysis has been performed for illustrative purpose.     

Maximum likelihood estimation,from doubly censored samples,of the mean of a normal distribution with known coeffucient of variation

Let x be a random variable having the normal distribution with mean μ and variance c²μ², where c is a known constant. The maximum likelihood estimation of μ when the lowest r₁ and the highest r₂ sample values censored have been given the asymptotic variance of the maximum likelihood estimator is obtained.     

A distribution function estimator for the difference of order statistics from two independent samples

For two independent populations X and Y we develop the empirical distribution function estimator for the difference of order statistics of the form X _(i)−Y _(j). The key practical application for this estimator pertains to inference between quantiles from two independent populations.     

Inference of R=P[X<Y] for generalized logistic distribution

This paper deals with the estimation of R=P[X<Y] when X and Y come from two independent generalized logistic distributions with different parameters. The maximum-likelihood estimator (MLE) and its asymptotic distribution are proposed. The asymptotic distribution is used to construct an asymptotic confidence interval of R. Assuming that the common scale parameter is known, the MLE, uniformly minimum variance unbiased estimator, Bayes estimation and confidence interval of R are obtained. The MLE of R, asymptotic distribution of R in the general case, is also discussed. Monte Carlo simulations are performed to compare the different proposed methods. Analysis of a real data set has also been presented for illustrative purposes.     

Estimation of a finite population distribution function based on a linear model with unknown heteroscedastic errors

The authors consider a finite population ρ = {(Y_k, x_k), k = 1,…,N} conforming to a linear superpopulation model with unknown heteroscedastic errors, the variances of which are values of a smooth enough function of the auxiliary variable X for their nonparametric estimation. They describe a method of the Chambers‐Dunstan type for estimation of the distribution of {Y_k, k = 1,…, N} from a sample drawn from without replacement, and determine the asymptotic distribution of its estimation error. They also consider estimation of its mean squared error in particular cases, evaluating both the analytical estimator derived by “plugging‐in” the asymptotic variance, and a bootstrap approach that is also applicable to estimation of parameters other than mean squared error. These proposed methods are compared with some common competitors in simulation studies.     

A remark on estimating the mean of a normal distribution with known coefficient of variation

Let X₁, X₂, …, X_n be iid N(μ, aμ²) (a>0) random variables with an unknown mean μ>0 and known coefficient of variation (CV) √a. The estimation of μ is revisited and it is shown that a modified version of an unbiased estimator of μ [cf. Khan RA. A note on estimating the mean of a normal distribution with known CV. J Am Stat Assoc. 1968;63:1039–1041] is more efficient. A certain linear minimum mean square estimator of Gleser and Healy [Estimating the mean of a normal distribution with known CV. J Am Stat Assoc. 1976;71:977–981] is also modified and improved. These improved estimators are being compared with the maximum likelihood estimator under squared-error loss function. Based on asymptotic consideration, a large sample confidence interval is also mentioned.     

Estimation of P(X>Y) with Topp–Leone distribution

We consider the estimation problem of the probability P=P(X>Y) for the standard Topp–Leone distribution. After discussing the maximum likelihood and uniformly minimum variance unbiased estimation procedures for the problem on both complete and left censored samples, we perform a Monte Carlo simulation to compare the estimators based on the mean square error criteria. We also consider the interval estimation of P.