Estimation of Pr(X < Y) Using Record Values in the One and Two Parameter Exponential Distributions

We consider the problem of estimating the stress-strength reliability when the available data is in the form of record values. The one parameter and two parameters exponential distribution are considered. In the case of two parameters exponential distributions we considered the case where the location parameter is common and the case where the scale parameter is common. The maximum likelihood estimators and the associated confidence intervals are derived.     

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.     

Unbiased Estimation of P(X > Y) for Exponential Populations Using Order Statistics with Application in Ranked Set Sampling

This paper addresses the problem of unbiased estimation of P[X > Y] = θ for two independent exponentially distributed random variables X and Y. We present (unique) unbiased estimator of θ based on a single pair of order statistics obtained from two independent random samples from the two populations. We also indicate how this estimator can be utilized to obtain unbiased estimators of θ when only a few selected order statistics are available from the two random samples as well as when the samples are selected by an alternative procedure known as ranked set sampling. It is proved that for ranked set samples of size two, the proposed estimator is uniformly better than the conventional non-parametric unbiased estimator and further, a modified ranked set sampling procedure provides an unbiased estimator even better than the proposed estimator.     

Asymptotic and Resampling-Based Confidence Intervals for P(X < Y)

ABSTRACT

We consider asymptotic and resampling-based interval estimation procedures for the stress-strength reliability P(X < Y). We developed and studied several types of intervals. Their performances are investigated using simulation techniques and compared in terms of attainment of the nominal confidence level, symmetry of lower and upper error rates, and expected length. Recommendations concerning their use are given.     

Estimation and testing of P(Y > X) in two-parameter exponential distributions

In this article, we obtain the UMVUE of the reliability function ξ=P(Y>X) and the UMVUE of ξ ^k =[P(Y>X)] ^k in the two-parameter exponential distributions with known scale parameters. We also derive the distribution of the UMVUE of ξ and further considering the tests of hypotheses regarding the reliability function ξ.     

The simplicity of likelihood based inferences for P(X < Y) and for the ratio of means in the exponential model

The profile likelihood of the reliability parameter θ = P(X < Y) or of the ratio of means, when X and Y are independent exponential random variables, has a simple analytical expression and is a powerful tool for making inferences. Inferences about θ can be given in terms of likelihood-confidence intervals with a simple algebraic structure even for small and unequal samples. The case of right censored data can also be handled in a simple way. This is in marked contrast with the complicated expressions that depend on cumbersome numerical calculations of multidimensional integrals required to obtain asymptotic confidence intervals that have been traditionally presented in scientific literature.     

A note on inference for P(X < Y) for right truncated exponentially distributed data

In this paper, a likelihood based analysis is developed and applied to obtain confidence intervals and p values for the stress-strength reliability R  =  P(X  <  Y) with right truncated exponentially distributed data. The proposed method is based on theory given in Fraser et al. (Biometrika 86:249–264, 1999) which involves implicit but appropriate conditioning and marginalization. Monte Carlo simulations are used to illustrate the accuracy of the proposed method.     

An Efficient Method of Parameter and Quantile Estimation for the Three-Parameter Weibull Distribution Based on Statistics Invariant to Unknown Location Parameter

The three-parameter Weibull distribution is widely used in life testing and reliability analysis. In this article, we propose an efficient method for the estimation of parameters and quantiles of the three-parameter Weibull distribution, which avoids the problem of unbounded likelihood, by using statistics invariant to unknown location. Through a Monte Carlo simulation study, we show that the proposed method performs well compared to other prominent methods based on bias and MSE. Finally, we present two illustrative examples.     

Inference of δ = P(X < Y) for Burr XII distributions with record samples

Let X and Y follow independent Burr type XII distributions, which share a common inner shape parameter. The maximum likelihood estimator of the parameter δ = P(X < Y) is studied based on record samples. The existence and uniqueness of the maximum likelihood estimator of δ based on record samples are established. When the inner shape parameter is known, an exact confidence interval of δ is derived; otherwise, the Fisher information matrix and two bootstrap methods are used to obtain three approximate confidence intervals of δ. The performances of the proposed methods are evaluated via Monte Carlo simulation. Two examples are provided for illustration.     

Estimation of P{X < Y} for geometric—exponential model based on complete and censored samples

This article deals with the estimation of R = P{X < Y}, where X and Y are independent random variables from geometric and exponential distribution, respectively. For complete samples, the MLE of R, its asymptotic distribution, and confidence interval based on it are obtained. The procedure for deriving bootstrap-p confidence interval is presented. The UMVUE of R and UMVUE of its variance are derived. The Bayes estimator of R is investigated and its Lindley's approximation is obtained. A simulation study is performed in order to compare these estimators. Finally, all point estimators for right censored sample from the exponential distribution, are obtained.     

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.     

Estimation of δ=P(X<Y) for Burr XII distribution based on the progressively first failure-censored samples

Let X and Y have two-parameter Burr XII distributions. The maximum-likelihood estimator of δ=P(X<Y) is studied under the progressively first failure-censored samples. Three confidence intervals of δ are constructed by using an asymptotic distribution of the maximum-likelihood estimator of δ and two bootstrapping procedures, respectively. Some computational results from intensive simulations are presented. An illustrative example is provided to demonstrate the application of the proposed method.     

Estimation of the Parameters of the Birnbaum–Saunders Distribution

Some alternative estimators to the maximum likelihood estimators of the two parameters of the Birnbaum–Saunders distribution are proposed. Most have high efficiencies as measured by root mean square error and are robust to departure from the model as well as to outliers. In addition, the proposed estimators are easy to compute. Both complete and right-censored data are discussed. Simulation studies are provided to compare the performance of the estimators.     

The Anderson–Darling Goodness-of-Fit Test Statistic for the Three-Parameter Lognormal Distribution

Anderson–Darling goodness-of-fit test percentage points are given for the three-parameter lognormal distribution for both the cases of positive skewness and a lower bound and negative skewness and an upper bound. The focus is on the most practical case when all parameters are unknown and must be estimated from the sample data. Fitted response functions for the critical values based on the shape parameter and sample size are reported to avoid using a vast array of tables.     

Convergence Rates of Empirical Bayes Estimation for the Parameter of the Uniform Distribution U(0, θ) Under Random Censorship

ABSTRACT

This article considers the empirical Bayes estimation problem in the uniform distribution U(0, θ) with censored data. For the parameter θ, using the empirical Bayes (EB) approach, we propose an EB estimation of θ which possesses a rate of convergence can be arbitrarily close to O(n ^?1/2) when the historical samples are randomly censored from the right, where n is the number of historical sample. A sample and some simulation results are also presented.     

A Bayesian inference of P(λ1<λ2) for two Poisson parameters

The statistical inference drawn from the difference between two independent Poisson parameters is often discussed in the medical literature. However, such discussions are usually based on the frequentist viewpoint rather than the Bayesian viewpoint. Here, we propose an index θ=P(λ_{1, post}<λ_{2, post}), where λ_{1, post} and λ_{2, post} denote Poisson parameters following posterior density. We provide an exact and an approximate expression for calculating θ using the conjugate gamma prior and compare the probabilities obtained using the approximate and the exact expressions. Moreover, we also show a relation between θ and the p-value. We also highlight the significance of θ by applying it to the result of actual clinical trials. Our findings suggest that θ may provide useful information in a clinical trial.     

Approximate Confidence Limits for a Proportion of the Pólya Distribution

The problem of constructing approximate confidence limits for a proportion parameter of the Pólya distribution is discussed. Three different methods for determining approximate one-sided and two-sided confidence limits for that parameter of the Pólya distribution have been proposed and compared. Particular cases of those confidence bounds are confidence intervals for the parameter of the binomial and the hypergeometric distributions.     

A Note on the Calculation of Pr{X 1 < X 2 < ··· < Xk }

Suppose that X_i are independent random variables, and that X_i has cdf F_i (x), 1 ≤ i ≤ k. Many statistical problems involve the probability Pr{X ₁ < X ₂ < ··· < X_k }. In this note a numerical method is proposed for computing this probability.     

Complement to a comparative analysis of heuristics for the <Emphasis Type=Italic>p</Emphasis>-median problem

A recent comparison of evolutionary, neural network, and scatter search heuristics for solving the p-median problem is completed by (i) gathering or obtaining exact optimal values in order to evaluate errors precisely, and (ii) including results obtained with several variants of a variable neighborhood search (VNS) heuristic. For a first, well-known, series of instances, the average errors of the evolutionary and neural network heuristics are over 10% and more than 1000 times larger than that of VNS. For a second series, this error is about 3% while the errors of the parallel VNS and of a hybrid heuristic are about 0.01% and that of parallel scatter search even smaller.     

Distribution of the Λ-Criterion for Sphericity Test and Its Connection to a Generalized Dirichlet Model

It is shown that the exact null distribution of the likelihood ratio criterion for sphericity test in the p-variate normal case and the marginal distribution of the first component of a (p ? 1)-variate generalized Dirichlet model with a given set of parameters are identical. The exact distribution of the likelihood ratio criterion so obtained has a general format for every p. A novel idea is introduced here through which the complicated exact null distribution of the sphericity test criterion in multivariate statistical analysis is converted into an easily tractable marginal density in a generalized Dirichlet model. It provides a direct and easiest method of computation of p-values. The computation of p-values and a table of critical points corresponding to p = 3 and 4 are also presented.