Pitman Closeness of kth Records Based on a Two-Sequence Case

Suppose upper kth records were observed from an X-sequence of iid continuous random variables, and kth upper records from another independent Y-sequence of iid variables from the same distribution are to be observed. The Pitman closeness probabilities of these statistics are derived. For symmetric distribution, the Pitman closeness probabilities of kth record statistics to the population median, are also examined and it is shown that these probabilities are distribution free. Numerical computations are conducted to illustrate the results developed here.     

On the Nearness of Records to Population Quantiles with Respect to Order Statistics in the Sense of Pitman Closeness

In this paper, Pitman closeness criterion is used to compare the nearness of record values and order statistics from two independent samples to a specific population quantile of the parent distribution while the underlying distributions are the same. General expressions for the associated Pitman closeness probability are obtained when the support of the parent distribution is bounded and also unbounded. Some distribution-free results are achieved for symmetric distributions. The exponential and uniform distributions are considered for illustrative proposes and exact expressions are obtained in each case.     

Closeness of Grassia's Transformed Gammas and the Beta Distribution

The flexibility and applicability of the Beta distribution are well known. In this note a comparison is made between the Beta and two other distributions on (0,1) which have been proposed by Grassia (1977) and which are derived through transformation of the Gamma distribution. The extreme closeness of the distributions is demonstrated graphically and explored more formally through a measure of closeness (Kolmogorov-Smirnov distance). Alternative measures of closeness are also briefly discussed. It is concluded that for many applications the distributions are sufficiently similar (that is, virtually identical) for a choice among them to rest solely on mathematical convenience.     

Comparison of order statistics in two-sample problem in the sense of Pitman closeness

In this paper, we study the Pitman measure of closeness of order statistics of two independent samples from the same distribution to population quantiles. We then derive various exact expressions of the probability closeness of order statistics from the X and Y samples. Some distribution-free results for the median of the sampling distribution are obtained. Exact and explicit expressions are presented for Uniform(?1, 1) and exponential distributions. Numerical results for illustrative purposes are also provided.     

Simultaneous Closeness of k-Records

In this article, we study the simultaneous Pitman closeness of upper (and lower) k-records to a common parameter of the parent distribution and obtain general expressions for the corresponding probabilities. Since the usual record values are contained in the k-records, the corresponding results for the usual records can be deduced as special cases. The results are then applied to location-scale families with an emphasis on population quantiles. Exact expressions are derived for some common distributions such as exponential and uniform. In each case, the simultaneous closest k-record to a population quantile is determined.     

Discriminating between the generalized Rayleigh and Weibull distributions: Some comparative studies

The generalized Rayleigh distribution was introduced and studied quite effectively in the literature. The closeness and separation between the distributions are extremely important for analyzing any lifetime data. In this spirit, both the generalized Rayleigh and Weibull distributions can be used for analyzing skewed datasets. In this article, we compare these two distributions based on the Fisher information measures and use it for discrimination purposes. It is evident that the Fisher information measures play an important role in separating between the distributions. The total information measures and the variances of the different percentile estimators are computed and presented. A real life dataset is analyzed for illustration purposes and a numerical comparison study is performed to assess our procedures in separating between these two distributions.     

Generalized exponential distribution: Existing results and some recent developments

Mudholkar and Srivastava [1993. Exponentiated Weibull family for analyzing bathtub failure data. IEEE Trans. Reliability 42, 299–302] introduced three-parameter exponentiated Weibull distribution. Two-parameter exponentiated exponential or generalized exponential distribution is a particular member of the exponentiated Weibull distribution. Generalized exponential distribution has a right skewed unimodal density function and monotone hazard function similar to the density functions and hazard functions of the gamma and Weibull distributions. It is observed that it can be used quite effectively to analyze lifetime data in place of gamma, Weibull and log-normal distributions. The genesis of this model, several properties, different estimation procedures and their properties, estimation of the stress-strength parameter, closeness of this distribution to some of the well-known distribution functions are discussed in this article.     

Bayesian inference for the variance components in general mixed linear models

The general mixed linear model, containing both the fixed and random effects, is considered. Using gamma priors for the variance components, the conditional posterior distributions of the fixed effects and the variance components, conditional on the random effects, are obtained. Using the normal approximation for the multiple t distribution, approximations are obtained for the posterior distributions of the variance components in infinite series form. The same approximation Is used to obtain closed expressions for the moments of the variance components. An example is considered to illustrate the procedure and a numerical study examines the closeness of the approximations.     

On Pitman's measure of closeness of k-records

Pitman closeness of both the upper and lower k-record statistics to the population quantiles of a location–scale family of distributions is studied. For the population median, the Pitman-closest k-record is also determined. In the case of symmetric distributions, the Pitman closeness probabilities of k-record statistics are shown to be distribution-free, and explicit expressions are also derived for these probabilities. Exact expressions are derived for the required probabilities for uniform and exponential distributions. Numerical results are given for these families and also the Pitman-closest k-record is determined.     

Estimation of parameters and reliability from generalized life models

Simultaneous estimation of parameters with p (≥ 2) components, where each component has a generalized life distribution, is considered under a sum of squared error loss function. Improved estimators are obtained which dominate the maximum likelihood and the niinimum mean square estimators. Robustness of the improved estimators is shown even when the component distributions are dependent. The result is extended to the estimation of the system reliability when the components are connected in series. Several numerical studies are performed to demonstrate the risk improvement and the Pitman closeness of the new estimators.     

Approximating distributions of random functionals of Ferguson-Dirichlet priors

We explore the possibility of approximating the Ferguson-Dirichlet prior and the distributions of its random functionals through the simulation of random probability measures. The proposed procedure is based on the constructive definition illustrated in Sethuraman (1994) in conjunction with the use of a random stopping rule. This allows us to set in advance the closeness to the distributions of interest. The distribution of the stopping rule is derived, and the practicability of the simulating procedure is discussed. Sufficient conditions for convergence of random functionals are provided. The numerical applications provided just sketch the idea of the variety of nonparametric procedures that can be easily and safely implemented in a Bayesian setting.     

Point estimation of lower quantiles based on two-sampling scheme

ABSTRACT

Consider a two-sampling scheme in which an initial sample is first taken from the underlying population and then by assuming a suitable restriction on this sample, some more data points are observed as a new restricted sample. This sampling scheme is used to do inference about the lower quantiles of the underlying distribution. The results are compared with those of simple random sampling in view of mean squared error and Pitman’s measure of closeness criteria for exponential and uniform distributions. It will be shown that the proposed sampling scheme would improve the performance of the point estimators of the lower quantiles of the population.     

Estimators of percentiles based on absolute loss

Estimators of percentiles of location and scale parameter distributions are optimized based on Pitman closeness and absolute risk. A median unbiased (MU) estimator and a minimum risk (MR) estimator are shown to exist within a class of estimators, and to depend upon the medians of two completely specified distributions.     

Current records and record range with some applications

In a sequence of independent and identically distributed (iid) random variables, the upper (lower) current records and record range are studied. We derive general recurrence relations between the single and product moments for the upper and lower current records based on Weibull and positive Weibull distributions, as well as Pareto and negative Pareto distributions, respectively. Moreover, some asymptotic results for general current records are established. In addition, a recurrence relation and an explicit formula for the moments of record range based on the exponential distribution are given. Finally, numerical examples are presented to illustrate and corroborate theoretical results.     

On approximating a linear combination of central wishart matrices with positive coefficients

Consider linear combinations of central Wishart matrices with positive coefficients. By equating the expected values and the generalized variance of variances and covariances, we propose approximating the distributions of those linear combinations by central Wishart distributions. Some Monte Carlo results are given to demonstrate the closeness of this approximation.     

Two-sample Pitman closeness comparison under progressive Type-II censoring

In this paper, we consider two problems concerning two independent progressively Type-II censored samples. We first consider the Pitman closeness (PC) of order statistics from two independent progressively censored samples to a specific population quantile. We then consider the point prediction of a future progressively censored order statistic and discuss the determination of the closest progressively censored order statistic from the current sample according to the simultaneous closeness probabilities. For both these problems, explicit expressions are derived for the pertinent PC probabilities, and then special cases are given as examples. For various censoring schemes, we also present numerical results for the standard uniform, standard exponential, and standard normal distributions. Finally, a distribution-free result for the median is obtained.     

Comparisons of the percentage points of distributions with the same first four moments,chosen from eight different systems of frequency curves

Probability integrals and percentage points of univariate distributions from up to eight different families, having common first four moments are compared. Among interesting observations is the remarkable consistency in the standardized upper and lower 5% points over considerable regions of the √β₁, β₂ plane; also the closeness of agreement between the log-normal and non-central t distributions and the Pearson Type VI and Type IV curves respectively.     

Estimation and prediction in the presence of an outlier under Type-II censoring

A single-outlier data set containing some independent random variables is considered such that all of observations expect one have the same distribution. To describe the model of interested, a location-scale family of distributions is used and the estimation problem of the parameters is studied when the data are collected under Type-II censoring scheme. Moreover, three different predictors are presented to predict the censored order statistics. They are also compared regarding both of mean squared prediction error and Pitman's measure of closeness criteria. The role of outlier parameter as well as censorship rate is studied on performance of proposed estimator and predictors. The results of the paper are illustrated via a real data set. Finally, some conclusions are stated.     

Pitman closeness of the class of isotonic estimators for ordered scale parameters of two Gamma distributions

The problem of estimating order-restricted scale parameters of two Gamma distributions is considered under the Pitman closeness criterion. A class of isotonic estimators including the MLE is proposed. Some properties of this class of isotonic estimators is given under the Pitman closeness criterion. In particular, some properties of the MLE naturally follow. We use simulation comparisons to examine the results.     

Comments on estimators based on error in the predicted distribution function

In estimation of percentiles in the exponential distribution, the distribution function evaluated at the estimated percentile is often evaluated for purposes of warranty considerations. Optimal estimators are discussed and compared on their error in the predicted distribution function. Inconsistency is shown to exist between measures of closeness and measures of risk in the predicted distribution function. An optimal estimator based on absolute loss in the predicted distribution function is obtained and shown to be superior in measures of closeness to the optimal estimator, which minimizes squared error loss in the predicted distribution function.