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This paper investigates interval estimation for measures of overlap, namely Matusita's measure, Weitzman's measure and based on Kullback–Leibler. Two types of sampling procedures, namely, Simple Random Sample and Upper Record Values from two exponential populations. Bootstrap method series approximation is used to construct confidence intervals for the overlap measures. To illustrate the performance of the likelihood confidence intervals obtained for this overlapping coefficient, under our proposal, we carry out some simulation studies that yield adequate coverage frequencies, this study conducts for comparing the performances of various competing estimators. A real data set is analysed to exemplify our proposal. [ABSTRACT FROM AUTHOR]

We study a type of double truncation where units are observed if and only if their death event occurs within a specific timespan. The resulting missing data mechanism is nonignorable and thus has to be reconsidered. Based on the density function of observed lifetimes and the random sample size, we derive a likelihood model that enables simultaneous estimation of the lifetime distribution and the parameters governing the birth process. In particular, knowledge of the population size is not required. We show that the model is identifiable under certain conditions by using results on exponential families. Bayesian estimators and corresponding standard errors for all involved parameters become available by using MCMC simulation. We describe how the simulation can be performed efficiently while maintaining sufficiently good mixing behaviour of the resulting chains. Both finite-sample and asymptotic properties of the investigated estimators are examined through a simulation study. The proposed method is applied to estimate the lifetime distribution of German companies. [ABSTRACT FROM AUTHOR]

Present investigation deals with the problem of random non response in estimation of population variance on current occasion in two-occasion successive sampling. To reduce the negative impact of random non response on both occasions, regression type imputation method has been suggested. Using auxiliary information, efficient estimation strategies have been developed for estimation of population variance on the current occasion. Estimator for the population variance is also derived as a special case when random non response occurs only on the current occasion. Empirical studies are carried out to show the dominance of suggested estimators over sample variance and exponential type estimators. Results are interpreted. [ABSTRACT FROM AUTHOR]

Let X be lognormal( μ, σ2) with density f( x); let θ > 0 and define [ABSTRACT FROM AUTHOR]

The problem of multiple outliers detection in one-parameter exponential family is considered. The outlier detection procedure involves two estimates of scale parameter which are obtained by maximizing two log-likelihoods; the complete data log-likelihood and its conditional expectation given suspected observations. The procedure is also applied to the exponential and normal samples. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we investigate the ordering properties of sample ranges arising from multiple-outlier models in terms of the reversed hazard rate order and the usual stochastic order. Under the setup of an exponential model, it is shown that the weak majorization order between the two hazard rate vectors is equivalent to the reversed hazard rate order between exponential sample ranges; the p-larger order between two hazard rate vectors implies the usual stochastic order between exponential sample ranges. Under the setup of a proportional hazard rate (PHR) model, we prove that the majorization order between two parameter vectors implies the usual stochastic order between sample ranges. The results established here strengthen and generalize some of the results known in the literature. Some numerical examples are provided to illustrate the theoretical results. [Copyright &y& Elsevier]

This paper proposes a new test for the hypothesis that two samples have the same distribution. The likelihood ratio test of Portnoy [Portnoy, S. (1988), 'Asymptotic Behaviour of Likelihood Methods for Exponential Families When the Number of Parameters Tends to Infinity', Annals of Statistics, 16, 356-366] is applied in the context of the consistent series density estimator of Crain [Crain, B.R. (1974), 'Estimation of Distributions Using Orthogonal Expansions', Annals of Statistics, 2, 454-463] and Barron and Sheu [Barron, A.R., and Sheu, C.-H. (1991), 'Approximation of Density Functions by Sequences of Exponential Families'. Annals of Statistics, 19, 1347-1369]. It is proven that the test, when suitably standardised, is asymptotically standard normal and consistent against any complementary fixed alternative. In comparison with established tests, such as the Kolmogorov-Smirnov, Cramer-von Mises and rank sum, median, and dispersion tests, the proposed tests enjoy broadly comparable finite sample size properties, but vastly superior power properties when considered over a range of different alternatives. [ABSTRACT FROM AUTHOR]

This paper discusses statistical inference in connection with a Weibull distribution model, using Type-II progressively censored data with random scheme. We used the maximum likelihood procedure to derive both point and interval estimates of the unknown parameters included in the model. The expected termination point to complete the censoring test is computed and analyzed under different censoring schemes. A numerical study is presented to illustrate the application of the theoretical results presented. [ABSTRACT FROM AUTHOR]

A new approach to sampling/interpolation problem for vector valued signals is proposed. For a function (signal) f from the Paley-Wiener space PWπ(CN) of entire vector valued functions, the sampling data are defined as the inner products (f(Ιn), αn) of f(Ιn) and vectors αn. It is proved that (αn, Ιn) is a sampling and interpolating sequence for PWπ (CN) if and only if the family (αnϱiΙnt) forms a Riesz basis in L²(—π, πCN). This relation allows to characterize all sampling and interpolating sequences in terms of a generating matrix function. To produce a wide class of matrix functions generating sampling and interpolating sequences we consider the matrix Sturm-Liouville problems and apply the Boundary Control Theory for hyperbolic dynamical systems. This approach allows to construct both asymptotically uniform and nonuniform sampling and interpolating sequences. [ABSTRACT FROM AUTHOR]

Abstract: We modify and extend George and Mudholkar''s [1981. A characterization of the logistic distribution by a sample median. Ann. Inst. Statist. Math. 33, 125–129] characterization result about the logistic distribution, which is in terms of the sample median and Laplace distribution. Moreover, we give some new characterization results in terms of the smallest order statistics and the exponential distribution. [Copyright &y& Elsevier]

Best linear unbiased estimation for parameters of a particular location-scale family based on progressively Type-II censored order statistics is considered and optimal censoring schemes are determined. As optimality criteria serve the ϕp-criteria from experimental design which are applied to the covariance matrix of the BLUEs. The results are supplemented by monotonicity properties of the trace and the determinant with respect to the sample size and the initial number of items in the experiment. [ABSTRACT FROM AUTHOR]

In this paper we consider the problem of unbiased estimation of the distribution function of an exponential population using order statistics based on a random sample. We present a (unique) unbiased estimator based on a single, say i th, order statistic and study some properties of the estimator for i = 2. We also indicate how this estimator can be utilized to obtain unbiased estimators when a few selected order statistics are available as well as when the sample is selected following an alternative sampling procedure known as ranked set sampling. It is further proved that for a ranked set sample of size two, the proposed estimator is uniformly better than the conventional nonparametric unbiased estimator, further, for a general sample size, a modified ranked set sampling procedure provides an unbiased estimator uniformly better than the conventional nonparametric unbiased estimator based on the usual ranked set sampling procedure. [ABSTRACT FROM AUTHOR]

In this article, we study a statistical model which features a finite population of exponentially distributed values and a length-biased, with-replacement sampling mechanism. This mechanism is such that units compete with one another for selection at each draw. It is shown how inference on a number of quantities can be performed using both frequentist and Bayesian strategies. A Monte Carlo study is used to assess the performance of the proposed point and interval estimators. [ABSTRACT FROM AUTHOR]

Summary. The task of estimating an integral by Monte Carlo methods is formulated as a statistical model using simulated observations as data. The difficulty in this exercise is that we ordinarily have at our disposal all of the information required to compute integrals exactly by calculus or numerical integration, but we choose to ignore some of the information for simplicity or computational feasibility. Our proposal is to use a semiparametric statistical model that makes explicit what information is ignored and what information is retained. The parameter space in this model is a set of measures on the sample space, which is ordinarily an infinite dimensional object. None-the-less, from simulated data the base-line measure can be estimated by maximum likelihood, and the required integrals computed by a simple formula previously derived by Vardi and by Lindsay in a closely related model for biased sampling. The same formula was also suggested by Geyer and by Meng and Wong using entirely different arguments. By contrast with Geyer's retrospective likelihood, a correct estimate of simulation error is available directly from the Fisher information. The principal advantage of the semiparametric model is that variance reduction techniques are associated with submodels in which the maximum likelihood estimator in the submodel may have substantially smaller variance than the traditional estimator. The method is applicable to Markov chain and more general Monte Carlo sampling schemes with multiple samplers. [ABSTRACT FROM AUTHOR]

Model diagnostic procedures in failure time models using hazard-based residuals rely on the assumption that the residual vector closely resembles a random sample from a unit exponential distribution when the model holds. This article formally investigates the validity of this critical assumption by deriving and examining the properties of parametrically, semiparametrically, and nonparametrically estimated residuals for complete and right-censored data. The joint distribution of the residual vector is characterized, and the behavior of some tests for exponentiality when applied to the residuals is examined analytically and through Monte Carlo methods. Findings reveal that the critical assumption of approximate unit exponentiality of the residual vector may not be viable and, consequently, the model diagnostic procedures considered, which revolve on checking the approximate unit exponentiality of the residual vector (specifically, hazard plotting and the use of spacings and total-time-on-test statistics on the residual vector) may have serious defects. This is especially evident in situations where the failure time distribution is not exponential or when the residuals are obtained nonparametrically in the no-covariate model or semiparametrically in the Cox proportional hazards model. [ABSTRACT FROM AUTHOR]

Parameterizations for natural exponential families (NEF's) with quadratic variance functions (QVF's) are compared according to the nearness to normality of the likelihood and posterior distribution. Non normality of the likelihood (posterior) is measured using two criteria The first is the magnitude of a standardised third derivative of the log-likelihood (logposterior density), the second is a comparison of the probability of particular tail regions under the normalized likelihood (posterior distribution) and under the corresponding normal approximation A relationship is given that links these two criteria Sample sizes are recommended for adequate normality in the likelihood for various parameterizations of the NEF-QVF models, and these results are extended to Bayesian models with a conjugate prior. [ABSTRACT FROM AUTHOR]

In one-parameter exponential families such as the binomial and Poisson, the variance is a function of the mean. Double exponential families allow the introduction of a second parameter that controls variance independently of the mean. Double families are used as constituent distributions in generalized linear regressions, in which both means and variances are allowed to depend on observed covariates. The theory is applied to two examples--a logistic regression and a large two-way contingency table. In such cases the binomial model of variance is often untrustworthy. For example, because genuine random sampling was infeasible, the subjects may have been obtained in clumps so that the statistician should really be using smaller sample sizes. Clumped sampling is just one of many possible causes of overdispersion, a habitual source of concern to users of binomial and Poisson models. This article concerns a class of regression families that allow the statistician to model overdispersion while carrying out the usual regression analyses for the mean as a function of the predictors. Close connections with previous ideas concerning generalized linear models are discussed. [ABSTRACT FROM AUTHOR]

The problem of estimating the mean of an exponential distribution is studied, when the data available are a random sample of time-censored rth-order statistics. Examples for such empirical situations are cited. Maximum likelihood estimators (MLE's) and moment-equation estimators (MEE's) are studied. Theoretical derivations are provided for the large sample variances and distributions of these estimators. The efficiency of the MEE compared to the MLE is studied. [ABSTRACT FROM AUTHOR]

Strong consistency and asymptotic normality of the maximum likelihood estimators are established in the context of jointly type-II censored samples from two exponential populations. Large-sample confidence intervals are derived for the individual scale parameters as well as their ratio, and some applications to series and parallel systems are discussed. For the ratio of the scale parameters, an exact confidence procedure is developed on the basis of the failure counts in the two samples, and its asymptotic efficiency is investigated. [ABSTRACT FROM AUTHOR]

This article investigates the use of individual cross-section data to describe macro functions. Necessary and sufficient conditions (denoted by AS) are found for ordinary least squares (OLS) slope coefficients from a cross section to consistently estimate the first derivatives of the macro function. AS embodies both sets of aggregation assumptions known; linear aggregation and sufficient statistics, and thus represents generalized aggregation conditions. A methodology is given for estimating second-order derivatives of the macro function from cross-section data for distributions of the exponential family, which extends to higher-order derivatives. Finally, a general test of linear aggregation schemes is presented. [ABSTRACT FROM AUTHOR]

In life testing, type I and type II censoring may be combined to form a hybrid life test. Epstein (1954) introduced this testing scheme and Epstein (1960b) proposed a two-sided confidence interval to estimate the mean lifetime, theta, of an exponential lifetime following such a test. His conjecture is modified slightly and a proof is given that establishes the validity of the proposed confidence interval. [ABSTRACT FROM AUTHOR]

The Bayesian sequential estimation problem for one-parameter exponential families is considered using loss related to the Fisher information and linear sampling cost. Tractible expressions for the Bayes estimator and the posterior expected loss are found, and the myopic, or one-step-ahead, stopping rule is defined. Sufficient conditions are given for optimality of the myopic procedure, and the myopic procedure is shown to be asymptotically optima! in all cases considered. [ABSTRACT FROM AUTHOR]

Consider a life-testing experiment subject to Type I censoring, where m items are put on test at the outset and are not replaced on failure. Assume an exponential failure distribution F(t) = 1 - exp (-t/theta). A test of H[sub 0]: theta is greater than or equal to versus H[sub 1]: theta < xi based on the maximum likelihood estimator of theta is proposed. It is shown that early termination of the test resulting in either acceptance or rejection of Ho is possible. Critical points are presented. The test is shown to compare favorably with a test presented by Epstein (1954) in terms of power and expected duration time. A comparison of the test with the untruncated and truncated sequential probability ratio test is also made. [ABSTRACT FROM AUTHOR]

Suppose a population is composed of I is greater than or equal to 2 distinct subpopulations, where pi, > 0 is the proportion of the population in the ith subpopulation, i = 1, ... , I. Given theta[sub t], 0 < theta[sub i], < 1, a sequence of independent Bernoulli (theta[sub t]) random variables can be observed from the ith subpopulation that are also independent across subpopulations. Of interest is the simultaneous Bayes sequential estimation of theta[sub 1], ..., theta[sub I], using component loss (theta [sub i] - theta[sub i])2/theta[sub t](1 - theta[sub t] and linear sampling cost. The estimation problem is solved under two sampling schemes. The first scheme requires that at each stage one observation be taken from the entire population without controlling the subpopulation sampled. The second scheme requires that at each stage one observation be taken from each subpopulation. The Bayes sequential estimation procedures are found for both schemes and compared in terms of expected total cost. These results are then extended from Bernoulli trials to a one-parameter exponential family with mean theta[sub i] and component loss (theta[sub i] - theta[sub i]) 2/var[sub theta[sub i]]U, where var[sub theta[sub i]U is the ith subpopulation variance. [ABSTRACT FROM AUTHOR]

This paper is a continuation of previous work concerning progressively censored sampling in the normal and the exponential distribution [1], in the Weibull distribution [4], and in the lognormal distribution [5]. Here maximum likelihood estimators and estimators which utilize the first order statistic are derived for the three-parameter gamma distribution (Pearson's Type Ill Distribution) when samples are progressively censored. Various special cases are also considered. Illustrative examples involving life test data are included. Some of the sampling properties of the proposed estimators are investigated. [ABSTRACT FROM AUTHOR]

Sufficient conditions for the validity of interchange between derivative and expectation, in the context of likelihood ratio gradient estimation, were given in L'Ecuyer (1990). The aim of this paper is to shed additional light on these conditions and introduce specific variants of them, which are often easier to check. Sufficient conditions for the derivative estimator to have finite moments up to a given order are also given and illustrated by examples. In particular, we give an example of an unbiased derivative estimator which satisfies the interchange conditions but which has infinite variance. [ABSTRACT FROM AUTHOR]

The estimation of functions of Bernoulli parameters is of interest in reliability theory. In this article, the theory of exponential families is used to obtain exact confidence limits for products and quotients of Bernoulli parameters when negative binomial observations are available from each population. Sampling methods based on compound Poisson distribution are suggested for estimating more general functions such as sums of products of Bernoulli parameters. Finally, it is shown that exact confidence limits for products can be obtained by a discrete analog of the Lieberman-Ross procedure, which exploits the independence of the minimum and difference of geometric variates. [ABSTRACT FROM AUTHOR]

All known theorems of the Darmois-Koopman-Pitman type are based on the assumption of absolutely continuous probabilities. In this article, a Darmois-Koopman-Pitman type theorem is proved for families of probability measures that are defined on discrete sample spaces. Two assumptions are imposed on the sufficient statistic for the family: (1) the range space of the sufficient statistic can be completely ordered, and (2) this ordering of the range space of the sufficient statistic has a certain regularity property. Under (1) and (2), it can be proved that the given family is exponential of order 1. [ABSTRACT FROM AUTHOR]

The paper presents the results of an inquiry concerning the small sample relative efficiency of maximum likelihood and best unbiased estimators of reliability functions of one-unit systems. Three cases are considered: The Poisson, exponential, and normal (standard deviation known). Two kinds of relative efficiency functions are studied. The first kind consists of the common ratio of the Cramer-Rao lower bound of the variances of unbiased estimators, to the mean-square-error of the considered estimator. The second kind is a new type of a relative efficiency function, which is called 'the closeness relative efficiency function.' This function is defined as the ratio of the probabilities that the maximum likelihood and the beat unbiased estimators yield estimates in a prescribed neighborhood of the unknown reliability value. A substantial part of the study is devoted to the derivation of the required moments of the estimators. [ABSTRACT FROM AUTHOR]

This paper investigates the properties of the minimum variance unbiased (M.V.U) and maximum likelihood (M.L.) estimators of the reliability functions of systems composed of two subsystems connected in series. The study falls into two parts, one for the Poisson case and one for the exponential case. In each of these cases the situations are distinguished between, where the two subsystems are identical and situations subsystems are different. In the Poisson case under minimum variance unbiased estimators a system A is considered which is composed of two subsystems connected in series. Failure time points of the subsystem follow a Poisson process with intensity. An experiment is performed on n independent replicates of each of the considered subsystems over a period of length. Under the exponential case, a system A is considered, same as Poisson case which consists of two subsystems connected in series. Failure time points of the two subsystems follow a Poisson process. Independent observations are available on the interfailure time lengths; namely, the life-lengths of the subsystems.