Showing papers in "Statistics in 2012"
TL;DR: This paper introduces a generalized measure of cumulative residual Renyi's entropy, and extends this measure into the bivariate set-up and proves certain characterizing relationships to identify different bivariate lifetime models.
Abstract: Recently, cumulative residual entropy (CRE) has been found to be a new measure of information that parallels Shannon's entropy (see Rao et al. [Cumulative residual entropy: A new measure of information, IEEE Trans. Inform. Theory. 50(6) (2004), pp. 1220–1228] and Asadi and Zohrevand [On the dynamic cumulative residual entropy, J. Stat. Plann. Inference 137 (2007), pp. 1931–1941]). Motivated by this finding, in this paper, we introduce a generalized measure of it, namely cumulative residual Renyi's entropy, and study its properties. We also examine it in relation to some applied problems such as weighted and equilibrium models. Finally, we extend this measure into the bivariate set-up and prove certain characterizing relationships to identify different bivariate lifetime models.
64 citations
TL;DR: In this article, a new method to estimate the mean of the number of persons possessing a rare sensitive attribute is proposed by utilizing the Poisson distribution in survey sampling, and the relative efficiencies of the proposed estimators over the direct question method estimator are investigated for different choices of parameters.
Abstract: In this paper, a new method to estimate the mean of the number of persons possessing a rare sensitive attribute is proposed by utilizing the Poisson distribution in survey sampling. Two situations are discussed: that when the proportion of persons possessing a rare unrelated attribute is known and that when it is unknown. Unbiased estimators of the mean number of persons possessing the rare sensitive attribute under two different situations are proposed. The variance expressions are derived in each situation. The relative efficiencies of the proposed estimators over the direct question method estimator are investigated for different choices of parameters and are discussed. A technical point is made that the traditional randomized response models cannot be used to estimate the mean of the Poisson random variable.
40 citations
TL;DR: This article introduced methods of constructing quantile functions as models of lifetimes with monotone and non-monotone hazard functions, based on the relationships the hazard quantile function has with the score function introduced by Parzen in connection with the tail heaviness of probability distributions.
Abstract: The present paper introduces methods of constructing quantile functions as models of lifetimes with monotone and nonmonotone hazard functions. This is accomplished on the basis of the relationships the hazard quantile function has with the score function introduced by Parzen in connection with the tail heaviness of probability distributions. Three models illustrated here contain several existing models as particular cases. The appropriateness of the models in real situations is also demonstrated.
39 citations
TL;DR: A family of bivariate copulas generated by a real-valued function is introduced and the obtained results are used in order to show that the Clayton family of copulas coincides with the class ofCopulas that are invariant under bivariate truncation and contains all exchangeable copulas which are invariants under univariate truncations.
Abstract: The class of all bivariate copulas that are invariant under univariate truncation is characterized. To this end, a family of bivariate copulas generated by a real-valued function is introduced. The obtained results are also used in order to show that the Clayton family of copulas (including its limiting elements) coincides with the class of copulas that are invariant under bivariate truncation and contains all exchangeable copulas which are invariant under univariate truncation.
37 citations
TL;DR: In this paper, generalized order statistics with conditional proportional hazard rates are used to derive maximum likelihood estimators for these parameters or functions of them along with several properties of the estimators.
Abstract: Generalized order statistics, and thus sequential order statistics with conditional proportional hazard rates, are shown to form a regular exponential family in the model parameters. This structure is utilized to derive maximum likelihood estimators for these parameters or functions of them along with several properties of the estimators. The Fisher information matrix is stated, and asymptotic efficiency is shown.
36 citations
TL;DR: In this article, the authors investigated the properties of the MPL in connection with other reliability measures, including hazard rate, mean residual lifetime, reversed hazard rate and the mean past lifetime.
Abstract: Let T be a lifetime random variable. In order to study the properties of T in reliability theory and survival analysis, several measures are proposed in the literature. Among these measures, hazard rate, mean residual lifetime, reversed hazard rate and the mean past lifetime (MPL) play important roles. In the present paper, we focus mainly on the MPL. We investigate its properties in connection with other reliability measures. Some results on partial ordering and characterization are also given. Finally, we deal with its statistical estimation.
28 citations
TL;DR: In this paper, the authors considered the problem of discriminating between the two distribution functions and obtained the asymptotic distribution of the discrimination statistic, which is used to determine the probability of correct selection in this discrimination process.
Abstract: Log-normal and Weibull distributions are the two most popular distributions for analysing lifetime data. In this paper, we consider the problem of discriminating between the two distribution functions. It is assumed that the data are coming either from log-normal or Weibull distributions and that they are Type-II censored. We use the difference of the maximized log-likelihood functions, in discriminating between the two distribution functions. We obtain the asymptotic distribution of the discrimination statistic. It is used to determine the probability of correct selection in this discrimination process. We perform some simulation studies to observe how the asymptotic results work for different sample sizes and for different censoring proportions. It is observed that the asymptotic results work quite well even for small sizes if the censoring proportions are not very low. We further suggest a modified discrimination procedure. Two real data sets are analysed for illustrative purposes.
28 citations
TL;DR: In this article, the authors proposed a summary metric for distributional asymmetry and spread based on the relative strengths of left and right-hand shifts, which is applicable even for long tail densities where distributional moments may not exist.
Abstract: Reflexive shifting of a given distribution, using its own distribution function, can reveal information. The shifts are changes in measure such that the separation of the resulting left and right unit shifted distributions reveals the binary entropy of position, called locational or partition entropy. This can be used for spread and asymmetry functions. Alternatively, summary metrics for distributional asymmetry and spread can be based on the relative strengths of left- and right-hand shifts. Such metrics are applicable even for long tail densities where distributional moments may not exist.
25 citations
TL;DR: In this article, the authors introduce a new tool to investigate some distributional properties of order statistics and records related by a random translation (contraction or dilation) scheme, based on the property of uniqueness of solutions of certain non-linear integral equations of Volterra type.
Abstract: We introduce a new tool to investigate some distributional properties of order statistics and records related by a random translation (contraction or dilation) scheme. This technique is based on the property of uniqueness of solutions of certain non-linear integral equations of Volterra type. We show how this tool is used to obtain new characterizations of distributions.
23 citations
TL;DR: In this paper, the authors derived the exact bivariate distribution of the MLE of the parameter vector of a two-parameter exponential model based on hybrid censored sample, and provided an alternate simpler expression for this distribution.
Abstract: Epstein [Truncated life tests in the exponential case, Ann. Math. Statist. 25 (1954), pp. 555–564] introduced a hybrid censoring scheme (called Type-I hybrid censoring) and Chen and Bhattacharyya [Exact confidence bounds for an exponential parameter under hybrid censoring, Comm. Statist. Theory Methods 17 (1988), pp. 1857–1870] derived the exact distribution of the maximum-likelihood estimator (MLE) of the mean of a scaled exponential distribution based on a Type-I hybrid censored sample. Childs et al. [Exact likelihood inference based on Type-I and Type-II hybrid censored samples from the exponential distribution, Ann. Inst. Statist. Math. 55 (2003), pp. 319–330] provided an alternate simpler expression for this distribution, and also developed analogous results for another hybrid censoring scheme (called Type-II hybrid censoring). The purpose of this paper is to derive the exact bivariate distribution of the MLE of the parameter vector of a two-parameter exponential model based on hybrid censored sample...
22 citations
TL;DR: In this paper, the risk function of a class of estimator for the mean parameter matrix of a matrix variate normal distribution is established, and the established result is useful in evaluating the performance of a set of shrinkage-pretest type estimators.
Abstract: In this paper, we establish the risk function of a class of estimator for the mean parameter matrix of a matrix variate normal distribution. In particular, the established result is useful in evaluating the performance of a class of shrinkage-pretest-type estimators.
TL;DR: In this paper, bias, variance and mean squared error (MSE) of the mean resultant length (MRL) was investigated for the population MRL, as an estimator of the average of random vectors on the unit circle.
Abstract: The mean resultant length (MRL) is the length of the average of random vectors on the unit circle. It is used to measure the concentration of unimodal circular distributions. The sample MRL, as an estimator for the population MRL, has not been investigated thoroughly yet. This work examines the bias, variance and mean- squared error (MSE) of the MRL. Unbiased or near unbiased estimators are developed wherever possible for the squared and non-squared MRL, as well as their variances and MSE. All estimators are tested numerically on four representative circular distributions.
TL;DR: This work presents a unifying framework which includes both Liebscher and Morillas copulas as special cases and above that, more general copulas may be constructed.
Abstract: Recently, Liebscher [Construction of asymetric multivariate copulas, J Multivariate Anal 99 (2008), pp 2234–2250] introduced a general construction scheme of d-variate copulas which generalizes the Archimedean family Similarly, Morillas [A method to obtain new copulas from a given one, Metrika 61 (2005), pp 169–184] proposed a method to obtain a variety of new copulas from a given d-copula Both approaches coincide only for the particular subclass of Archimedean copulas Within this work, we present a unifying framework which includes both Liebscher and Morillas copulas as special cases Above that, more general copulas may be constructed First examples are given
TL;DR: In this paper, a theory of non-parametric statistical estimation of decreasing alpha-percentile residual life (DPRL(alpha)) functions was proposed, and the authors studied the relationship between the DPRL(α) and the increasing failure rate ageing notions.
Abstract: Earlier researchers have studied some aspects of the classes of distribution functions with decreasing alpha-percentile residual life (DPRL(alpha)), 0 < alpha < 1. The purpose of this paper is to note some further properties of these classes, and to initiate a theory of non-parametric statistical estimation of DPRL(alpha) functions. Specifically, the close relationship between the DPRL(alpha) and the increasing failure rate ageing notions is studied. Other close relationships, between the DPRL(alpha) ageing notions and the percentile residual life stochastic orders, are described, and further properties of the above classes of distributions are derived. Finally, we introduce an estimator of the percentile residual life function, under the condition that it decreases, and we prove its strongly uniform consistency.
TL;DR: In this article, a new test of equality between two dependence structures is introduced, where the new statistics are functionals of a suitably integrated two-sample empirical copula process and the limiting behaviours of the proposed statistics are established under the null hypothesis.
Abstract: We introduce a new test of equality between two dependence structures. The new statistics are functionals of a suitably integrated two-sample empirical copula process. The limiting behaviours of the proposed statistics are established under the null hypothesis. Emphasis is placed on the explanation of the strong approximation methodology.
TL;DR: In this paper, it was shown that it is possible to make a given Finsler metric both forward and backward complete by a trivial projective change and that the existence of such a change is equivalent to the global hyperbolicity of the space time.
Abstract: A trivial projective change of a Finsler metric F is the Finsler metric F + d f. I explain when it is possible to make a given Finsler metric both forward and backward complete by a trivial projective change.Though the problem is purely Finslerian, it was inspired by Lorentz geometry and mathematical relativity: it was observed that it is possible to understand the light-like geodesics of a (normalized, standard) stationary 4-dimensional space time as geodesics of a certain Finsler Randers metric on a 3-dimensional manifold. The trivial projective change of the Finsler metric corresponds to the choice of another 3-dimensional slice, and the existence of a trivial projective change that is forward and backward complete is equivalent to the global hyperbolicity of the space time.
TL;DR: In this article, a hazard rate based representation of the Fisher information in generalized order statistics is derived under mild regularity conditions, and sufficient conditions for the validity of this representation in location and scale family settings are given.
Abstract: A representation of the Fisher information in generalized order statistics in terms of the hazard rate of the underlying distribution function is derived under mild regularity conditions. This expression supplements results for complete, Type-II censored, and progressively Type-II censored data. As a byproduct, we find a hazard rate based representation for samples of k-records which apparently has not been known so far. Moreover, sufficient conditions for the validity of this representation in location and scale family settings are given. The result is illustrated by considering generalized order statistics based on logistic, Laplace, and extreme value distributions.
TL;DR: In this paper, a consistent test for heteroscedasticity for nonlinear semi-parametric regression models with nonparametric variance function based on the kernel method is presented.
Abstract: It is important to detect the variance heterogeneity in regression models. Heteroscedasticity tests have been well studied in parametric and nonparametric regression models. This paper presents a consistent test for heteroscedasticity for nonlinear semi-parametric regression models with nonparametric variance function based on the kernel method. The properties of the test are investigated through Monte Carlo simulations. The test methods are illustrated with a real example.
TL;DR: In this article, the Gaussian quasi-maximum likelihood estimator (QMLE) for random coefficient autoregressions is examined and consistency and asymptotic normality are established for general random coefficients and general correlation structure between these coefficients and the noise.
Abstract: We examine the Gaussian quasi-maximum likelihood estimator (QMLE) for random coefficient autoregressions. Consistency and asymptotic normality are established for general random coefficients and general correlation structure between these coefficients and the noise. In particular, the obtained results apply even if the stationary solution has infinite absolute mean or infinite variance. Next an application to the integer-valued times series modelling is given which provides a novel alternative for traditional INAR-like models for these series.
TL;DR: The main goal of this paper is to establish the consistency and the rate of convergence of such a modified kernel estimator for strong mixing functional data with only E|Y|<∞, which weakens the moment assumption on the response variable Y.
Abstract: Assume that the explanatory variable X is valued in some abstract semi-metric functional space and the response variable Y is real-valued In this paper, we investigate a modified kernel estimation of the regression function r(x)=E(Y|X=x) The main goal of this paper is to establish the consistency and the rate of convergence of such a modified kernel estimator for strong mixing functional data with only E|Y|<∞, which weakens the moment assumption on the response variable Y
TL;DR: In this paper, the problem of obtaining Bayesian prediction bounds of future observables from a finite mixture of Burr type XII distribution with its reciprocal based on type-I censored data is addressed.
Abstract: This paper is concerned with the problem of obtaining Bayesian prediction bounds of future observables from a finite mixture of Burr type XII distribution with its reciprocal based on type-I censored data. We consider the one-sample and two-sample prediction schemes using the Markov chain Monte Carlo algorithm. Numerical examples are given to illustrate the procedures and the accuracy of prediction intervals is investigated via extensive Monte Carlo simulation.
TL;DR: Subsample bootstrap as discussed by the authors is a modified version of the 0.632-bootstrap, where each resample has exactly the same number m ∼eq ⌊0.6 32 n⌋ of distinct observations.
Abstract: In the classical Bootstrap approach the number of distinct observation in the resample is random. To overcome this hitch Rao et al. [Bootstrap by sequential resampling, J. Statist. Plan. Inference 64 (1997), pp. 257–281] have proposed a modified resampling procedure – the so-called Sequential Bootstrap or 0.632-Bootstrap – in which each resample has exactly the same number m ∼eq ⌊0.632 n⌋ of distinct observations. Motivated by this idea we introduce an akin procedure, the Subsample Bootstrap, where additionally even the size of each resample is equal. It will turn out that the Subsample Bootstrap empirical process is consistent for a wide class of Donsker classes.
TL;DR: In this article, a stochastic restricted k-d class estimator for the vector of parameters in a linear model when additional linear restrictions on the parameter vector are assumed to hold.
Abstract: In this paper, we introduce a stochastic restricted k–d class estimator for the vector of parameters in a linear model when additional linear restrictions on the parameter vector are assumed to hold. The stochastic restricted k–d class estimator is a generalization of the ordinary mixed estimator and the k–d class estimator. We show that our new biased estimator is superior in the mean squared error matrix sense to the k–d class estimator [S. Sakallioglu and S. Kaciranlar, A new biased estimator based on ridge estimation, Statist. Papers 49 (2008), pp. 669–689] and the stochastic restricted Liu estimator [H. Yang and J.W. Xu, An alternative stochastic restricted Liu estimator in linear regression, Statist. Papers 50 (2009), pp. 639–647]. Finally, a numerical example is given to show the theoretical results.
TL;DR: In this article, the authors present the asymptotic properties of maximum quasi-likelihood estimators (MQLEs) in generalized linear models with adaptive designs under some mild regular conditions.
Abstract: In this paper, we present the asymptotic properties of maximum quasi-likelihood estimators (MQLEs) in generalized linear models with adaptive designs under some mild regular conditions. The existence of MQLEs in quasi-likelihood equation is discussed. The rate of convergence and asymptotic normality of MQLEs are also established. The results are illustrated by Monte-Carlo simulations.
TL;DR: For the first time, explicit closed-form expressions for the characteristic functions for the Burr III and Burr XII distributions were derived in this paper, and the expressions involve the Fox -function and the Wright generalized 2Ψ0-function.
Abstract: For the first time, explicit closed-form expressions are derived for the characteristic functions for the Burr III and Burr XII distributions. The expressions involve the Fox -function and the Wright generalized 2Ψ0-function. An application is illustrated for insurance.
TL;DR: In this article, a nonparametric estimator of a probability density function defined on the positive real line is proposed, and the comparison with traditional kernel density estimator is discussed.
Abstract: An unknown moment-determinate cumulative distribution function or its density function can be recovered from corresponding moments and estimated from the empirical moments. This method of estimating an unknown density is natural in certain inverse estimation models like multiplicative censoring or biased sampling when the moments of unobserved distribution can be estimated via the transformed moments of the observed distribution. In this paper, we introduce a new nonparametric estimator of a probability density function defined on the positive real line, motivated by the above. Some fundamental properties of proposed estimator are studied. The comparison with traditional kernel density estimator is discussed.
TL;DR: A method for generating n-dimensional copulas starting with an (n−1)-dimensional copula already known is introduced and investigated.
Abstract: For every n≥3, a method is introduced and investigated for generating n-dimensional copulas starting with an (n−1)-dimensional copula already known. These copulas are particularly useful when the behaviour of a random vector (X 1, X 2, …, X n−1) formed by n−1 components is known, but another random variable, say X n , should be included into the model. An illustration of the usefulness of this construction is presented, showing some of its computational features.
TL;DR: In this article, it was shown that the proportions of observations falling in the left and right vicinity of the k n th order statistic converge in probability to some population quantities, and they then proved that this convergence converges to a population quantity.
Abstract: In this paper, we show that the proportions of observations falling in the left and right vicinity of the k n th order statistic converge in probability to some population quantities. We then prove...
TL;DR: In this article, a moment identity applicable to a general class of discrete probability distributions was derived for modified power series, Ord and Katz families, which has potential applications in different fields.
Abstract: In this paper, we obtain a moment identity applicable to a general class of discrete probability distributions. We then derive the corresponding identities for modified power series, Ord and Katz families. It is noted that the proposed identity has potential applications in different fields.
TL;DR: In this paper, the preliminary test approach for the estimation of the regression parameter in a multiple regression model under a multicollinearity situation is considered and the conditions of superiority of the proposed estimators are obtained.
Abstract: In this paper, we consider the preliminary test approach for the estimation of the regression parameter in a multiple regression model under a multicollinearity situation. The preliminary test two-parameter estimators based on the Wald (W), likelihood ratio, and Lagrangian multiplier tests are given, when it is suspected that the regression parameter may be restricted to a subspace and the regression error is distributed with multivariate Student's t distribution. The bias and mean square error of the proposed estimators are derived and compared. The conditions of superiority of the proposed estimators are obtained. Finally, we conclude that the optimum choice of the level of significance becomes the traditional choice by using the Wald test.