Increasing failure rate and decreasing reversed hazard rate properties of the minimum and maximum of multivariate distributions with log-concave densities

For a multivariate random vector X = (X ₁,...,X _n ) with a log-concave density function, it is shown that the minimum min{X ₁,...,X _n } has an increasing failure rate, and the maximum max{X ₁,...,X _n } has a decreasing reversed hazard rate. As an immediate consequence, the result of Gupta and Gupta (in Metrika 53:39–49, 2001) on the multivariate normal distribution is obtained. One error in Gupta and Gupta method is also pointed out.      

GMM estimation of a maximum entropy distribution with interval data

We develop a generalized method of moments (GMM) estimator for the distribution of a variable where summary statistics are available only for intervals of the random variable. Without individual data, one cannot calculate the weighting matrix for the GMM estimator. Instead, we propose a simulated weighting matrix based on a first-step consistent estimate. When the functional form of the underlying distribution is unknown, we estimate it using a simple yet flexible maximum entropy density. Our Monte Carlo simulations show that the proposed maximum entropy density is able to approximate various distributions extremely well. The two-step GMM estimator with a simulated weighting matrix improves the efficiency of the one-step GMM considerably. We use this method to estimate the U.S. income distribution and compare these results with those based on the underlying raw income data.     

Monotonicity of the (reversed) hazard rate of the (maximum) minimum in bivariate distributions

It is well known that in the case of independent random variables, the (reversed) hazard rate of the (maximum) minimum of two random variables is the sum of the individual (reversed) hazard rates and hence the onotonicity of the (reversed) hazard rate of the marginals is preserved by the monotonicity of the (reversed) hazard rate of the (maximum) minimum. However, for the bivariate distributions this property is not always preserved. In this paper, we study the monotonicity of the (reversed) hazard rate of the (maximum) minimum for two well known families of bivariate distributions viz the Farlie-Gumbel-Morgenstern (FGM) and Sarmanov family. In case of the FGM family, we obtain the (reversed) hazard rate of the (maximum) minimum and provide several examples in some of which the (reversed) hazard rate is monotonic and in others it is non-monotonic. In the case of Sarmanov family the (reversed) hazard rate of the (maximum) minimum may not be expressed in a compact form in general. We consider some examples to illustrate the procedureResearch of the second author is supported by a grant from Natural Sciences and Engineering Research Council and the research of the other two authors is partially supported by a travel grant from the Canadian American Center of the University of Maine     

Characterization of the multivariate normal distribution by conditional normal distributions

Summary LetX andY be two random vectors with values in ℝ ^k and ℝ∝, respectively. IfZ=(X ^T,Y ^T) ^T is multivariate normal thenX givenY=y andY givenX=x are (multivariate) normal; the converse is wrong. In this paper simple additional conditions are stated such that the converse is true, too. Furthermore, the case is treated that the random vectorZ=(X ₁ ^T , …,X _t ^T ) ^T is splitted intot≥3 partsX ₁, …,X _t.     

On the analysis of multivariate growth curves

Growth curve data arise when repeated measurements are observed on a number of individuals with an ordered dimension for occasions. Such data appear frequently in almost all fields in which statistical models are used, for instance in medicine, agriculture and engineering. In medicine, for example, more than one variable is often measured on each occasion. However, analyses are usually based on exploration of repeated measurements of only one variable. The consequence is that the information contained in the between-variables correlation structure will be discarded.  In this study we propose a multivariate model based on the random coefficient regression model for the analysis of growth curve data. Closed-form expressions for the model parameters are derived under the maximum likelihood (ML) and the restricted maximum likelihood (REML) framework. It is shown that in certain situations estimated variances of growth curve parameters are greater for REML. Also a method is proposed for testing general linear hypotheses. One numerical example is provided to illustrate the methods discussed. Received: 22 February 1999     

A generalized maximum likelihood characterization of the normal distribution

LetP be a probability measure on ℝ andI _x be the set of alln-dimensional rectangles containingx. If for allx ∈ ℝⁿ and θ ∈ ℝ the inequality holds,P is a normal distributioin with mean 0 or the unit mass at 0. The result generalizes Teicher’s (1961) maximum likelihood characterization of the normal density to a characterization ofN(0, σ²) amongall distributions (including those without density). The m.l. principle used is that of Scholz (1980).     

A note on the characterization of the multivariate normal distribution

Summary The multivariate normal distribution is characterized in the class of infinitely divisible distributions.     

A property of Stein's estimator for the mean of a multivariate normal distribution

A uniform bound on the risk (under squared error loss) of Stein's estimator Ψ₁ for the mean of the multivariate normal distribution is given. Using the bound, the asymptotic behaviour of the risk of Ψ₁ under a Bayesian assumption is obtained.     

Multivariate t and beta distributions associated with the multivariate F distribution

Relationships between F, skew t and beta distributions in the univariate case are in this paper extended in a natural way to the multivariate case. The result is two new distributions: a multivariate t/skew t distribution (on ℜ^m) and a multivariate beta distribution (on (0,1)^m). A special case of the former distribution is a new multivariate symmetric t distribution. The new distributions have a natural relationship to the standard multivariate F distribution (on (ℜ⁺)^m) and many of their properties run in parallel. We look at: joint distributions, mathematically and graphically; marginal and conditional distributions; moments; correlations; local dependence; and some limiting cases. Received: March 2001     

Record values of the Lomax distribution

In this paper, firstly an introduction to the idea of record values for a sequence of independent and identically (Lomax or Pareto II) distributed random variables is given. Some of the distributional properties of these record values and moments up to second order are derived. These moments clearly depend on the location, scale and shape parameter of the Lomax distribution and two types of estlmators of these parameters, based on a series of observed record values are presented.     

Sequential decisions under uncertainty and the maximum theorem

For sequential decision problems in which the decision-maker observes a process of state variables and chooses an adapted process of action variables, the paper defines a topology on the space of measures of processes of state variables which ensures the applicability of Berge's maximum theorem to the decision-maker's optimal behavior. The topology controls for the information available to the decision-maker at each decision date. The paper also discusses the implications of the analysis for the dynamic-programming approach to sequential decision-making under uncertainty, and for equilibrium existence proof strategies in sequential-market models and games.     

Gamma-minimax estimation of a multivariate normal mean

Summary The mean vector of a multivariate normal distribution is to be estimated. A class Γ of priors is considered which consists of all priors whose vector of first moments and matrix of second moments satisfy some given restrictions. The Γ-minimax estimator under arbitrary squared error loss is characterized. The characterization follows from an application of a result of Browder and Karamardian published in Ichiishi (1983) which is a special version of a minimax inequality due to Ky Fan (1972). In particular, it is shown that within the set of all estimators a linear estimator is Γ-minimax. The authors would like to thank the Deutsche Forschungsgemeinschaft for financial support.     

Use of minimum risk approach in the estimation of regression models with missing observations

This article considers a linear regression model with some missing observations on the response variable and presents two estimators of regression coefficients employing the approach of minimum risk estimation. Small disturbance asymptotic properties of these estimators along with the traditional unbiased estimator are analyzed and conditions, that are easy to check in practice, for the superiority of one estimator over the other are derived. Received May 2001     

Right censoring in a discrete life model

Although there are many sophisticated models for estimation of failure rate based on censored data in continuous distributions, not much work has been done in the discrete case. We introduce a discrete model for life lengths and consider its properties. For this model, we derive the corresponding maximum likelihood estimators of the parameters under Type I and Type II right-censoring. Received May 2000     

Periodical Replacement Model for a Multi-Unit System Subject to Failure Rate Interaction

A periodical replacement model for a multi-unit system subject to failure rate interaction between units is presented. In a N-unit system, one is dominant unit and the others are secondary units. When a secondary unit fails, it increases the failure rate of the dominant unit by some amount, while the failure of dominant unit make the system into total failure. Without failure rate interaction, the failure rates of all units also increase with age. All single failures of secondary units are assumed to be corrected by minimal repairs. The system is replaced at age T, or on total failure, whichever occurs first. The aim of this paper is to derive the expected cost rate per unit time as a criterion of optimality and to seek the optimal period T ^* to minimize that cost rate.     

A note on multinomial maximum likelihood estimation under ordered restrictions and the EM algorithm

A frequently occurring problem is to find the maximum likelihood estimation (MLE) of p subject to p∈C (C⊂ P the probability vectors in R ^k ). The problem has been discussed by many authors and they mainly focused when p is restricted by linear constraints or log-linear constraints. In this paper, we construct the relationship between the the maximum likelihood estimation of p restricted by p∈C and EM algorithm and demonstrate that the maximum likelihood estimator can be computed through the EM algorithm (Dempster et al. in J R Stat Soc Ser B 39:1–38, 1997). Several examples are analyzed by the proposed method.     

Asymptotic considerations for selecting the best component of a multivariate normal population

Summary Several two-stage and sequential procedures have been developed for selecting the best component in a multivariate normal population through the classical indifference zone approach ofBechhofer. The works inMukhopadhyay/Chou will constitute to be very basic in this regard. A number of first-order and second-order asymptotic analysis of these procedures has also been reported. In the case of a bivariate population, the normality assumption has been dropped, and the rate of convergence of the probability of correct selection is studied.     

中国城市最低生活保障标准的比较与评价

认为名义最低生活保障标准难以真实反映各地区最低生活保障救助的实际效果,引入新的指标,对中国城市近年来最低生活保障标准和实质救助水平及其变化趋势进行比较,同时为政府有关部门对最低生活保障的政策选择和评估提供一个新工具。研究结果显示,名义最低生活保障标准较高的城市,如深圳、上海等,其实质救助水平并不高。     

城中村改造中的"市场失灵"和"政府失灵"及防止途径

城中村是中国快速城市化进程中出现的特有现象。城中村改造问题的本质是利益问题,实质上是与城中村改造相关的利益各方利益调整的过程。由于市场失灵的存在,城中村改造基本都在政府主导下进行,但由于政府的失灵,往往也会产生很多弊端。但大多数学者仍认为政府应该在改造的过程中发挥主导作用,问题是政府应该在城中村改造中发挥怎样的作用,或者说市场的边界又在哪里。本文初步明确了城中村改造中政府干预的重点和市场的边界,阐明了城中村改造的总体思路,最后提出了防止城中村改造失败的对策。     

Periodic inspection plans: The case of Weibull distribution

In estimating the lifetime distribution of a product, the efficiency of a periodic inspection plan vis-á-vis with the continuous inspection plan is discussed. The periodic inspection plan envisages inspecting the units in the sample periodically at m successive equi-distant time points, for a given choice of m. The issue of choosing the value of m is also discussed. The discussions are carried out when the underlying lifetime distribution is Weibull.