A note on complete families of distributions

Some methods to establish completeness of families of distributions are discussed. Several well known parametric families are shown to be complete.     

A new continuous distribution and two new families of distributions based on the exponential

Recent work on social status led to derivation of a new continuous distribution based on the exponential. The new variate, termed the ring(2)-exponential, in turn leads to derivation of two closely related new families of continuous distributions, the mirror-exponential and the ring-exponential. Both the standard exponential and the ring(2)-exponential are special cases of both the new families. In this paper, we first focus on the ring(2)-exponential, describing its derivation and examining its properties, and next introduce the two new families, describing their derivation and initiating exploration of their properties. The mirror-exponential arises naturally in the study of status; the ring-exponential arises from the mathematical structure of the ring(2)-exponential. Both have the potential for broad application in diverse contexts across science and engineering. Within sociobehavioral contexts, the new mirror-exponential may have application to the problem of approximating the form and inequality of the wage distribution.     

A note on the covariance matrix of the maximum likelihood estimator in constrained multivariate linear regression

A useful result concerning variances and covariances of a linear function of a random matrix is applied to find the variance–covariance matrix of the maximum likelihood estimator in multivariate linear regression subject to zero constraints.     

A note on three-stage confidence intervals for the difference of locations: The exponential case

Fixed-width confidence intervals for the difference of location parameters of two independent negative exponential distributions are constructed via triple sampling when the scale parameters are unknown and unequal. The present three-stage estimation methodology is put forth because (i) it is operationally more convenient than the existing purely sequential counterpart, and (ii) the three-stage and the purely sequential estimation techniques have fairly similar asymptotic second-order characteristics.     

A note on the maximum deviation of the scale-contaminated normal to the best normal distribution

In this paper we consider the case of the scale-contaminated normal (mixture of two normals with equal mean components but different component variances: (1−p)N(μ,σ²)+pN(μ,τ²) with σ and τ being non-negative and 0≤p≤1). Here is the scale error and p denotes the amount with which this error occurs. It's maximum deviation to the best normal distribution is studied and shown to be montone increasing with increasing scale error. A closed-form expression is derived for the proportion which maximizes the maximum deviation of the mixture of normals to the best normal distribution. Implications to power studies of tests for normality are pointed out. Received May 2001     

A note on the value of crime control: Evidence from the property market

This note provides evidence on the amount people are willing to pay for crime control. The theory of hedonic price indexes is used to derive estimates from property values. The results are consistent with both theory and intuition.     

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.     

The rate of convergence in law of the maximum of an exponential sample

Summary  We derive a uniform rate of convergence of (1– n^-1x)ⁿ to e^-x(x < 0). It provides a uniform rate of convergence for the distribution of the largest order statistic in a sample from an exponential distribution to the "double exponential" extreme value distribution. It likewise provides a rate of convergence for the distribution of the smallest order statistic from a uniform distribution.     

On maximum likelihood prediction based on Type II doubly censored exponential data

In this paper, the maximum likelihood predictor (MLP) of the kth ordered observation, t _k, in a sample of size n from a two-parameter exponential distribution as well as the predictive maximum likelihood estimators (PMLE's) of the location and scale parameters, θ and β, based on the observed values t _r, …, t _s (1≤r≤s<k≤n), are obtained in closed forms, contrary to the belief they cannot be so expressed. When θ is known, however, the PMLE of β and MLP of t _k do not admit explicit expressions. It is shown here that they exist and are unique; sharp lower and upper bounds are also provided. The derived predictors and estimators are reasonable and also have good asymptotic properties. As applications, the total duration time in a life test and the failure time of a k-out-of-n system may be predicted. Finally, an illustrative example is included. Received: August 1999     

A probability inequality for ranges and its application to maximum range test procedures

Summary It is proved that for any fixed argument the sequence (P _k) of the distribution functions of the ranges ofk i.i.d. univariate random variables is log-concave if the random variables have a log-concave density. If the support of the distribution is an infinite interval and the density is monotonous then the theorem holds also with “log-convex” instead of “log-concave”. The resulting inequalities can be used by a quick algorithm for closed maximum range test procedures for all pairwise comparisons (Royen 1988, 1989a, 1989b). Under the above assumptions the application of this algorithm can be extended e.g. to pairwise comparisons of variances.     

Nonlinear regression and curved exponential families. Improvement of the approximation to the asymptotic distribution

Inference in nonlinear models is usually based on the asymptotic normal distribution, based on linearizing the model. The accuracy of this approximation can in many cases be improved by a reparametrization. Systematic methods for doing this will be described. Sometimes a saddlepoint approximation can be used, and this offers several advantages compared to the asymptotic distribution and the Edgeworth expansion. The improved methods are unfortunately not commonly used. It will be discussed why this is so. The methods will be illustrated by a series of examples.     

Nonparametric maximum likelihood estimation of a probability density via mathematical programming

Based on sample values of a one-dimensional random variable a nonparametric maximum likelihood estimate for the unknown probability density is introduced as the solution of an optimization problem in an appropriate Hilbert space. This solution turns out to be a polynomial spline function, and a complete characterization is given using recent results on the differentiability of the optimal value of a parametrized family of optimization problems. An important feature of this estimate is that its support interval results in a quite natural way from the formulation of the problem and is not fixed in advance. The estimator is shown to have a certain consistency property for a special class of density functions. Numerical results will be given in a subsequent paper.     

A note on the microeconomics of migration

"The purpose of this note is to demonstrate in a simple model that an individual's migration from a small town to a large city may be rationalized purely by a consumption motive, rather than the motive of obtaining a higher income. More specifically, it is shown that in a large city an individual may derive a higher utility from spending a given amount of income than in a small town." A formal model is first developed that includes the principal forces at work and is then illustrated using a graphic example. The theoretical and empirical issues raised are considered in the concluding section.