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 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 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.     

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.     

A note on redistribution

A consequence of income redistribution may well be to make everyone (including lower-income groups to which redistribution takes place) worse off after a period of years. Possibly income redistribution might gain consent because of risk aversion. The paper also draws attention to the tendency for people to be more concerned about poverty close at hand rather than far away.     

A remark on serial correlation in maximum likelihood

Maximum likelihood estimation can be consistent and asymptotically normal despite serial correlation in the residuals. The usual estimator of the asymptotic covariance of the parameter estimator is inconsistent, but an alternative consistent estimator is derived.     

A note on odds in the cattle futures market

This paper investigates the odds of generating a 100-fold return in the cattle futures market. We employ cattle futures data for the period October 11, 1978, through July 31, 1979, to compute the probability of obtaining such a return. The tests are constructed to give the investor the benefit of the doubt whenever doubt exists. The most conservative finding is that the probability is one in approximately thirty-one trillion. Assuming that the return is made in the most efficient way possible, this probability falls to approximately 1.5×10⁻¹⁶.     

A note on the optimality ofk-stage tests

Summary This note deals with sequential tests requiring at mostk observations. First a lemma is proved concerning the form of Bayes solutions. Then it is demonstrated by giving a counter-example that the theorem ofWald andWolfowitz has no correspondence fork-stage tests.

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Zusammenfassung Es wird gezeigt, daß man beik-stufigen Tests zwar für die Gestalt vonBayes-Lösungen ein Analogon zu den unbeschränkten Sequenztests hat, daß aber der Satz vonWald undWolfowitz über die gleichmäßige Optimalität bzgl. des Stichprobenumfangs keine Entsprechung besitzt.