Efficient estimators for the good family

We consider the problem of estimating the two parameters of the discrete Good distribution. We first show that the sufficient statistics for the parameters are the arithmetic and the geometric means. The maximum likelihood estimators (MLE's) of the parameters are obtained by solving numerically a system of equations involving the Lerch zeta function and the sufficient statistics. We find an expression for the asymptotic variance-covariance matrix of the MLE's, which can be evaluated numerically. We show that the probability mass function satisfies a simple recurrence equation linear in the two parameters, and propose the quadratic distance estimator (QDE) which can be computed with an ineratively reweighted least-squares algorithm. the QDE is easy to calculate and admits a simple expression for its asymptotic variance-covariance matrix. We compute this matrix for the MLE's and the QDE for various values of the parameters and see that the QDE has very high asymptotic efficiency. Finally, we present a numerical example.     

Smooth goodness of fit tests for the weibull distribution with singly censored data

The smooth goodness of fit tests are generalized to singly censored data and applied to the problem of testing Weibull (or extreme value) fit. Smooth tests, Pearson-type tests, and the spacings tests proposed by Mann, Schemer, and Fertig (1973) are compared on the basis of local asymptotic relative efficiency with respect to the asymptotic best test against generalized gamma alternatives, The smooth test of order one Is found to be most efficient for the generalized gamma alternatives.     

Tests for symmetry about an unknown value based on the empirical distribution function

A class of Kolmogorov-Smirnov and Cramér-von Mises type statistics for testing symmetry about an unknown value is described. These statistics are not distribution-free, however, and, indeed, are not readily amenable to calculation. A linear rank statistic analog of the first component of the Cramér-von Mises type statistic is investigated. Asymptotic non-null properties of these procedures in the normal case are studied, and an efficiency comparison of the Cramér-vonMises statistic, the linear rank statistic analog, the modified Wil-coxon statistic, and the likelihood ratio test is reported.     

Nonparametric tests for scale shift at an unknown time point

A class of linear rank tests is suggested for testing a shift in scale at an unknown time point in a sequence of independent observations. The tests,based on inverse normal scores and on ordered exponential scores,are shown to be asymptotically as efficient as their distribution-oriented competitors. Critical values and powers for these two rank tests are also discussed.     

On the asymptotic efficiency of moment and maximum likelihood estimators in the three-parameter inverse gaussian distribution

The asymptotic distribution of estimators generated by the methods of moments and maximum likelihood are considered. Simple formulae are provided which enable comparisons of asymptotic relative efficiency to be effected.     

Combining Univariate Tests for Multivariate Location Problem

In this article, three methods of combining dependent univariate tests are studied. The Bahadur approximate efficiencies are derived under the asymptotic normal assumption. These procedures are applied to the multivariate location problem and compared with two Hotelling-type tests. A Monte Carlo study indicates that in certain cases the powers of the combination methods are much better than Hotelling's T ² and other multivariate nonparametric tests.     

Experimental designs and their efficiencies for spatially correlated observations in two dimensions

Optimality of experimental designs for spatially correlated observations is investigated.come two dimensional correlation structures are discussed and an attempt has been made to find optimal or nearly optimal design for each sitution.The solution lend to designs similar to that used for repeated measurements.The relative efficiency of the proposed designs in comparison to randomized latin square designs is tabulated for some cases.     

Pseudo maximum likelihood estimation for the dirichlet-multinomial distribution

Pseudo maximum likelihood estimation (PML) for the Dirich-let-multinomial distribution is proposed and examined in this pa-per. The procedure is compared to that based on moments (MM) for its asymptotic relative efficiency (ARE) relative to the maximum likelihood estimate (ML). It is found that PML, requiring much less computational effort than ML and possessing considerably higher ARE than MM, constitutes a good compromise between ML and MM. PML is also found to have very high ARE when an estimate for the scale parameter in the Dirichlet-multinomial distribution is all that is needed.     

On tail-ordering and comparison of failure rates

In this paper, we have studied some implications between tail-ordering (also known as dispersive ordering) and failure rate ordering (also called TP₂ ordering) of two probability distribution functions. Based on independent random samples from these distributions, a class of distribution-free tests has been proposed for testing the null hypothesis that the two life distributions are identical against the alternative that one failure rate is uniformly smaller than the other. The tests have good efficiencies as compared to their competitors.     

An asymptotic distribution-free selection procedure for a two-way layout problem

This paper deals with an asymptotic distribution-free subset selection procedure for a two-way layout problem. The treatment effect with the largest unknown value is of interest to us. The block effect is a nuisance parameter in this problem. The proposed procedure is based on the Hodges-Lehmann estimators of location parameters. The asymptotic relative efficiency of the proposed procedure with the normal means procedure is evaluated. It is shown that the proposed procedure has a high efficiency.