Inferences on the lognormal mean for complete samples

In many engineering problems it is necessary to draw statistical inferences on the mean of a lognormal distribution based on a complete sample of observations. Statistical demonstration of mean time to repair (MTTR) is one example. Although optimum confidence intervals and hypothesis tests for the lognormal mean have been developed, they are difficult to use, requiring extensive tables and/or a computer. In this paper, simplified conservative methods for calculating confidence intervals or hypothesis tests for the lognormal mean are presented. In this paper, “conservative” refers to confidence intervals (hypothesis tests) whose infimum coverage probability (supremum probability of rejecting the null hypothesis taken over parameter values under the null hypothesis) equals the nominal level. The term “conservative” has obvious implications to confidence intervals (they are “wider” in some sense than their optimum or exact counterparts). Applying the term “conservative” to hypothesis tests should not be confusing if it is remembered that this implies that their equivalent confidence intervals are conservative. No implication of optimality is intended for these conservative procedures. It is emphasized that these are direct statistical inference methods for the lognormal mean, as opposed to the already well-known methods for the parameters of the underlying normal distribution. The method currently employed in MIL-STD-471A for statistical demonstration of MTTR is analyzed and compared to the new method in terms of asymptotic relative efficiency. The new methods are also compared to the optimum methods derived by Land (1971, 1973).     

Optimal testing in a three way anova model

We consider likelihood ratio, score and Wald tests for a three-way random effects ANOVA model. Competitor tests are compared using criteria such as small sample power, asymptotic relative efficiency, and convenient null distribution. The final choice is between a new test and two tests long used in practice.     

Small sample properties of parameter estimation in the dirichlet distribution

A numerically feasible algorithm is proposed for maximum likelihood estimation of the parameters of the Dirichlet distribution. The performance of the proposed method is compared with the method of moments using bias ratio and squared errors by Monte Carlo simulation. For these criteria, it is found that even in small samples maximum likelihood estimation has advantages over the method of moments.     

Hellinger distance as a penalized log likelihood

The present paper studies the minimum Hellinger distance estimator by recasting it as the maximum likelihood estimator in a data driven modification of the model density. In the process, the Hellinger distance itself is expressed as a penalized log likelihood function. The penalty is the sum of the model probabilities over the non-observed values of the sample space. A comparison of the modified model density with the original data provides insights into the robustness of the minimum Hellinger distance estimator. Adjustments of the amount of penalty leads to a class of minimum penalized Hellinger distance estimators, some members of which perform substantially better than the minimum Hellinger distance estimator at the model for small samples, without compromising the robustness properties of the latter.     

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.     

The estimation of treatment effect for the censored bivariate data under the univariate censoring

This paper establishes a nonparametric estimator for the treatment effect on censored bivariate data under unvariate censoring. This proposed estimator is based on the one from Lin and Ying(1993)'s nonparametric bivariate survival function estimator, which is itself a generalized version of Park and Park(1995)' quantile estimator. A Bahadur type representation of quantile functions were obtained from the marginal survival distribution estimator of Lin and Ying' model. The asymptotic property of this estimator is shown below and the simulation studies are also given     

Response transformations in repeated measures and growth curve models

This article describes estimation and inference procedures for the parameters of the Box-Cox and foided-power transformations in repeated measures and growth curve models. Procedures for computing maximum likelihood estimates of the transformation and covariance parameters under several covanance structures (omnibus sphericity, local sphericity, and unstructured) are described. Lack of fit statistics and hypothesis tests for comparing these structures also are described. The procedures are illustrated on three data sets. Software for performing the analyses in the SAS System is described and is available from the authors.     

Influence measure for the L1 regression

Because outliers and leverage observations unduly affect the least squares regression, the identification of influential observations is considered an important and integrai part of the analysis. However, very few techniques have been developed for the residual analysis and diagnostics for the minimum sum of absolute errors, L₁ regression. Although the L₁ regression is more resistant to the outliers than the least squares regression, it appears that outliers (leverage) in the predictor variables may affect it. In this paper, our objective is to develop an influence measure for the L₁ regression based on the likelihood displacement function. We illustrate the proposed influence measure with examples.     

Generalized additive mixed models

Following the extension from linear mixed models to additive mixed models, extension from generalized linear mixed models to generalized additive mixed models is made, Algorithms are developed to compute the MLE's of the nonlinear effects and the covariance structures based on the penalized marginal likelihood. Convergence of the algorithms and selection of the smooth param¬eters are discussed.     

Quasi-maximum exponential likelihood estimation for a non stationary GARCH(1,1) model

ABSTRACT

This article investigates a quasi-maximum exponential likelihood estimator(QMELE) for a non stationary generalized autoregressive conditional heteroscedastic (GARCH(1,1)) model. Asymptotic normality of this estimator is derived under a non stationary condition. A simulation study and a real example are given to evaluate the performance of QMELE for this model.