Anderson's classification statistic based on a post-stratified training sample

The performance of Anderson's classification statistic based on a post-stratified random sample is examined. It is assumed that the training sample is a random sample from a stratified population consisting of two strata with unknown stratum weights. The sample is first segregated into the two strata by post-stratification. The unknown parameters for each of the two populations are then estimated and used in the construction of the plug-in discriminant. Under this procedure, it is shown that additional estimation of the stratum weight will not seriously affect the performance of Anderson's classification statistic. Furthermore, our discriminant enjoys a much higher efficiency than the procedure based on an unclassified sample from a mixture of normals investigated by Ganesalingam and McLachlan (1978).     

Tests of hypotheses based on ranks in the general linear model

A unified approach is developed for testing hypotheses in the general linear model based on the ranks of the residuals. It complements the nonparametric estimation procedures recently reported in the literature. The testing and estimation procedures together provide a robust alternative to least squares. The methods are similar in spirit to least squares so that results are simple to interpret. Hypotheses concerning a subset of specified parameters can be tested, while the remaining parameters are treated as nuisance parameters. Asymptotically, the test statistic is shown to have a chi-square distribution under the null hypothesis. This result is then extended to cover a sequence of contiguous alternatives from which the Pitman efficacy is derived. The general application of the test requires the consistent estimation of a functional of the underlying distribution and one such estimate is furnished.     

On estimating the dimensionality in discriminant analysis

In discriminant analysis, the dimension of the hyperplane which population mean vectors span is called the dimensionality. The procedures commonly used to estimate this dimension involve testing a sequence of dimensionality hypotheses as well as model fitting approaches based on (consistent) Akaike's method, (modified) Mallows' method and Schwarz's method. The marginal log-likelihood (MLL) method is developed and the asymptotic distribution of the dimensionality estimated by this method for normal populations is derived. Furthermore a modified marginal log-likelihood (MMLL) method is also considered. The MLL method is not consistent for large samples and two modified criteria are proposed which attain asymptotic consistency. Some comments are made with regard to the robustness of this method to departures from normality. The operating characteristics of the various methods proposed are examined and compared.     

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.     

Smoothed estimates for nonlinear time series models with irregular data

Time series data observed at unequal time intervals (irregular data) occur quite often and this usually poses problems in its analysis. A recursive form of the exponentially smoothed estimated is here proposed for a nonlinear model with irregularly observed data and its asymptotic properties are discussed An alternative smoother to that of Wright (1985) is also derived. Numerical comparison is made between the resulting estimates and other smoothed estimates.     

Normalité asymptotique d'estimateurs convergents du mode conditionnel

We state sufficient conditions for asymptotic normality of convergent estimates of the conditional mode, irrespective of data dependence, and give an application to α-mixing stationary processes.     

关于两类二阶非线性系统的全局渐近稳定性

本文讨论了两类二阶非线性系统的零解的全局渐近稳定性，得到了一组充分条件．     

Linear Discriminant Analysis of Multivariate Spatial–Temporal Regressions

Abstract.  We consider classification of the realization of a multivariate spatial–temporal Gaussian random field into one of two populations with different regression mean models and factorized covariance matrices. Unknown means and common feature vector covariance matrix are estimated from training samples with observations correlated in space and time, assuming spatial–temporal correlations to be known. We present the first-order asymptotic expansion of the expected error rate associated with a linear plug-in discriminant function. Our results are applied to ecological data collected from the Lithuanian Economic Zone in the Baltic Sea.     

Joint Initial Stocking and Transshipment—Asymptotics and Bounds

A common problem faced by many firms in their supply chains can be abstracted as follows. Periodically, or at the beginning of some selling season, the firm needs to distribute finished goods to a set of stocking locations, which, in turn, supply customer demands. Over the selling season, if and when there is a supply‐demand mismatch somewhere, a re‐distribution or transshipment will be needed. Hence, there are two decisions involved: the one‐time stocking decision at the beginning of the season and the supply/transshipment decision throughout the season. Applying a stochastic dynamic programming formulation to a two‐location model with compound Poisson demand processes, we identify the optimal supply/transshipment policy and show that the optimal initial stocking quantities can be obtained via maximizing a concave function whereas the contribution of transshipment is of order square‐root‐of T. Hence, in the context of high‐volume, fast‐moving products, the initial stocking quantity decision is a much more important contributor to the overall profit. The bounds also lead to a heuristic policy, which exhibits excellent performance in our numerical study; and we further prove both the bounds and the heuristic policy are asymptotically optimal when T approaches infinity. Extension to multiple locations is also discussed.     

Bivariate log Birnbaum–Saunders distribution

Univariate Birnbaum–Saunders distribution has received a considerable amount of attention in recent years. Rieck and Nedelman (A log-linear model for the Birnbaum–Saunders distribution. Technometrics, 1991;33:51–60) introduced a log Birnbaum–Saunders distribution. The main aim of this paper is to introduce bivariate log Birnbaum–Saunders distribution. The proposed model is symmetric and it has five parameters. It can be obtained using Gaussian copula. Different properties can be obtained using copula structure. It is observed that the maximum likelihood estimators (MLEs) cannot be obtained explicitly. Two-dimensional profile likelihood approach may be adopted to compute the MLEs. We propose some alternative estimators also, which can be obtained quite conveniently. The analysis of one data set is performed for illustrative purposes. Finally, it is observed that this model can be used as a bivariate log-linear model, and its multivariate generalization is also quite straight forward.