Invariance of linearity of regression and related characterizations of some classical discrete distributions

Rao (1963) introduced what we call an additive damage model. In this model, original observation is subjected to damage according to a specified probability law by the survival distribution. In this paper, we consider a bivariate observation with second component subjected to damage. Using the invariance of linearity of regression of the first component on the second under the transition of the second component from the original to the damaged state, we obtain the characterizations of the Poisson, binomial and negative binomial distributions within the framework of the additive damage model.     

Sequential procedures in identification

In this paper we study the problem of identifying a popula-tion with one of the two populations, with an aim to control both types of errors. We assume that the populations are normal with unknown means, but with unit variance. We have cited examples from anthropological studies where our formulation of the problem fits in quite nicely. We observe that SPRT’s based on the maximal invariant may not terminate with probability one. Simulation studies reported here show a substantial saving in the average number of samples compared to the best invariant fixed sample test.     

Empirical bayes interval estimates involving uniform distributions

Bayesian and empirical Bayesian decision rules are exhibited for the interval estimation of the parameter 0 of a Uniform (0,θ) distribution. The estimate ?,δ>resulting in the interval [?,?+δ]suffers loss given by L(?,δ>,θ)=1-[?≦e≦?+δ]+c₁((?-θ)²+(?+δ?θ)²))+c₂δ. The solution is presented for prior distributions G which have bounded support, no point masses,∫θ^?mdG(θ)<∞ and for some integer m. An example is presented involving a particular parametric form for G and rates of risk convergence in the empirical Bayes problem for this example are calculated.     

Age-period-cohort analysis: an illustration of the problems in assessing interaction in one observation per cell data

This paper discusses the specific problems of age-period-cohort (A-P-C) analysis within the general framework of interaction assessment for two-way cross-classified data with one observation per cell. The A-P-C multiple classification model containing the effects of age groups (rows), periods of observation (columns), and birth cohorts (diagonals of the two-way table) is characterized as one of a special class of models involving interaction terms assumed to have very specific forms. The so-called A-P-C identification problem, which results from the use of a particular interaction structure for detecting cohort effects, is shown to manifest itself in the form of an exact linear dependency among the columns of the design matrix. The precise relationship holding among these columns is derived, as is an explicit formula for the bias in the parameter estimates resulting from an incorrect specification of an assumed restriction on the parameters required to solve the normal equations. Current methods for modeling A-P-C data are critically reviewed, an illustrative numerical example is presented, and one potentially promising analysis strategy is discussed. However, gien the large number of possible sources for error in A-P-C analyses, it is strongly recommended that the results of such analyses be interpreted with a great deal of caution.     

Asymptotic non-null distribution of a test connected with exponential populations

In this paper asymptotic expansions of the non-null distribution of the likelihood ratio criterion for testing the equality of several one parameter exponential distributions are obtained under local alternatives. These expansions are in terms of Chi-square distributions.     

Tests for the two-sample problem based on empirical probability measures

Tests are proposed for the equality of two unknown distributions. For empirical probability measures that are defined for samples from the two distributions, the proposed tests are based on the supremum of the absolute differences between the corresponding empirical probabilities, the supremum being taken over all possible events (Borel sets). In contrast, competing EDF tests compare only empirical probabilities of a subclass of Borel sets. The proposed tests are compared for simulated samples to the Kolmogorov-Smirnov, Cramér-von Mises, Kuiper, and Mann-Whitney-Wilcoxon tests     

Weighted combinations of rank tests for location-scale alternatives

All existing location-scale rank tests use equal weights for the components. We advocate the use of weighted combinations of statistics. This approach can partly be substantiated by the theory of locally most powerful tests. We specifically investi= gate a Wilcoxon-Mood combination. We give exact critical values for a range of weights. The asymptotic normality of the test statistic is proved under a general hypothesis and Chernoff-Savage conditions. The asymptotic relative efficiency of this test with respect to unweighted combinations shows that a careful choice of weights results in a gain in efficiency.     

On detection of outliers in symmetric normal models

Extensions of recent results for detection of mean slippage type outliers from i.i.d. multivariate normal and elliptically symmetric distributions are made to symmetric case, that is, when the observations are equicorrelated. The main tool used is Wijsman's (1967) representation theorem. The results obtained can be viewed as a robustness property of the use of Mardia's multivariate kurtosis as a locally optimal test statistic to detect outliers against equicorrelated distributions.