The centered normal conditionals distribution

In this paper, we discuss some aspects of the distribution theory associated with the centered bivariate normal conditionals distribution including discussion of its marginal distributions. We calculate the maximum likelihood and pseudolikelihood estimators. We propose a simplified moment based method of estimation. Finally, we discuss generalizations to higher dimensions.     

On a pragmatic modification of survey sampling in three stages

Presented are formulae for an unbiased estimator of a finite population total and an unbiased variance estimator for it when samples are taken by usual procedures in the first two stages with varying probabilities but the third stage units are sampled for economy and convenience in a non-standard way from the pool of all sampled second stage units rather than independently from each of the latter separately containing the former.     

Reference prior bayes estimator for bivariate normal covariance matrix with risk comparison

The explicit form of the reference prior bayes estimator due to Yang and Ber-ger (1994) for bivariate normal covariance matrix under entropy loss is given in terms of Legendre polynomials when degrees of freedom is even and in terms of hypergeometric functions in general case. The finite series expression of the density function of the ratio of latent roots of bivariate Wishart matrix is obtained and the exact risk is compared with those of James-Stein minimax estimator and other orthogonally equivariant estimators. It is found numerically that the reference prior bayes estimator has the smallest risk among the class of equivariant estimators compared, when the ratio of the largest to the smallest population latent roots of covariance matrix lies in the middle of the interval [1, ∞]. It has larger risk than that of James-Stein minimax estimator when the ratio is large. Moreover it has larger risk than that of MLE when, for instance, degrees of freedom is 20 and the ratio lies between 4 and 8.     

Estimation for the semipareto processes

This paper proposes different estimators for the parameters of SemiPareto and Pareto autoregressive minification processes The asymptotic properties of the estimators are established by showing that the SemiPareto process is α-mixing. Asymptotic variances of different moment and maximum likelihood estimators are compared.     

A Note on Robustness of Normal Variance Estimators Under t-Distributions

ABSTRACT

In many real life problems one assumes a normal model because the sample histogram looks unimodal, symmetric, and/or the standard tests like the Shapiro-Wilk test favor such a model. However, in reality, the assumption of normality may be misplaced since the normality tests often fail to detect departure from normality (especially for small sample sizes) when the data actually comes from slightly heavier tail symmetric unimodal distributions. For this reason it is important to see how the existing normal variance estimators perform when the actual distribution is a t-distribution with k degrees of freedom (d.f.) (t _k -distribution). This note deals with the performance of standard normal variance estimators under the t _k -distributions. It is shown that the relative ordering of the estimators is preserved for both the quadratic loss as well as the entropy loss irrespective of the d.f. and the sample size (provided the risks exist).     

Nearest Neighbor Adjusted Best Linear Unbiased Prediction

Statistical inference for linear models has classically focused on either estimation or hypothesis testing of linear combinations of fixed effects or of variance components for random effects. A third form of inference—prediction of linear combinations of fixed and random effects—has important advantages over conventional estimators in many applications. None of these approaches will result in accurate inference if the data contain strong, unaccounted for local gradients, such as spatial trends in field-plot data. Nearest neighbor methods to adjust for such trends have been widely discussed in recent literature. So far, however, these methods have been developed exclusively for classical estimation and hypothesis testing. In this article a method of obtaining nearest neighbor adjusted (NNA) predictors, along the lines of “best linear unbiased prediction,” or BLUP, is developed. A simulation study comparing “NNABLUP” to conventional NNA methods and to non-NNA alternatives suggests considerable potential for improved efficiency.     

A Comparative Study of Some Modified Chi-Squared Tests

Some recent results in the theory and applications of modified chi-squared goodness-of-fit tests are briefly discussed. It seems that for the first time power of modified chi-squared type tests for the logistic and three-parameter Weibull distributions based on moment type estimators is studied. Power of different modified tests against some alternatives for equiprobable fixed or random grouping intervals, and for Neyman–Pearson classes is investigated. It is shown that power of test statistic essentially depends on the quantity of Fisher's sample information this statistic uses. Some recommendations on implementing modified chi-squared type tests are given.     

A Statistical Reanalysis of the Classical Rutherford's Experiment

Modified chi-squared and some newly developed tests for the Poisson, binomial, and an approximated Feller's distribution are discussed. A reanalysis of the classical Rutherford's experimental data on alpha decay is done. Previous analyses of the data were not correct from the point of view of the theory of statistical testing. Tests used show that the data contradict to both Poisson and binomial distribution and do not contradict to a precise “binomial” approximation of Feller's distribution that takes into account a counter's dead time. This gives a plausible statistically correct confirmation of the well-established exponential law of radioactive decay.     

Classical Estimator of Parameters of the Gamma Case With Presence of Two Outliers

The authors derive the moment, maximum likelihood, and mixture estimators of parameters of the gamma distribution with presence of two outliers generated from uniform distribution. These estimators are compared empirically when all the parameters are unknown; their bias and mean squared error are investigated with the help of numerical technique. The authors shown that these estimators are asymptotically unbiased. At the end, they conclude that mixture estimators are better than the maximum likelihood and moment estimators.