A Characterization of the Normal Family of Distributions Based on the Bias of the Maximum Likelihood Estimator

This paper characterizes the family of Normal distributions within the class of exponential families of distributions, via the structure of the bias of the maximum likelihood estimator Θ _n of the canonical parameter Θ . More specifically, when E _θ ( Θ n ) – Θ = (1/ n ) Q ( Θ ) + o (1/ n ), the equality Q ( Θ ) = 0 proves to be a property of the Normal distribution only. The same conclusion is obtained for the one-dimensional case bt assuming that Q ( Θ ) is a polynomial of Θ .     

The limitations of using school league tables to inform school choice

Summary. In England, so-called 'league tables' based on examination results and test scores are published annually, ostensibly to inform parental choice of secondary schools. A crucial limitation of these tables is that the most recent published information is based on the current performance of a cohort of pupils who entered secondary schools several years earlier, whereas for choosing a school it is the future performance of the current cohort that is of interest. We show that there is substantial uncertainty in predicting such future performance and that incorporating this uncertainty leads to a situation where only a handful of schools' future performances can be separated from both the overall mean and from one another with an acceptable degree of precision. This suggests that school league tables, including value-added tables, have very little to offer as guides to school choice.     

Central Limit Theorem approximations for the number of runs in Markov-dependent binary sequences

We consider Markov-dependent binary sequences and study various types of success runs (overlapping, non-overlapping, exact, etc.) by examining additive functionals based on state visits and transitions in an appropriate Markov chain. We establish a multivariate Central Limit Theorem for the number of these types of runs and obtain its covariance matrix by means of the recurrent potential matrix of the Markov chain. Explicit expressions for the covariance matrix are given in the Bernoulli and a simple Markov-dependent case by expressing the recurrent potential matrix in terms of the stationary distribution and the mean transition times in the chain. We also obtain a multivariate Central Limit Theorem for the joint number of non-overlapping runs of various sizes and give its covariance matrix in explicit form for Markov dependent trials.     

Characterizations of Student's t-distribution via regressions of order statistics

Utilizing regression properties of order statistics, we characterize a family of distributions introduced by Akhundov et al. [New characterizations by properties of midrange and related statistics, Commun. Stat. Theory Methods 33(12) (2004), pp. 3133–3143], which includes the t-distribution with two degrees of freedom as one of its members. Then we extend this characterization result to t-distribution with more than two degrees of freedom.     

Estimable functions in the nonlinear model

When the method of least squares is used to estimate the parameters in a general model and the generated system of normal equations is linearly dependent, the estimate of the vector of parameters which satisfies the criterion is not unique. However, there exist certain functions of the estimated vector of parameters which are invariant to the least squares solution obtained from the normal equations. We define those invariant functions to be estimable, and present a technique to determine the functions of the parameters which are estimable for the general model. The method results in solving either a linear first order partial differential equation or a system of linear first order partial differential equations corresponding, respectively, to a single or multiple dependency between columns of the Jacobian matrix of the mean of the model. The usual results concerning estimability for linear models are a special case of the general results developed.     

Multiple Time Series Modeling and Extended Sample Cross-Correlations

This article provides an expository account of the multivariate autoregressive moving average models and proposes an extended sample cross-correlation approach for practical model identification. An iterative model building procedure for applying these models to real data is discussed and demonstrated by analyzing the 5-series U.S. Hog Data.     

Using regression mixture models with non-normal data: examining an ordered polytomous approach

Mild to moderate skew in errors can substantially impact regression mixture model results; one approach for overcoming this includes transforming the outcome into an ordered categorical variable and using a polytomous regression mixture model. This is effective for retaining differential effects in the population; however, bias in parameter estimates and model fit warrant further examination of this approach at higher levels of skew. The current study used Monte Carlo simulations; 3000 observations were drawn from each of two subpopulations differing in the effect of X on Y. Five hundred simulations were performed in each of the 10 scenarios varying in levels of skew in one or both classes. Model comparison criteria supported the accurate two-class model, preserving the differential effects, while parameter estimates were notably biased. The appropriate number of effects can be captured with this approach but we suggest caution when interpreting the magnitude of the effects.     

Extensions of Samuelson's Inequality

In this note we obtain upper and lower bounds for the kth largest number in a set of real numbers in terms of their mean and standard deviation. For each inequality necessary and sufficient conditions for equality are given.     

The Planning Stage in Statistical Reasoning

An important step in the statistical problem-solving process is the selection of the appropriate statistical procedure for the real-world situation under analysis. A decision-tree term project has been found to be an effective teaching device to help MBA students understand this step. The project requires the students to construct a decision-tree structure, which, through a series of questions and responses, will lead from the statement of a statistical question to the appropriate sampling distribution to use in addressing the question.     

On Necessary and Sufficient Conditions for Ordinary Least Squares Estimators to Be Best Linear Unbiased Estimators

Two often-quoted necessary and sufficient conditions for ordinary least squares estimators to be best linear unbiased estimators are described. Another necessary and sufficient condition is described, providing an additional tool for checking to see whether the covariance matrix of a given linear model is such that the ordinary least squares estimator is also the best linear unbiased estimator. The new condition is used to show that one of the two published conditions is only a sufficient condition.