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.     

Reliability estimation for the exponential distribution

This paper is concerned with classical statistical estimation of the reliability function for the exponential density with unknown mean failure time θ, and with a known and fixed mission time τ. The minimum variance unbiased (MVU) estimator and the maximum likelihood (ML) estimator are reviewed and their mean square errors compared for different sample sizes. These comparisons serve also to extend previous work, and reinforce further the nonexistence of a uniformly best estimator. A class of shrunken estimators is then defined, and it produces a shrunken quasi-estimator and a shrunken estimator. The mean square errors for both these estimators are compared to the mean square errors of the MVU and ML estimators, and the new estimators are found to perform very well. Unfortunately, these estimators are difficult to compute for practical applications. A second class of estimators, which is easy to compute is also developed. Its mean square error properties are compared to the other estimators, and it outperforms all the contending estimators over the high and low reliability parameter space. Since, for all the estimators, analytical mean square error comparisons are not tractable, extensive numerical analyses are done in obtaining both the exact small sample and large sample results.     

Use of the 21% rulf in experimental design

When the probabilities underlying a contingency table are assumed to be symmetric, it is advantageous to combine the frequencies of equiprobable cells (Haber, 1981, 1982).The present work introduces models, as well as exact and asymptotic methods, for analyzing such ‘intraclass’ contingency tables in several dimensions.     

On a confidence interval about a parameter estimated by a ratio of normal random variables

A confidence interval is geometrically constructed about a parameter estimated by the ratio of bivariate normal random variables. The resulting confidence interval is equivalent to that of Fieller's theorem. The geometric construction shown that such intervals are conservative. Bioassay examples are used to demonstrate the technique.