Fred C. Leone

A Family of J-Shaped Frequency Functions

THE usual procedure in obtaining theoretical distributions is to obtain a statement of the frequency function, f(x). The distribution function, F(x), may be expressed as a definite integral. One may adopt the opposite viewpoint and obtain a distribution function, F(x), from which the associated frequency function, f(x), may be obtained by differentiation. F(x) will be referred to as a cumulative frequency function (c.f.f.). The latter viewpoint was discussed by Burr in [1]. It is the purpose of this paper to give a family of c.f.f.s which is believed to be new, which is readily handled from the calculational standpoint, which has useful values of a3 and a4, the standard third and fourth moments, including the range of values for failure data mentioned below, and which yields J-shaped frequency functions. A graph is presented which shows the values of a32 and

Sampling Distributions of Variance Components I. Empirical Studies of Balanced Nested Designs

A development of the mathematical structure of nested designs is presented using, for a general case, a four-stage nested design. For the casea where the variables at each stage are normally distributed, analytical results can be obtained. This is done for the unbalanced “staggered” and “inverted” designns. Empirical estimattas of variance for the non-normal case are obtained. These are compared with the analytical solutions. Also considered is the probability of negative variance estimates. It ia interesting to note that for these alternatives to balanced nested designs, one can decrease the probabilities of negative estimates of some variances at the cost of increasing them for others.

Eficiency of Ordinary Least Squares Estimators from Trimmed and Winsorized Samples in Linear Regression

The problem of estimating the parameters of the simple linear regression model when the samples are symmetrically trimmed or Winsorized is considered. The efficiency of the ordinary least squares estimators (OLSE) relative to the best linear unbiased estimators (O-BLUE) from trimmed, Winsorized and complete samples is discussed. The exact relative efficiencies are given for some symmetric distributions and for values of 5 ≤ n ≤ 20. It is found that the (OLSE) estimators based on trimmed data are almost as efficient as the (O-BLUE) estimators for a variety of distributional specifications, while Winsorizing performs rather less well in several cases and marginally better in a few. It is found that for the standard normal and the scale contaminated normal distributions, the (OLSE) from trimmed samples maintain very high efficiencies relative to the (O-BLUE). Similarly, for the Winsorized (OLSE). But as the tails of the distribution become heavier—e.g. the double exponential distribution—the loss in efficien...

Relative Efficiencies of ‘O-BLUE’ Estimators in Simple Linear Regression

Abstract This article is concerned with the estimators of the parameters of the linear model . The estimation procedure consists of using the Best Linear Unbiased Estimation method applied to ordered samples. These are denoted as “O-BLUE” estimators. The exact efficiences of such linear estimators based on censored samples relative to those based upon complete samples are given for some symmetric distributions as well as some scale contaminated distributions. Further, some percentage relative efficiencies of the estimators are given when normal assumptions are made, but do not hold due to specific scale contamination.

The Robustness of Efficiency of Adjusted Trimmed Estimators in Linear Regression

Simple estimators of the parameters σ, β and σ of the simple linear regression model, based upon trimmed samples, and called adjusted Trimmed Estimators (ATE) are introduced. They are very easy to compute, and their coefficients may be chosen so that their efficiency relative to the (O-BLUE) from complete samples is equal to one. The coefficients are given for five symmetric distributions, namely, the normal distribution, the scale contaminated normal distributions with a scale factor of σ = 3.0 and contamination proportions ∊ = 0.01, .05 and .1O and the double exponential distribution, for values of 5 ≤ n ≤ 20 and different amounts of trimming (g). The estimators are shown to be biased, but the exact amounts of bias are given. Adjusted Efficiency Robust Estimators (AERE) are also introduced. They are shown to maintain high efficiency relative to the (O-BLUE) under changes in the distribution of the errors.

TEACHING SERVICE COURSES AND SHORT COURSES IN STATISTICS - A RESPONSE

Publisher Summary This chapter discusses the views of L. H. Koopmans, Watts, and Folks regarding teaching service courses. These courses are indeed a major portion of almost every department of statistics and should be given the prominence they deserve. L. H. Koopmans has described a particular introductory course, its goals, its compromises and constraints, use of calculators as opposed to computers, and the course content. D. G. Watts, on the other hand, has enumerated the tasks from which the objectives—within the context of a course in regression analysis—evolve. J. L. Folks has been concerned with answering the question of whether the service course is in fact a servant or master. The chapter presents remarks primarily to Folks on servant or master and then highlights a few additional concepts.

The Effect of Trimming and Normality Assumptions on the Efficiency of the O-BLUE Estimators

Abstract The seriousness of trimming samples, and considering the trimmed samples as if they are complete samples of the retained size, from a normal distribution is explored for the parameters of the simple linear regression model. The exact efficiencies of the O-BLUE estimators of the parameters under these conditions are investigated relative to the O-BLUE estimators based upon the trimmed samples from the actual distribution G. Five symmetric distributions are considered. It is found that the overall loss in relative efficiency is quite substantial especially when the true distribution has heavier tails than the normal distribution, and in particular for larger amounts of trimming.