Computation of the central and noncentral f distributions

Recursion relations suitable for rapid computation are derived for the cumulative distribution of F′ = (X/m)/(Y/n) where X is χ²(λ, m) and Y is independently χ²(n). When n is even no complicated function evaluations are needed. For n odd, a special doubly noncentral t distribution is needed to start the computation. Series representations for this t distribution are given with rigorous bounds on truncation errors. Proper recursion techniques for numerical evaluation of the special functions are given.     

A monte carlo investigation of a statistic for a bivariate missing data problem

Testing the equal means hypothesis of a bivariate normal distribution with homoscedastic varlates when the data are incomplete is considered. If the correlational parameter, ρ, is known, the well-known theory of the general linear model is easily employed to construct the likelihood ratio test for the two sided alternative. A statistic, T, for the case of ρ unknown is proposed by direct analogy to the likelihood ratio statistic when ρ is known. The null and nonnull distribution of T is investigated by Monte Carlo techniques. It is concluded that T may be compared to the conventional t distribution for testing the null hypothesis and that this procedure results in a substantial increase in power-efficiency over the procedure based on the paired t test which ignores the incomplete data. A Monte Carlo comparison to two statistics proposed by Lin and Stivers (1974) suggests that the test based on T is more conservative than either of their statistics.     

Tests of hypotheses based on ranks in the general linear model

A unified approach is developed for testing hypotheses in the general linear model based on the ranks of the residuals. It complements the nonparametric estimation procedures recently reported in the literature. The testing and estimation procedures together provide a robust alternative to least squares. The methods are similar in spirit to least squares so that results are simple to interpret. Hypotheses concerning a subset of specified parameters can be tested, while the remaining parameters are treated as nuisance parameters. Asymptotically, the test statistic is shown to have a chi-square distribution under the null hypothesis. This result is then extended to cover a sequence of contiguous alternatives from which the Pitman efficacy is derived. The general application of the test requires the consistent estimation of a functional of the underlying distribution and one such estimate is furnished.     

On a statistic useful in dimensionality reduction in multivariable linear stochastic system

In this paper we study the sampling properties of a test statistic which has important applications in the area of linear stochastic control systems with multi-inputs and multi-outputs. The statistic is the ratio of a partial sum of the eigenvalues of a sample covariance matrix and its trace. It turns out that using a method due to Sugiura we may derive a useful approximation for its distribution up to and including terms of order l/n, where n denotes the appropriate size. Numerical illustrations using real data are given.     

On a characterization of the dispersion matrix based on the properties of regression

This paper provides a partial solution to a problem posed by J. Neyman (1965) regarding the characterization of multivariate negative binomial distribution based on the properties of regression. It is shown that some of the properties of regression characterize the form of the nonsingular dispersion matrix of the parent distribution, which, interestingly enough, corresponds to only two types viz. those of positive and negative multivariate binomial distributions.     

The rank distribution of the maximin in a random matrix

Let S be a set of tm distinct real numbers and R a random t × m matrix of these tm numbers with rows {r_i} and columns (c_i}. Define b = Max Min x. l≤i≤t x?r_i. Let c be the event Max Min x = Min Max x. l≤i≤t x?r_i l≤i≤m x?c_i. This paper derives the probability distribution of the rank of b in S, as well as the same distribution conditional on c.     

Forecasting Accuracy of Alternative Techniques: A Comparison of U.S. Macroeconomic Forecasts

This article introduces and discusses a new measure of the relative economic affluence (REA) between income distributions with different means. The REA measure D is applied to the U.S. white and black household income distributions of 1967 and 1979. The measure D shows that the REA of the white households with respect to the black households decreased from 1967 to 1979. This conclusion contrasts with those obtained by applications of distance or quasi-distance functions. It is shown in this study that REA measures and distance functions address different and relevant issues. An REA measure deals with the relation “more affluent than” and defines a partial strict ordering over the set of pairs of income distributions—that is, the relation is asymmetric and transitive—whereas a distance function accounts for the dissimilarity between distributions without imposing an ordering relation and hence fulfills the symmetry property.     

A Case Study in Nonresponse: Plaintiff vs. California State Board of Equalization

In any sample survey, nonresponse bias is a potential issue. Even with a moderately high nonresponse rate, however, covariates can sometimes be used to show that the nonresponse bias is likely to be small. This note presents such an argument, which was used by the winning side in a tax case.     

The Detection of Influential Observations for Allocation,Separation, and the Determination of Probabilities in a Bayesian Framework

Normal theory separation and allocation problems are discussed from a predictive point of view. Influence statistics are defined and employed to ascertain the impact that particular observations will have on the inferential goals—allocation of future observations, separation between populations, and the determination of probabilities for future cases. Methods are illustrated on a collection of financial data taken from Johnson and Wichern (1982).     

Simulation of the p-generalized Gaussian distribution

In this paper, we introduce the p-generalized polar methods for the simulation of the p-generalized Gaussian distribution. On the basis of geometric measure representations, the well-known Box–Muller method and the Marsaglia–Bray rejecting polar method for the simulation of the Gaussian distribution are generalized to simulate the p-generalized Gaussian distribution, which fits much more flexibly to data than the Gaussian distribution and has already been applied in various fields of modern sciences. To prove the correctness of the p-generalized polar methods, we give stochastic representations, and to demonstrate their adequacy, we perform a comparison of six simulation techniques w.r.t. the goodness of fit and the complexity. The competing methods include adapted general methods and another special method. Furthermore, we prove stochastic representations for all the adapted methods.