John G. Saw

Chebyshev Inequality With Estimated Mean and Variance

Abstract Chebyshevs inequality is investigated when the population mean and variance are estimated from a sample. The necessary modification to the inequality is simple and is actually valid when (a) the population moments do not exist and (b) the sample is exchangeably distributed. The latter case would include, for example, a sample taken without replacement from a finite population and the independent and identically distributed case.

Reliability Analysis of the k-out-of-n:F System

A class of repairable systems known as k-out-of-n:F systems, 1 ? k ? n, consists of n units in parallel redundancy which are serviced by a single repairman; system failure occurs when k units are simultaneously inoperable for the first time. In this paper, assuming constant failure rates and general repair distributions, reliability characteristics of the k-out-of-n:F system are treated using two different methods. In Part I, a conditional transform approach is applied to the 2-out-of-n:F system. Transforms of distributions are obtained for T (the time to system failure), the time spent on repairs during (0, T) and the free time of the repairman during (0, T). In Part II, the supplementary variable technique is used to investigate time to failure characteristics of the k-out-of-n:F system for k = 2 and k = 3. A model of an airport limousine service illustrates the use of the results.

A multivariate one-way classification model with random effects

In this paper a multivariate generalization of the one-way random effects model is investigated, maximum likelihood estimators are obtained, and the likelihood ratio test is derived for an hypothesis on the rank of the covariance matrix of the random effect vectors. Properties of the likelihood ratio test are investigated. A sequential procedure for determining the rank of the covariance matrix of the random effect vectors is presented.

The Distribution of Linear Combinations of t-Variables

Abstract The distribution of an arbitrary linear combination of Student-t random variables with odd degrees of freedom is derived. An easy method of expressing this distribution as a mixture of t-distributions is demonstrated, enabling the calculation of percentage points using only tables of the t-distribution.

Zonal polynomials: An alternative approach

Let (LX) = (01~) ova ,..., a,) be a sequence and L+(U) be the space of all homogeneous polynomials of degree f and symmetric in the elements of (a). A partition off into positive, integral parts will be written p = (pr , p, ,...) with the convention that pr > pa > ... . The set of all such partitions will be denoted by P(f ). The multiplicity of the positive integer j in p will be denoted by mj: 1 < j < f. We define a lexicon on S(f) by listing p E P(f) ahead of q E P(f) (and writing p > q) if, for some t, pj = qj , 1 < j < t 1 and Pt > 4t * Given the sets MP, q): P, q E S(f 1); {d,: P E S(f >> and HP, P): P E p(f >I, we will denote by [c(p, q)] the matrix; by [&,I the column vector and by MP,P)l h d g t e ia onal matrix formed from the sets in which the order of the elements corresponds to the lexicon on Y(f) in the obvious way. The augmented monomial symmetric function in n variables corresponding to the partition p E 9(f) is defined by David and Kendall [I] as

Jacobians of singular transformations with applications to statistical distribution theory

In the statistical distribution theory, frequent use is made of one-to-one transformations from an n×1 vector, x, of random variables to an n×1 vector, u, of random variables. In the context of this note, such a transformation would be described as nonsingular. It is sometimes advantageous to transform from a set of n variables to a set of n+m variables. With m positive, the transformation is described as singular The purpose of this note is to give the analog of the jacobian of the transformation in the singular case and to establish a method by means of which it can usually be found quite easily.

On comparing two poisson intensity functions

Given realizations of two possion processes with unknown intensities A(·) and F(·) observed over the interval (t1,t2), we suppose that it is desired to distinution between H0 Ξ(·)/λ(·) is constant on (t1,t2) versus H+:Ξ(·)/λ(·) increases on (t1,t2). We propose a decision rule which uses the percentage points of the Mann-Whitney U-distribution. We show that the decision rule is unbiased and that the set of alternatives in H+ can be weakly ordered, specifically: if Ξ(·)/λ(·), β(·)/λ(·) and Ξ(·)/β(·) are increasing on (t1, t2) then P{H0 is rejected |Ξ(·)}≧P{H0 is rejected|B(·)}≧P{H0 is rejected|H0}.

Ultraspherical polynomials and statistics on the m-sphere

The purpose of this note is to establish the connection between the ultraspherical polynomials and distributions on the m-sphere. Certain functions defined on the m-sphere have an ultraspherical decomposition which can be used to advantage. Examples of their use are given.

A lower bound for the distribution of a partial product of latent roots

A lower bound is given for the cumulative distribution of T=θ1 θ1…θs , the product of the s largest solutions to |E-θ(E+H)|=0 wherein E .and H are Wishart variables with equal dispersion; E is centrally distributed; H has noncentrality matrix Δ and rank(Δ)<r=m−s. The application is in tests of dimensionality, that is, tests concerning the rank Δ on those occasions where large sample theory may not be pertinent.

Spatial distribution of the tawny mole cricket, Scapteriscus vicinus

The tawny mole cricket, Scapteriscus vicinus Scudder, Gryllotalpidae, is the most important pest of turf and pasture grasses in the state of Florida. The subterranean lifestyle and great mobility of these insects make ecological and behavioral studies difficult. Direct observation of movement and intraspecific interactions in the field are impossible, and yet information of these types is vital to understanding the ecology of mole crickets. Certainly, any sampling program should take into account the spatial dispersion of the insects underground. Evidence from several sources indicates that mole cricket distribution in the field tends to be clumped rather than random. Damage to pastures appears first as isolated areas of grass loss, although damage may quickly spread. Flush samples from damaged areas produce more crickets than samples from adjacent undamaged areas (Hudson, 1985). Kleyla & Dodson (1978) found that calling males in a small (60 m x 20 m) field were clumped. Females and non-calling males were not studied. On a smaller scale, damage studies in 1.5 m diameter field cages planted with grass plugs showed that mole crickets tend to concentrate feeding damage on one or a few plugs rather than feeding equally on all available food (Hudson, 1985). If the apparent aggregation is a fact and extends to non-feeding periods, then knowledge of this clumping will be helpful in evaluating sampling data and planning sampling programs. Materials and methods