Effect of outliers on the estimation of parameters

Summary The estimation of parameter in the type of distributionf(x)=b x ^{–1 /b} exp (–x ^b (/b),x>0, is considered, when several outliers of the type , ,r=1,2, ...,k, are present in the data. The estimates of as well as of 's are put in the closed form. Special cases, Weibull, Gamma and Exponential are considered for the case of single outlier. Actual estimates are calculated from the generated samples of size 2 and 3 for the Weibull and Exponential.     

On the power of nonparametric changepoint-tests

We consider a sequenceX _1n,..., X_nn, n N, of independent random elements. Suppose there exists a [0, 1) such thatX _1n,...,X _(n),n have the distribution v₁ andX _[n]+1.n ,...,X _nn have the distribution v₂v₁. We construct consistent level- tests forH ₀:=0 versusH ₁:(0, 1), which are based on certainU-statistic type processes. A detailed investigation of the power function is also provided.     

A curtailed test for the shape parameter of the Weibull distribution

Summary A procedure is proposed in this paper for testing the shape parameter, of the Weibull distribution. The test statistic which is based on the extremal quotient, possesses a monotone property which makes it possible for rejection earlier than the last planned observation of the null hypothesis,H ₀: =₀ when the alternative hypothesis isH _a: <₀ and early acceptance ofH ₀ whenH _a: >₀. The test being scale-free, does not require the scale parameter to be known.     

On monotone optimal decision rules and the stay-on-a-winner rule for the two-armed bandit

Summary Consider the following optimization problem: Find a decision rule such thatw(x, (x))=max _a w(x, a) for allx under the constraint (x)D (x). We give conditions for the existence of monotone optimal decision rules . The term monotone is used in a general sense. The well-known stay-on-a-winner rules for the two-armed bandit can be characterized as monotone decision rules by including the stage number intox and using a special ordering onx. This enables us to give simple conditions for the existence of optimal rules that are stay-on-a-winner rules. We extend results ofBerry andKalin/Theodorescu to the case of dependent arms.     

Realization of closed,convex, and symmetric subsets of the unit square as regions of risk for testing simple hypotheses

It is well-known that the region of risk for testing simple hypotheses is some closed, convex, and (1/2, 1/2)-symmetric subset of the unit square, which contains the points (0, 0) and (1, 1). It is shown that for any such subsetR of the unit square and any atomless probability measureP on some -algebra there exists some probability measureQ on the same -algebra such thatR is the corresponding region of risk for testingP againstQ. This generalizes a result of [4] and is as a first step derived here for the special case, whereP is equal to the uniform distribution on the unit interval. The corresponding distributionQ is given explicitly in this case and the general case is treated by some well-known measure-isomorphism. This method of proof shows thatQ might be chosen to be of typeQ=Q ₁+(1–)Q ₂ for some satisfying 01, whereQ ₁ is a probability measure, which is absolutely continuous with respect toP andQ ₂ is a one-point mass.     

Minimal efficiency of designs under the class of orthogonally invariant information criteria

Consider the linear regression model with uncorrelated errors and an experimental design . In the article, we address the problem of calculating the minimal efficiency of with respect to the class of orthogonally invariant information criteria, containing all Kiefers criteria of _p-optimality, among others. We show that the -minimal efficiency of is equal to the minimal efficiency of with respect to a finite class of criteria which generalize the criterion of E-optimality. We also formulate conditions under which a design is maximin efficient, i.e. the most efficiency-stable for criteria from . To illustrate the results, we calculated the -minimal efficiency of _p (in particular D, A and E) optimal designs for polynomial regression on [–1,1] up to degree 4. Moreover, for the quadratic model we explicitly constructed the -maximin efficient design.Acknowledgement. The author would like to thank prof. Pázman as well as an anonymous referee for useful and inspiring comments on earlier versions of this article.Supported by: VEGA grant of the Slovak Agency No. 1/0264/03     

Estimating a distribution function in the presence of auxiliary information

For estimating the distribution functionF of a population, the empirical or sample distribution functionF _n has been studied extensively. Qin and Lawless (1994) have proposed an alternative estimator for estimatingF in the presence of auxiliary information under a semiparametric model. They have also proved the point-wise asymptotic normality of . In this paper, we establish the weak convergence of to a Gaussian process and show that the asymptotic variance function of is uniformly smaller than that ofF _n . As an application of , we propose to employ the mean and varianceŜ _n ² of to estimate the population mean and variance in the presence of auxiliary information. A simulation study is presented to assess the finite sample performance of the proposed estimators , andŜ _n ² .     

On estimating the variance of a finite population

Summary Observing that the estimator for a finite population variance as recommended byLiu [1974a, b] can sometimes become negative, we suggest a few non-negative alternative estimators and note some of their properties. UnlikeLiu we follow the conventional Bayesian approach to get another estimator with an optimal property of uniform admissibility.This paper, however, was prepared when the author worked in the Department of Economic Statistics, University of Sydney.     

One-Step and Two-Step Estimation of the Effects of Exogenous Variables on Technical Efficiency Levels

Consider a stochastic frontier model with one-sided inefficiency u, and suppose that the scale of u depends on some variables (firm characteristics) z. A one-step model specifies both the stochastic frontier and the way in which u depends on z, and can be estimated in a single step, for example by maximum likelihood. This is in contrast to a two-step procedure, where the first step is to estimate a standard stochastic frontier model, and the second step is to estimate the relationship between (estimated) u and z.In this paper we propose a class of one-step models based on the scaling property that u equals a function of z times a one-sided error u ^* whose distribution does not depend on z. We explain theoretically why two-step procedures are biased, and we present Monte Carlo evidence showing that the bias can be very severe. This evidence argues strongly for one-step models whenever one is interested in the effects of firm characteristics on efficiency levels.     

Minimal sufficient statistics for the group divisible partially balanced incomplete block design (GD-PBIBD) with interaction under an Eisenhart Model II

Summary In this paper, an Eisenmhart Model II with interaction for a GD-PBIB design withp replicates per cell is considered. Specifically the Model Y_ijl=µ+_i+_j+()_ij+e_ijl is assumed, wherei=1, 2, ...,b; j=1, 2, ...,t andl=0, 1, 2, ...p s _ij wheres _ij=1, if treatmentj appears in blocki, 0, otherwise.If _i, _j, ()_ij ande _ijl are normally and independently distributed, then a minimal sufficient (Vector-valued) statistic for the class of densities for this model is found, together with the distribution of each component in the minimal sufficient statistic. It is also shown that the minimal sufficient statistic for this class densities is not complete. Hence the solution of the problem of finding minimum variance unbiased estimators of the variance components is not straightforward.     

Reproducibility in the one-parameter exponential family

Summary LetX ₁, ...,X _n be i.i.d. random variables with common distribution an element of a linear one-parameter exponential family indexed by a natural parameter . It is proved that the distribution of is an element ofF, for all andn=1, 2, ... if and only ifF is a family of scale transformed Poisson distributions.     

A numerical representation of semiorders on a countable set

Let be a semiorder on a countable setX and letx⁰ y if and only if either there existsx withxxy or there existsx withxxy. Then ⁰ is a preference relation with transitive indifference, which can be represented by a utility functionf of the usual sort. It is well known that is represented by a pair of real-valued functionsu, v, in the sense thatxy if and only ifu(x)>v(y). We prove that there exists a pair of functionsu, v, representing , such thatu+v is the utility function which represents in the usual sense. Moreover it is easily seen that, for such a pair of functionsu, v, we havex⁰ y if and only if eitheru(x)>u(y) or (u(x)=u(y) andv(x)>v(y)).

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Sommario Consideriamo unsemiordine su un insiemeX numerabile e poniamox⁰ y se e solo se esistex tale chexxy, oppure esistex tale chexxy. In questo caso ⁰ è unordine debole, che può essere rappresentato da una funzione di utilitàf nel senso usuale. D'altra parte è rappresentato da una coppia di funzioniu, v, nel senso chexy se e solo seu(x)>v(y). In questo lavoro si prova che ammette una rappresentazioneu, v tale chex⁰ y se e solo seu(x)+v(x)>u(y)+v(y). Si dimostra altresì che, con riguardo ad una siffatta rappresentazioneu, v di , riescex⁰ y se e solo seu(x)>u(y) oppure (u(x)=u(y) ed anchev(x)>v(y)).

     

A characterization of the exponential distribution by higher order gap

Summary SupposeX is a non-negative random variable with an absolutely continuous (with respect to Lebesgue measure) distribution functionF (x) and the corresponding probability density functionf(x). LetX ₁,X ₂,...,X _n be a random sample of sizen fromF andX _i,n is thei-th smallest order statistics. We define thej-th order gapg _i,j(n) asg _i,j(n)=X _i+j,n–X_i,n 1i<n, 1nn–i. In this paper a characterization of the exponential distribution is given by considering a distribution property ofg _i,j(n).     

Sequential approach to simultaneous estimation of the mean and variance

The problem of simultaneous estimation of the mean and variance of a normal distribution has been studied. We propose a semi-circular region _n ={(a,b)':b>0} of radiusd, which has approximately a preassigned coverage probability. Asymptotic efficiency and asymptotic consistency (asd0) of our proposed sequential procedures have been proved.Research partially supported by U.S. Army Research Grant No. DAAG29-76-G-0038.     

Characterizations of distributions via linearity of regression of generalized order statistics

Let X (r, n, m, k), 1 r n, denote generalized order statistics based on an absolutely continuous distribution function F. We characterize all distribution functions F for which the following linearity of regression holds E(X(r+l,n,m,k) | X(r,n,m,k))=aX(r,n,m,k)+b.We show that only exponential, Pareto and power distributions satisfy this equation. Using this result one can obtain characterizations of exponential, Pareto and power distributions in terms of sequential order statistics, Pfeifers records and progressive type II censored order statistics. Received July 2001/Revised August 2002     

Estimation of a regression coefficient after two preliminary tests of significance

Summary The estimation of the regression coefficient of a population, defined byE (y)= +x, incorporating two preliminary tests of significance has been discussed. The experimenter has two random samples of different sizes from two such populations, as defined above, with regression coefficients ₁ and ₂ respectively, where ₂ may possibly be equal to ₁. Besides this, it is also conjectured that the common conditional variance ² of the two populations has a specified value ₀ ² . The two preliminary tests are used to resolve these two uncertainties.The author has rejoined Lucknow University, Lucknow India on Oct. 4, 1976 after availing leave for two years.     

Clarifying the Boundaries of Sexual Harassment and Employer Liability: Judicial Application of Harris v. Forklift Systems, Inc.

Many had anticipated that Harris v. Forklift Systems, Inc., the first significant Supreme Court ruling on sexual harassment since 1986, would clarify the boundary between merely offensive conduct and unlawful conduct, and would offer greater guidance on when an employer is liable for the creation of an abusive (or hostile) work environment. However, the Harris court handed down only a short, narrow decision that left these areas largely unresolved, and, accordingly, it was widely criticized for circumventing contentious issues and for perpetuating the vague framework under which abusive work environment claims are now analyzed. This article examines a representative cross-section of federal and state decisions that have interpreted and applied Harris to evaluate whether courts have achieved any consensus on what constitutes an abusive work environment and on when an employer is liable for this type of sexual harassment.     

Asymptotic considerations for selecting the best component of a multivariate normal population

Summary Several two-stage and sequential procedures have been developed for selecting the best component in a multivariate normal population through the classical indifference zone approach ofBechhofer. The works inMukhopadhyay/Chou will constitute to be very basic in this regard. A number of first-order and second-order asymptotic analysis of these procedures has also been reported. In the case of a bivariate population, the normality assumption has been dropped, and the rate of convergence of the probability of correct selection is studied.     

An application of the renewal theoretic selection principle: The first divisible sum

Summary This paper deals with the distribution of the first partial sum of a sequence of independent and identically distributed random variables which is divisible by some integerk>1. We give a formula for its characteristic function and obtain limit results fork. A new characterization of the geometric distribution is also given. The results are applicable to models such as periodically observed self-renewing aggregates.     

Tests for location-scale families based on the empirical characteristic function

A class of procedures is presented for using random samples to test the fit of location-scale families—distributions F(·;θ₁,θ₂) such that Z=(X−θ₁)/θ₂ has distribution Working with empirically standardized data, the test statistic is a measure of distance between the empirical characteristic function, and the c.f. of Z under the null hypothesis, ϕ₀(t). The closed-form test statistic is derived by integrating over the product of a weight function times Using as weight function for each location-scale family the squared modulus of ϕ₀ itself presents a unified test procedure. Included as special cases are well-known tests for normal and Cauchy laws. Small-sample powers are compared with those of Anderson-Darling tests for each of seven univariate location-scale families.