Runs in an ordered sequence of random variables

Let be independent and identically distributed random variables with continuous distribution function. Denote by the corresponding order statistics. In the present paper, the concept of -neighbourhood runs, which is an extension of the usual run concept to the continuous case, is developed for the sequence of ordered random variables      

On characterizations of distributions by regression of adjacent generalized order statistics

Let , r ≥ 1, denote generalized order statistics, with arbitrary parameters , based on distribution function F. In this paper we characterize continuous distributions F by the regression of adjacent generalized order statistics, i.e. where are continuous and increasing functions and ψ is strictly increasing. Further we investigate in detail the case when ψ(x) = x and g is a linear function of the form g(x) = cx + d for some .     

$$\mathcal{D}$$-optimal Minimum Support Mixture Designs in Blocks

An easy method to construct efficient blocked mixture experiments in the presence of fixed and/or random blocks is presented. The method can be used when qualitative variables are involved in a mixture experiment as well. The resulting designs are -optimal in the class of minimum support designs. It is illustrated that the minimum support designs are more efficient than orthogonally blocked mixture experiments presented in the literature and only slightly less efficient than -optimal designs.     

A note on minimum variance

Summary Minimizing is discussed under the unbiasedness condition: and the condition (A):f _i (x) (i=1, ..., p) are linearly independent , and .     

A motivation for $$1/ \sqrt{n}$$-shrinking neighborhoods

In this paper we give a motivation for the shrinking rate let p ₀ and q _n be the outlier probability under the ideal model, and some member of a neighborhood about this ideal model of radius r _n , respectively. Assuming n i.i.d. observations, the critical rate of r _n may be defined such that the minimax test for outlier probability q _n =p ₀ versus q _n >p ₀ has asymptotic error probabilities bounded away from 0 and 1/2. Summarizing the neighborhoods to their upper probability, this leads to r _n of the exact rate . The result makes precise and simplifies ideas in Bickel (1981), Rieder (1994), and Huber (1997). Considering general probabilities of exact Hellinger distance r _n to P, this shrinking rate translates into , but leads to the same optimality theory as in the corresponding setup.     

Berry–Esseen bounds for density estimates under NA assumption

Let {X _j } be a strictly stationary sequence of negatively associated random variables with the marginal probability density function f(x). The recursive kernel estimators of f(x) are defined by

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 .     

Asymptotic normality of linear functions of concomitants of order statistics

Let T( ) be a linear function of concomitants of order statistics, whereT (·) denotes a statistical functional depending on some distribution function (df)F and is an estimator ofF. Under an auxiliary model approach we consider statistics of the form , where denotes a weighted empirical df and a finite population df (t denotes a triangular array). The results can be used to estimate income inequality in finite populations and especially when the survey is based on some design. The paper was written when the author was working at the Statistical Research Unit, Statistics Sweden, Stockholm, Sweden The research was supported by the Joint Committé of the Nordic Social Research Council.     

Moving averages of ergodic processes

A necessary and sufficient condition for the almost everywhere convergence of the moving ergodic averages is given. The result is then generalized to ergodic flows, and finally constrasted with earlier results ofPfaffelhuber andJain.     

$${\phi_p}$$-optimal designs for a linear log contrast model for experiments with mixtures

A mixture experiment is an experiment in which the k ingredients are nonnegative and subject to the simplex restriction on the (k − 1)-dimensional probability simplex S ^k-1. In this work, an essentially complete class of designs under the Kiefer ordering for a linear log contrast model with a mixture experiment is presented. Based on the completeness result, -optimal designs for all p,−∞ ≤ p ≤ 1 including D- and A-optimal are obtained, where the eigenvalues of the design moment matrix are used. By using the approach presented here, we gain insight on how these -optimal designs behave. Mong-Na Lo Huang was supported in part by the National Science Council of Taiwan, ROC under grant NSC 93-2118-M-110-001.     

Transformationen,die die Normalverteilung charakterisieren

Zusammenfassung Es sei A: R ⁿ R ⁿ eine Abbildung mit für jedes sei einn-dimensionaler Zufallsvektor. Wir beschreiben die Klasse aller TransformationenA, für die unabhängige, nachN(0, 1) verteilte Komponenten hat, sofern nur die KomponentenX ₁,...,X _n des Zufallsvektors ebenfalls unabhängig und identish Gaußisch verteilt sind mit Erwartungswert Null und Varianz 1. Weiter sind Bedingungen angegeben, die sicherstellen, daß nachN(O, ²) verteilte KomponentenX ₁,...,X _n hat, sofern dieX ₁,...,X _n unabhängig und und identisch verteilt sind. Zwei vonBeer undLukacs behandelte Transformationen sind Spezialfälle der hier untersuchten Transformationen.

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Summary Let A: R ⁿ R ⁿ be a transformation with the property for every . We consider a random vector and characterize the class of all transformationsA such that has independentN (0, 1) distributed componentsY ₁,...,Y _n if has the same distribution. Furthermore in the paper there are given conditions which ensure that hasN(O, ² distributed components if and are identically distributed and the componentsX ₁,...,X _n are independent, identically distributed random variables. Two of the transformations tried byBeer andLukacs are special cases of our transformations.

     

A characterization of entropies of degree α

The functional equation has been studied by several authors under various assumptions onf and onk, l. Here we solve this equation iff is measurable andk 3,l 2 are fixed integers. Using the solution we characterize the entropies of degree for all real . Our results generalize the results ofBehara/Nath [1973],Kannappan [1974] andMittal [1976].     

Rate of convergence in the central limit theorem and in the strong law of large numbers for von mises statistics

This paper provides the rate of convergence in the central limit theorem and in the strong law of large numbers forvon Mises statistics , based on i.i.d. random variablesX ₁ ,..., X _N .The proofs rely on a decomposition ofvon Mises statistics into a linear combination ofU-statistics and then use (generalized) results on the convergence rates forU-statistics obtained byGrams/Serfling [1973] andCallaert/Janssen [1978].     

The exact density of a statistic related to the shape parameter of a gamma variate

The exact density of the statistic ln , where and denote, respectively, the arithmetic and the geometric means of a random sample from a two-parameter gamma distribution, is obtained in a computable form using the technique of the inverse Mellin transform. This statistic is related to the maximum likelihood estimator of the shape parameter of a gamma distribution.     

Contributions to nonparametric generalized failure rate function estimation

For a specified distribution functionG with densityg, and unknown distribution functionF with densityf, the generalized failure rate function (x)=f(x)/gG ^–1 F(x) may be estimated by replacingf andF byf _n and , wheref _n is an empirical density function based on a sample of sizen from the distribution functionF, and . Under regularity conditions we show and, under additional restrictions whereC is a subset ofR and _n. Moreover, asymptotic normality is derived and the Berry-Esséen type bound is shown to be related to a theorem which concerns the sum of i.i.d. random variables. The order boundO(n^–1/2+c _n ^1/2 ) is established under mild conditions, wherec _n is a sequence of positive constants related tof _n and tending to 0 asn.Research was supported in part by the Army, Navy and Air Force under Office of Naval Research contract No. N00014-76-C-0608. AMS 1970 subject classifications. Primary 62G05. Secondary 60F15.     

Asymptotically optimal nonparametric one- and two-way analysis of variance tests for the log linear model under censoring

We considerr ×c populations with failure ratesλ _ij(t) satisfying the condition

Variation diminishing property of densities of uniform generalized order statistics

Let f _*,r , r ≥ 1, denote the density function of rth uniform generalized order statistics as defined by Kamps (1995) or Cramer and Kamps (2003). We prove the following variation diminishing property: the number of zeros in (0,1) of any linear combination does not exceed the number of sign changes in the sequence (a ₁, . . . ,a _r ). This result is applied to study monotonicity and convexity properties of f _*,r .