Eugene F. Schuster

Incorporating support constraints into nonparametric estimators of densities

Let fn ∗ (x) be the usual Parzen-Rosenblatt kernel estimator of the pdf f of a random variable X based on a sample X1,…,Xn from X.In many practical applications,it is knownt hat X>c and/or X<d for given constants c and d.Additionally, one might know the values of(c)and/or f(d).“mirrorimage”and“tieddown”modifications of fn ∗incorporate this additional information into an estimator fn which has support [c,d].This estimatoris interpreted in a manner which allows one to use most of the known convergence properties of kernel estimates in studying the behavior of fn.

On the Nonconsistency of Maximum Likelihood Nonparametric Density Estimators

One criterion proposed in the literature for selecting the smoothing parameter(s) in RosenblattParzen nonparametric constant kernel estimators of a probability density function is a leave-out-one-at-a-time nonparametric maximum likelihood method. Empirical work with this estimator in the univariate case showed that it worked quite well for short tailed distributions. However, it drastically oversmoothed for long tailed distributions. In this paper it is shown that this nonparametric maximum likelihood method will not select consistent estimates of the density for long tailed distributions such as the double exponential and Cauchy distributions. A remedy which was found for estimating long tailed distributions was to apply the nonparametric maximum likelihood procedure to a variable kernel class of estimators. This paper considers one data set, which is a pseudo-random sample of size 100 from a Cauchy distribution, to illustrate the problem with the leave-out-one-at-a-time nonparametric maximum likelihood method and to illustrate a remedy to this problem via a variable kernel class of estimators.

Classification of Probability Laws by Tail Behavior

Abstract In this article I refine Parzens density-quantile tail exponent classification of probability laws by tail behavior by subdividing his medium-tailed class into medium-short, medium-medium, and medium-long. I give a physical interpretation of short-, medium-, and long-tailed distributions in terms of the limiting size of the extreme spacings in a random sample from the distribution. I show that this classification, by the limiting size of extreme spacings, fits nicely within the framework of my refinement of Parzens density-quantile classification.

On the Goodness-of-Fit Problem for Continuous Symmetric Distributions

Abstract This article considers the problem of testing a completely specified continuous symmetric distribution against alternatives which are also symmetric about the same point. The symmetry is utilized in obtaining a new distribution-free statistic of the Kolomogorov-Smirnov type which can be used to halve the width of the Kolomogorov-Smirnov confidence band for the unknown distribution function.

On the negative hypergeometric distribution

A negative hypergeometric random variable, Yr, records the waiting time in trials until the rth success is obtained in repeated random sampling without replacement from a dichotomous population of N containing n ( ≥ r) successes S, and m failures F. In this paper we give a probability space for Yr and a representation of Yr in terms of exchangeable random variables from which one can easily deduce the probability distribution of Yr, E(Yr ) and σ2 (Yr ). We then indicate a compound (Beta) binomial derivation of the distribution and moments of Yr and give the maximum likelihood estimates of n and m and p = n/N and N in the general case when n and m are unknown.

On a universal strong law of large numbers for conditional expectations

A number of generalizations of the Kolmogorov strong law of large numbers are known including convex combinations of random variables (rvs) with random coefficients. In the case of pairs of i.i.d. rvs (X1, Y1), ..., (Xn, Yn), with y being the probability distribution of the xs, the averages of the Ys for which the accompanying Xs are in a vicinity of a given point x may converge with probability 1 (w.p. 1) and for j-almost everywhere (y a.e.) x to conditional expectation r(x)= E(Y|X = x). We consider the Nadaraya-Watson estimator of E(Y|X = x) where the vicinities of x are determined by window widths An. Its convergence towards r(x) w.p. I and for Y a.e. x under the condition E| Y| < oc is called a strong law of large numbers for conditional expectations (SLLNCE). If no other assumptions on y except that implied by EI Y < o0 are required then the SLLNCE is called universal. In the present paper we investigate the minimal assumptions for the SLLNCE and for the universal SLLNCE. We improve the best-known results in this direction.

On the conditional and unconditional distributions of the number of runs in a sample from a multisymbol alphabet

Let be a randomly ordered vector of length where Ni is the number of symbols of type i,i = 1, .k in . In the unconditional problem, S is the outcome of N independent trials of a multinomial experiment with k classes and has the multinomial distribution. In the conditional problem, each is a known fixed number and is a random arrangement of the N symbols. The main results in this paper are new recursion formulas for the pdf of the total number of runs, say R, in for both the unconditional and conditional problems. We also give formulas for the pdf of the number of runs of a given symbol type. Finally, we demonstrate the utility of our recursion algorithms for the pdf of R in the software system Mathematica (Mathematica is a registered trademark of Wolfram Research, Incorporated),discuss reasons that our algorithm in the conditional problem is much faster and easier to program than the 1957 algorithm of Barton and David, and correct three errors in their table.

Distribution theory of runs via exchangeable random variables

The conditional distribution theory of the number of runs R in a randomly ordered sequence of length N = m + n of two types of symbols, say m of type F (failures) and n of type S (successes), is studied via the representation R = 1 + [summation operator]m + 1k = 1[alpha]kIk where I1,..., Im + 1 are exchangeable Bernoulli random variables with [alpha]1 = [alpha]m + 1 = 1 and [alpha]k = 2, otherwise. This exchangeable representation of R, and related statistics, considerably facilitates the study of distribution theory of these statistics.

The conditional distribution of the longest run in a sample from a multiletter alphabet

Let be an ordered sequence of length of the members of a fixed set consisting of ni letters of type i,i ‐ 1,…,k. The main result in this paper is a recursion based algorithm to compute the distribution function and/or p-values for the (conditional) distribution function of the longest run of any type in under the null hypothesis of random ordering in when nl,…, nk are known. We demonstrate the utility of our algorithm using recur-sion and exact arithmetic in the software system Mathematica (Mathernatica is a registered trademark of Wolfram Research Incorporated).

Accuracy of proportion estimators: a simple rule

AbstractWe consider the sample survey type problem of estimating the proportionp of a finite population of sizeN having a given attribute by the proportion