Data Dependent Cells Chi‐Square Test With Recurrent Events

We consider a recurrent event wherein the inter‐event times are independent and identically distributed with a common absolutely continuous distribution function F. In this article, interest is in the problem of testing the null hypothesis that F belongs to some parametric family where the q‐dimensional parameter is unknown. We propose a general Chi‐squared test in which cell boundaries are data dependent. An estimator of the parameter obtained by minimizing a quadratic form resulting from a properly scaled vector of differences between Observed and Expected frequencies is used to construct the test. This estimator is known as the minimum chi‐square estimator. Large sample properties of the proposed test statistic are established using empirical processes tools. A simulation study is conducted to assess the performance of the test under parameter misspecification, and our procedures are applied to a fleet of Boeing 720 jet planes' air conditioning system failures.     

Estimating the Size of Street-Dwelling Populations Using Mark-Resighting Counts

Mark-resighting constitutes an advanced technology for estimating animal abundance. Joint hypergeometric maximum likelihood, Minta-Mangel and Bowden estimators are usually adopted with mark-resighting data. In presence of any tendency of animals to aggregate into groups, the Bowden estimator is the sole reliable method, providing that marks are quite evenly distributed among groups. In some cetacean surveys, marking disturbances are avoided through natural marking. Natural marking with Bowden criterion is used to estimate the abundance of street-dwelling populations. The marked individuals are persons identified and recorded in the initial part of the survey and recognizable in subsequent occasions. A simulation helps determine the performance of the Bowden estimator under a wide set of situations, taking into account key features of street-dwelling populations. When marked individuals are evenly distributed among groups, the strategy is efficient.     

Estimators for the Horvitz-Thompson Statistic Based on Some Posterior Distributions

The knowledge of first-order inclusion probabilities characterizing a sampling scheme is essential in design-based estimation of finite population totals. Sometimes the scheme is so complex that these probabilities cannot be computed exactly. Instead, both inclusion probabilities and corresponding sampling weights are simulated. One empirical Horvitz-Thompson estimator for a population total using simulation-based range-preserving estimates of sampling weights is obtained by applying the restricted maximum likelihood principle directly to each inclusion probability. The assumption of a prior distribution and the assessment of resulting posterior for a weight lead to two other estimators. One of them is the posterior mean estimator of the Horvitz-Thompson statistic. In a simulation involving Polish agricultural census data and a sequential fixed-cost sampling scheme, this estimator has attractive properties also from a frequentist point of view.     

ESTIMATION OF THE PARAMETER OF A POISSON DISTRIBUTION USING A LINEX LOSS FUNCTION

This paper considers estimation of the parameter of a Poisson distribution using Varian's (1975) asymmetric LINEX loss function L (δ) = b{exp(aδ) - aδ - 1}, where δ is the estimation error and b > 0, a 0. It is shown that for a < 0, the sample mean X¯ is admissible whereas for a > 0, X¯ is dominated by c*X¯, where c*= (n/a)log(1+a/n). Practical implications of this result are indicated. More general results, concerning the admissibility of estimators of the form cX¯+ d are also presented.     

Conditional statistical inference and quantification of relevance

Summary. We argue that it can be fruitful to take a predictive view on notions such as the precision of a point estimator and the confidence of an interval estimator in frequentist inference. This predictive approach has implications for conditional inference, because it immediately allows a quantification of the concept of relevance for conditional inference. Conditioning on an ancillary statistic makes inference more relevant in this sense, provided that the ancillary is a precision index. Not all ancillary statistics satisfy this demand. We discuss the problem of choice between alternative ancillary statistics. The approach also has implications for the best choice of variance estimator, taking account of correlations with the squared error of estimation itself. The theory is illustrated by numerous examples, many of which are classical.     

Root n consistent and optimal density estimators for moving average processes

Abstract.  The marginal density of a first order moving average process can be written as a convolution of two innovation densities. Saavedra & Cao [Can. J. Statist. (2000), 28, 799] propose to estimate the marginal density by plugging in kernel density estimators for the innovation densities, based on estimated innovations. They obtain that for an appropriate choice of bandwidth the variance of their estimator decreases at the rate 1/ n . Their estimator can be interpreted as a specific U -statistic. We suggest a slightly simplified U -statistic as estimator of the marginal density, prove that it is asymptotically normal at the same rate, and describe the asymptotic variance explicitly. We show that the estimator is asymptotically efficient if no structural assumptions are made on the innovation density. For innovation densities known to have mean zero or to be symmetric, we describe improvements of our estimator which are again asymptotically efficient.     

Improved estimators for the selected location parameters

_i , i = 1, 2, ..., k be k independent exponential populations with different unknown location parameters θ _i , i = 1, 2, ..., k and common known scale parameter σ. Let Y _i denote the smallest observation based on a random sample of size n from the i-th population. Suppose a subset of the given k population is selected using the subset selection procedure according to which the population π _i is selected iff Y _i ≥Y ₍₁₎−d, where Y ₍₁₎ is the largest of the Y _i 's and d is some suitable constant. The estimation of the location parameters associated with the selected populations is considered for the squared error loss. It is observed that the natural estimator dominates the unbiased estimator. It is also shown that the natural estimator itself is inadmissible and a class of improved estimators that dominate the natural estimator is obtained. The improved estimators are consistent and their risks are shown to be O(kn ⁻²). As a special case, we obtain the coresponding results for the estimation of θ₍₁₎, the parameter associated with Y ₍₁₎. Received: January 6, 1998; revised version: July 11, 2000     

Simple random sample equivalent survey designs reducing undesirable units from a finite population

In this paper, we consider fixed size sampling plans for which the first order inclusion probabilities are identical for all units and the second order inclusion probabilities are constant for every pair-wise unit. The statistical conditions are identified under which these plans are equivalent to the usual simple random sampling plan. These sampling plans are constructed to reduce undesirable units.     

A Berry–Esséen Bound for M-estimators

We prove a Berry–Esséen bound for general M-estimators under optimal regularity conditions on the score function and the underlying distribution. As an application we obtain Berry–Esséen bounds for the sample median, the L_p -median, p > 1 and Huber's estimator of location