Tukey-Type Distributions in the Context of Financial Data

Abstract

In one-parameter (θ) families, we were not aware of explicit hypothesis testing scenarios where maximal invariant statistics failed to distinguish the models. We start with a concrete example (Sec. 2.2) to highlight such a hypothesis testing problem involving markedly different models. In this problem, because of the absence of a nontrivial uniformly most powerful invariant (UMPI) test, we briefly suggest two approaches to test the hypothesis. The first resolution (Sec. 3.1) is frequentist in nature. It utilizes a weight function on the parameter space and compares “average” distributions obtained under the null and alternative models in the sense of Wald (1947 Wald , A. ( 1947 ). Sequential Analysis . New York : Wiley . [Google Scholar] 1950 Wald , A. ( 1950 ). Statistical Decision Functions . New York : Wiley . [Google Scholar]). In contrast, a fully Bayesian resolution (Sec. 3.2) is also included. The note ends with a series of other interesting examples involving one-parameter families where maximal invariant statistics fail to distinguish the hypothesized models. The examples include easy-to-construct families of probability models involving only a single location or scale parameter θ.     

A Prediction Interval for M Estimators*

There has been much work in the area of estimating the center of a symmetric population. If one allows for the possibility that the population may be heavy-tailed then robust procedures, and in particular M estimators, have proven quite popular. In this paper we consider the following problem: given a random sample, produce an interval such that the M estimator derived from a future random sample (from the same population) will lie in that interval with some preassigned probability. Clearly such an interval is of use, especially in quality control where prediction is vital. In this paper such an interval is proposed based on asymptotic theory. A simulation study was run for a variety of sample sizes (the sizes of the observed and future samples need not be equal) and distributions. The particular M estimator of choice is that based on the biweight ψ function. The proposed interval performs reasonably well relative to the best that can be achieved asymptotically.     

Estimation of Stigmatized Characteristics of a Hidden Gang in a Finite Population

This paper considers the problem of estimating the size and mean value of a stigmatized quantitative character of a hidden gang in a finite population. The proposed method may be applied to solve domestic problems in a particular country or across countries: for example, a government may be interested in estimating the average income of victims or perpetrators of domestic violence. The proposed method is based on the technique introduced by Warner (1965) to estimate the proportion of a sensitive attribute in a finite population without threatening the privacy of the respondents. Expressions for the bias and variance of the proposed estimators are given, to a first order of approximation. Circumstances in which the method can be applied are studied and illustrated using a numerical example.     

An evaluation of the bootstrap for model validation in mixture models

Bootstrapping has been used as a diagnostic tool for validating model results for a wide array of statistical models. Here we evaluate the use of the non-parametric bootstrap for model validation in mixture models. We show that the bootstrap is problematic for validating the results of class enumeration and demonstrating the stability of parameter estimates in both finite mixture and regression mixture models. In only 44% of simulations did bootstrapping detect the correct number of classes in at least 90% of the bootstrap samples for a finite mixture model without any model violations. For regression mixture models and cases with violated model assumptions, the performance was even worse. Consequently, we cannot recommend the non-parametric bootstrap for validating mixture models.

The cause of the problem is that when resampling is used influential individual observations have a high likelihood of being sampled many times. The presence of multiple replications of even moderately extreme observations is shown to lead to additional latent classes being extracted. To verify that these replications cause the problems we show that leave-k-out cross-validation where sub-samples taken without replacement does not suffer from the same problem.     

Ulster Protestants in New Zealand: A Bibliographic Essay

One of the most noticeable aspects of recent studies which examine Irish migration to New Zealand has been the identification of a sizeable contingent of Ulster Protestant settlers within that migrant stream. Their presence has proven to be a complicating factor in how the history of the Irish in New Zealand has been written and has made easy assumptions about the loyalty, identity, politics and ethnicity of that population impossible. This essay surveys the existing literature on New Zealand's Ulster Protestant population across a wide range of subjects while at the same time considering their apparent disappearance as a distinct ethnic grouping in the new world.     

Back-to-back mixtures of discrete distributions

A method for expanding the choices for fits of discrete data is given. The method is very simple: a breakpoint is chosen for the data set on either side of which two separate discrete distributions are fit. Thus, the method is a mixture of two discrete distributions. The method is appealing in light of the ease with which the likelihood equations simplify. For illustrative purposes, the method is used on the data set that motivated its conception.