LIKELIHOOD MOMENT ESTIMATION FOR THE GENERALIZED PARETO DISTRIBUTION

Traditional methods for estimating parameters in the generalized Pareto distribution have theoretical and computational defects. The moment estimator and the probability‐weighted moment estimator have low asymptotic efficiencies. They may not exist or may give nonsensical estimates. The maximum likelihood estimator, which sometimes does not exist, is asymptotically efficient, but its computation is complex and has convergence problems. The likelihood moment estimator is proposed, which is computationally easy and has high asymptotic efficiency.     

Simultaneous estimation of the hardy-weinberg proportions

We consider the problem of simultaneously estimating k + 1 related proportions, with a special emphasis on the estimation of Hardy-Weinberg (HW) proportions. We prove that the uniformly minimum-variance unbiased estimator (UMVUE) of two proportions which are individually admissible under squared-error loss are inadmissible in estimating the proportions jointly. Furthermore, rules that dominate the UMVUE are given. A Bayesian analysis is then presented to provide insight into this inadmissibility issue: The UMVUE is undesirable because the two estimators are Bayes rules corresponding to different priors. It is also shown that there does not exist a prior which yields the maximum-likelihood estimators simultaneously. When the risks of several estimators for the HW proportions are compared, it is seen that some Bayesian estimates yield significantly smaller risks over a large portion of the parameter space for small samples. However, the differences in risks become less significant as the sample size gets larger.     

A Note on the Inverse Birthday Problem With Applications

The classical birthday problem considers the probability that at least two people in a group of size N share the same birthday. The inverse birthday problem considers the estimation of the size N of a group given the number of different birthdays in the group. In practice, this problem is analogous to estimating the size of a population from occurrence data only. The inverse problem can be solved via two simple approaches including the method of moments for a multinominal model and the maximum likelihood estimate of a Poisson model, which we present in this study. We investigate properties of both methods and show that they can yield asymptotically equivalent Wald-type interval estimators. Moreover, we show that these methods estimate a lower bound for the population size when birth rates are nonhomogenous or individuals in the population are aggregated. A simulation study was conducted to evaluate the performance of the point estimates arising from the two approaches and to compare the performance of seven interval estimators, including likelihood ratio and log-transformation methods. We illustrate the utility of these methods by estimating: (1) the abundance of tree species over a 50-hectare forest plot, (2) the number of Chlamydia infections when only the number of different birthdays of the patients is known, and (3) the number of rainy days when the number of rainy weeks is known. Supplementary materials for this article are available online.     

Estimation of gene/haplotype frequencies in genetic marker systems based on phenotype data

Two equivalent methods (gene counting and maximum likelihood) for estimating gene frequencies in a general genetic marker system based on observed phenotype data are derived. Under the maximum likelihood approach, an expression is given for the estimated covariance matrix from which estimated standard errors of the estimators can be found. In addition, consideration is given to the problem of estimating gene frequencies when there are available several independent population data sets.     

ONE-SIDED ESTIMATING AND TESTING PROBLEMS FOR LOCATION MODELS FROM GROUPED SAMPLES

This article deals with one-sided problems for location models from grouped samples. Suppose the support region of a density function, which does not depend on parameters, is divided into some disjointed intervals, grouped samples are the number of observations falling in each intervals respectively. The studying of grouped samples may be dated back to the beginning of the century, in which only one sample location and/or scale models is considered. This article considers one-sided estimating and testing problems for location models. Some methods for computing the maximum likelihood estimates of the parameters subject to order restrictions are proposed and a numerical example by the method is given.     

Bayes least squares linear estimation of densities

Linear, least squares statistical methods in which the "parameters" are interpreted as random variables were introduced by Whittle, and further developed by Hartigan and others. They are applied here to the problem of estimating the coefficients in an orthogonal expansion of a multivariate density, given a simple random sample.     

A nonparametric mixture approach to density and null proportion estimation in large-scale multiple comparison problems

A new method for estimating the proportion of null effects is proposed for solving large-scale multiple comparison problems. It utilises maximum likelihood estimation of nonparametric mixtures, which also provides a density estimate of the test statistics. It overcomes the problem of the usual nonparametric maximum likelihood estimator that cannot produce a positive probability at the location of null effects in the process of estimating nonparametrically a mixing distribution. The profile likelihood is further used to help produce a range of null proportion values, corresponding to which the density estimates are all consistent. With a proper choice of a threshold function on the profile likelihood ratio, the upper endpoint of this range can be shown to be a consistent estimator of the null proportion. Numerical studies show that the proposed method has an apparently convergent trend in all cases studied and performs favourably when compared with existing methods in the literature.     

Optimal bounded influence regression and scale m-estimators in the context of experimental desing

The problem of simultaneous robust estimation of regression and scale parameters in the linear regression model is studied in the context of experimental design. Optimal M-estimates are given for a modified optimization problem of minimizing the asymptotic variances under bounded influence functions. This is done by reducing the multidimensional regression problem to the problem of estimating one-dimensional location and scale. For the location-scale case two subfamilies of optimal score functions are described in detail along with comparisons of the asymptotic variances and gross-error-sensitivities of the corresponding M-estimators. It turns out that, even for small gross-error-sensitivities, one of the subfamilies provides variances which are close to those of the nonrobust maximum likelihood estimators.     

Estimating the Cost of a Warranty

The cost of certain types of warranties is closely related to functions that arise in renewal theory. The problem of estimating the warranty cost for a random sample of size n can be reduced to estimating these functions. In an earlier paper, I gave several methods of estimating the expected number of renewals, called the renewal function. This answered an important accounting question of how to arrive at a good approximation of the expected warranty cost. In this article, estimation of the renewal function is reviewed and several extensions are given. In particular, a resampling estimator of the renewal function is introduced. Further, I argue that managers may wish to examine other summary measures of the warranty cost, in particular the variability. To estimate this variability, I introduce estimators, both parametric and nonparametric, of the variance associated with the number of renewals. Several numerical examples are provided.     

Parameter Orthogonality and Bias Adjustment for Estimating Functions

Abstract.  We consider an extended notion of parameter orthogonality for estimating functions, called nuisance parameter insensitivity, which allows a unified treatment of nuisance parameters for a wide range of methods, including Liang and Zeger's generalized estimating equations. Nuisance parameter insensitivity has several important properties in common with conventional parameter orthogonality, such as the nuisance parameter causing no loss of efficiency for estimating the interest parameter, and a simplified estimation algorithm. We also consider bias adjustment for profile estimating functions, and apply the results to restricted maximum likelihood estimation of dispersion parameters in generalized estimating equations.     

Semiparametric estimation of employment duration models

There are a variety of economic areas, such as studies of employment duration and of the durability of capital goods, in which data on important variables typically are censored. The standard techinques for estimating a model from censored data require the distributions of unobservable random components of the model to be specified a priori up to a finite set of parameters, and misspecification of these distributions usually leads to inconsistent parameter estimates. However, economic theory rarely gives guidance about distributions and the standard estimation techniques do not provide convenient methods for identifying distributions from censored data. Recently, several distribution-free or semiparametric methods for estimating censored regression models have been developed. This paper presents the results of using two such methods to estimate a model of employment duration. The paper reports the operating characteristics of the semiparametric estimators and compares the semiparametric estimates with those obtained from a standard parametric model.     

A survey and comparison of methods for estimating extreme right tail-area quantiles

The purpose of this paper is to survey many of the methods for estimating extreme right tail-area quantiles in order to determine which method or methods gives the best approximations. The problem is to find a good estimate of x_p defined by 1 - F(x _p) = p where p is a very small number for a random sample from an unknown distribution. An extension of this problem is to determine the number of largest order statistics that should be used to make an estimate. From extensive computer simulations trying to minimize relative error, conclusions can be drawn based on the value of p. For p = .02, the exponential tail method by Breiman, et al using a method by Pickands for determining the number of order statistics to use works best for light to heavy tailed distributions. For extremely heavy tailed distributions, a method proposed by Hosking and Wallis seems to be the most accurate at p = .02 and p = .002. The quadratic tail method by Breiman, et al appears best for light to moderately heavy tailed distributions at p = .002 and for all distributions at p = .0002.     

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.     

Optimal Change-point Estimation in Inverse Problems

We develop a method of estimating a change-point of an otherwise smooth function in the case of indirect noisy observations. As two paradigms we consider deconvolution and non-parametric errors-in-variables regression. In a similar manner to well-established methods for estimating change-points in non-parametric regression, we look essentially at the difference of one-sided kernel estimators. Because of the indirect nature of the observations we employ deconvoluting kernels. We obtain an estimate of the change-point by the extremal point of the differences between these two-sided kernel estimators. We derive rates of convergence for this estimator. They depend on the degree of ill-posedness of the problem, which derives from the smoothness of the error density. Analysing the Hellinger modulus of continuity of the problem we show that these rates are minimax     

On linear lon-odds and estimation of discriminant coefficients

Fisher's Linear Discriminant Function Can be used to classify an individual who has sampled from one of two multivariate normal Populations. In the following, this function is viewed as the other given his data vector it is assumed that the Population means and common covariance matrix are unknown. The vector of discriminant coeffients β(p×1) is the gradient of posterior log-odds and certain of its lineqar functions are directional derivatives which have a practical meaning. Accordingly, we treat the problems of estimating several linear functions of β The usual estimatoes of these functions are scaled versions of the unbiased estmators. In this Paper, these estimators are domainated by explicit alterenatives under a quadratic loss function. we reduce the problem of estimating β to that of estimating the inverse convariance matrix.     

Nonparametric point estimators for the change-point problem

Three nonparametric methods for estimating a change-point, and the mean function Pefore and after the change has occurred are developed for a restricted class of processes. The estimators which are developed are intuitive, and their asymptotic behavior is studied, Konte Cario cornparisoris are undertaKen for smali and moderate samples.     

Estimating Desired Sample Size for Simple Random Sampling of a Skewed Population

A simulation study was conducted to assess how well the necessary sample size to achieve a stipulated margin of error can be estimated prior to sampling. Our concern was particularly focused on performance when sampling from a very skewed distribution, which is a common feature of many biological, economic, and other populations. We examined two approaches for estimating sample size—one being the commonly used strategy aimed at regulating the average magnitude of the stipulated margin of error and the second being a previously proposed strategy to control the tolerance probability with which the stipulated margin of error is exceeded. Results of the simulation revealed that (1) skewness does not much affect the average estimated sample size but can greatly extend the range of estimated sample sizes; and (2) skewness does reduce the effectiveness of Kupper and Hafner's sample size estimator, yet its effectiveness is negatively impacted less by skewness directly, and to a much greater degree by the common practice of estimating the population variance via a pilot sampling from the skewed population. Nonetheless, the simulations suggest that estimating sample size to control the probability with which the desired margin of error is achieved is a worthwhile alternative to the usual sample size formula that controls the average width of the confidence interval only.     

Covariance Matrix Estimation via Network Structure

In this article, we employ a regression formulation to estimate the high-dimensional covariance matrix for a given network structure. Using prior information contained in the network relationships, we model the covariance as a polynomial function of the symmetric adjacency matrix. Accordingly, the problem of estimating a high-dimensional covariance matrix is converted to one of estimating low dimensional coefficients of the polynomial regression function, which we can accomplish using ordinary least squares or maximum likelihood. The resulting covariance matrix estimator based on the maximum likelihood approach is guaranteed to be positive definite even in finite samples. Under mild conditions, we obtain the theoretical properties of the resulting estimators. A Bayesian information criterion is also developed to select the order of the polynomial function. Simulation studies and empirical examples illustrate the usefulness of the proposed methods.     

Generalized confidence intervals compatible with the Min test for simultaneous comparisons of one subpopulation to several other subpopulations

A problem where one subpopulation is compared with several other subpopulations in terms of means with the goal of estimating the smallest difference between the means commonly arises in biology, medicine, and many other scientific fields. A generalization of Strass-burger-Bretz-Hochberg approach for two comparisons is presented for cases with three and more comparisons. The method allows constructing an interval estimator for the smallest mean difference, which is compatible with the Min test. An application to a fluency-disorder study is illustrated. Simulations confirmed adequate probability coverage for normally distributed outcomes for a number of designs.     

Adaptive Isotonic Estimation of the Minimum Effective and Peak Doses in the Presence of Covariates

We consider a problem of estimating the minimum effective and peak doses in the presence of covariates. We propose a sequential strategy for subject assignment that includes an adaptive randomization component to balance the allocation to placebo and active doses with respect to covariates. We conclude that either adjusting for covariates in the model or balancing allocation with respect to covariates is required to avoid bias in the target dose estimation. We also compute optimal allocation to estimate the minimum effective and peak doses in discrete dose space using isotonic regression.