SB-robustness of directional mean for circular distributions

In this paper we study the robustness of the directional mean (a.k.a. circular mean) for different families of circular distributions. We show that the directional mean is robust in the sense of finite standardized gross error sensitivity (SB-robust) for the following families: (1) mixture of two circular normal distributions, (2) mixture of wrapped normal and circular normal distributions and (3) mixture of two wrapped normal distributions. We also show that the directional mean is not SB-robust for the family of all circular normal distributions with varying concentration parameter. We define the circular trimmed mean and prove that it is SB-robust for this family. In general the property of SB-robustness of an estimator at a family of probability distributions is dependent on the choice of the dispersion measure. We introduce the concept of equivalent dispersion measures and prove that if an estimator is SB-robust for one dispersion measure then it is SB-robust for all equivalent dispersion measures. Three different dispersion measures for circular distributions are considered and their equivalence studied.     

An alternative estimator for randomized response sampling with continuous distributions from a dichotomous population

In this paper, we introduce an alternative estimator of a population proportion from a dichotomous population when using randomized response sampling with continuous randomizing distributions. We also propose the alternative use of exponential randomizing densities. The estimator is obtained by method of moments and is compared with Franklin's (1989) estimator using normal and exponential distributions. The proposed estimator is more efficient than Franklin's (1989) estimator under suitable conditions for the two distributions.     

Inference about permutation parameter in large samples

The estimation problem of a permutation parameter on the basis of a random sample of increasing size is considered. A necessary and sufficient condition for the existence of an estimator, asymptotically fully efficient for two different distributions families, is derived. We also study the application of this result to cyclic groups of order two and three.     

Gamma-minimax results for the class of unimodal priors

Olman and Shmundak proved 1985 that in estimating a bounded normal mean under squared error loss the Bayes estimator with respect to the uniform distribution on the parameter interval is gamma-minimax when the parameter interval is sufficiently small and the class of priors consists of all symmetric and unimodal distributions. Recently, one of the authors showed that this result remains valid for quite general families of distributions which satisfy some regularity conditions. In the present paper a generalization to the class of unimodal priors with fixed mode is derived. It is proved that the Bayes estimator with respect to a suitable mixture of two uniform distributions is gamma-minimax for sufficiently small parameter intervals. To that end appropriate characterizations of a saddle point in the corresponding statistical games are established. Some results of a numerical study are presented.     

Estimation of the parameters in a truncated normal distribution

This paper deals with the estimation of the parameters of doubly truncated and singly truncated normal distributions when truncation points are known. We derive, for these families, a necessary and sufficient condition for the maximum likelihood estimator(MLE) to be finite. Furthermore, the probability of the MLE being infinite is positive. A simulation study for single truncation is carried out to compare the modified maximum likelihood estimator, and the mixed estimator.     

Identifiability and estimation of recursive max‐linear models

We address the identifiability and estimation of recursive max‐linear structural equation models represented by an edge‐weighted directed acyclic graph (DAG). Such models are generally unidentifiable and we identify the whole class of DAG s and edge weights corresponding to a given observational distribution. For estimation, standard likelihood theory cannot be applied because the corresponding families of distributions are not dominated. Given the underlying DAG, we present an estimator for the class of edge weights and show that it can be considered a generalized maximum likelihood estimator. In addition, we develop a simple method for identifying the structure of the DAG. With probability tending to one at an exponential rate with the number of observations, this method correctly identifies the class of DAGs and, similarly, exactly identifies the possible edge weights.     

Minimum variance unbiased estimators of the ratio of means of two lognormal variates

The minimum variance unbiased estimators from independent samples of the ratio of the means of two lognormal distributions with equal and unequal shape parameters are derived using a method due to Finney (1941). The like estimator for two gamma distributions of known shape is given. A numerical example from a recent cloud-seeding experiment is also given.     

Adaptive Estimation Using Weighted Least Squares

This paper proposes an adaptive estimator that is more precise than the ordinary least squares estimator if the distribution of random errors is skewed or has long tails. The adaptive estimates are computed using a weighted least squares approach with weights based on the lengths of the tails of the distribution of residuals. Smaller weights are assigned to those observations that have residuals in the tails of long-tailed distributions and larger weights are assigned to observations having residuals in the tails of short-tailed distributions. Monte Carlo methods are used to compare the performance of the proposed estimator and the performance of the ordinary least squares estimator. The estimates that were studied in this simulation include the difference between the means of two populations, the mean of a symmetric distribution, and the slope of a regression line. The adaptive estimators are shown to have lower mean squared errors than those for the ordinary least squares estimators for short-tailed, long-tailed, and skewed distributions, provided the sample size is at least 20. The ordinary least squares estimator has slightly lower mean squared error for normally distributed errors. The adaptive estimator is recommended for general use for studies having sample sizes of at least 20 observations unless the random errors are known to be normally distributed.     

On the bias and variance of some proportion estimators

Providing certain parameters are known, almost any linear map from R^P to R¹ can be adjusted to yield a consistent and unbiased estimator in the context of estimating the mixing proportion θ on the basis of an unclassified sample of observations taken from a mixture of two p-dimensional distributions in proportions θ and 1-θ. Attention is focused on an estimator proposed recently, θ, which has minimum variance over all such linear maps. Unfortunately, the form of θ depends on the means of the component distributions and the covariance matrix of the mixture distribution. The effect of using appropriate sample estimates for these unknown parameters in forming θ is investigated by deriving the asymptotic mean and variance of the resulting estimator. The relative efficiency of this estimator under normality is derived. Also, a study is undertaken of the performance of a similar type of estimator appropriate in the context where an observed data vector is not an observation from either one or the other onent distributions, but is recorded as an integrated measurement over a surface area which is a mixture of two categories whose characteristics have different statistical distributions.The asymptotic bias in this case is compared with some available practical results.     

The precision of regression-type estimator for incremental cost–effectiveness ratio

The estimation of incremental cost–effectiveness ratio (ICER) has received increasing attention recently. It is expressed in terms of the ratio of the change in costs of a therapeutic intervention to the change in the effects of the intervention. Despite the intuitive interpretation of ICER as an additional cost per additional benefit unit, it is a challenge to estimate the distribution of a ratio of two stochastically dependent distributions. A vast literature regarding the statistical methods of ICER has developed in the past two decades, but none of these methods provide an unbiased estimator. Here, to obtain the unbiased estimator of the cost–effectiveness ratio (CER), the zero intercept of the bivariate normal regression is assumed. In equal sample sizes, the Iman–Conover algorithm is applied to construct the desired variance–covariance matrix of two random bivariate samples, and the estimation then follows the same approach as CER to obtain the unbiased estimator of ICER. The bootstrapping method with the Iman–Conover algorithm is employed for unequal sample sizes. Simulation experiments are conducted to evaluate the proposed method. The regression-type estimator performs overwhelmingly better than the sample mean estimator in terms of mean squared error in all cases.     

Penalized likelihood approach for the four-parameter kappa distribution

The four-parameter kappa distribution (K4D) is a generalized form of some commonly used distributions such as generalized logistic, generalized Pareto, generalized Gumbel, and generalized extreme value (GEV) distributions. Owing to its flexibility, the K4D is widely applied in modeling in several fields such as hydrology and climatic change. For the estimation of the four parameters, the maximum likelihood approach and the method of L-moments are usually employed. The L-moment estimator (LME) method works well for some parameter spaces, with up to a moderate sample size, but it is sometimes not feasible in terms of computing the appropriate estimates. Meanwhile, using the maximum likelihood estimator (MLE) with small sample sizes shows substantially poor performance in terms of a large variance of the estimator. We therefore propose a maximum penalized likelihood estimation (MPLE) of K4D by adjusting the existing penalty functions that restrict the parameter space. Eighteen combinations of penalties for two shape parameters are considered and compared. The MPLE retains modeling flexibility and large sample optimality while also improving on small sample properties. The properties of the proposed estimator are verified through a Monte Carlo simulation, and an application case is demonstrated taking Thailand’s annual maximum temperature data.     

On families of beta- and generalized gamma-generated distributions and associated inference

A general family of univariate distributions generated by beta random variables, proposed by Jones, has been discussed recently in the literature. This family of distributions possesses great flexibility while fitting symmetric as well as skewed models with varying tail weights. In a similar vein, we define here a family of univariate distributions generated by Stacy’s generalized gamma variables. For these two families of univariate distributions, we discuss maximum entropy characterizations under suitable constraints. Based on these characterizations, an expected ratio of quantile densities is proposed for the discrimination of members of these two broad families of distributions. Several special cases of these results are then highlighted. An alternative to the usual method of moments is also proposed for the estimation of the parameters, and the form of these estimators is particularly amenable to these two families of distributions.     

On estimating extremal dependence structures by parametric spectral measures

Estimation of extreme value copulas is often required in situations where available data are sparse. Parametric methods may then be the preferred approach. A possible way of defining parametric families that are simple and, at the same time, cover a large variety of multivariate extremal dependence structures is to build models based on spectral measures. This approach is considered here. Parametric families of spectral measures are defined as convex hulls of suitable basis elements, and parameters are estimated by projecting an initial nonparametric estimator on these finite-dimensional spaces. Asymptotic distributions are derived for the estimated parameters and the resulting estimates of the spectral measure and the extreme value copula. Finite sample properties are illustrated by a simulation study.     

Tail exponent estimation via broadband log density-quantile regression

Heavy tail probability distributions are important in many scientific disciplines such as hydrology, geology, and physics and therefore feature heavily in statistical practice. Rather than specifying a family of heavy-tailed distributions for a given application, it is more common to use a nonparametric approach, where the distributions are classified according to the tail behavior. Through the use of the logarithm of Parzen's density-quantile function, this work proposes a consistent, flexible estimator of the tail exponent. The approach we develop is based on a Fourier series estimator and allows for separate estimates of the left and right tail exponents. The theoretical properties for the tail exponent estimator are determined, and we also provide some results of independent interest that may be used to establish weak convergence of stochastic processes. We assess the practical performance of the method by exploring its finite sample properties in simulation studies. The overall performance is competitive with classical tail index estimators, and, in contrast, with these our method obtains somewhat better results in the case of lighter heavy-tailed distributions.     

The unit extended Weibull families of distributions and its applications

In this paper, two new general families of distributions supported on the unit interval are introduced. The proposed families include several known models as special cases and define at least twenty (each one) new special models. Since the list of well-being indicators may include several double bounded random variables, the applicability for modeling those is the major practical motivation for introducing the distributions on those families. We propose a parametrization of the new families in terms of the median and develop a shiny application to provide interactive density shape illustrations for some special cases. Various properties of the introduced families are studied. Some special models in the new families are discussed. In particular, the complementary unit Weibull distribution is studied in some detail. The method of maximum likelihood for estimating the model parameters is discussed. An extensive Monte Carlo experiment is conducted to evaluate the performances of these estimators in finite samples. Applications to the literacy rate in Brazilian and Colombian municipalities illustrate the usefulness of the two new families for modeling well-being indicators.     

Kernel density estimation for heavy-tailed distributions using the champernowne transformation

When estimating loss distributions in insurance, large and small losses are usually split because it is difficult to find a simple parametric model that fits all claim sizes. This approach involves determining the threshold level between large and small losses. In this article, a unified approach to the estimation of loss distributions is presented. We propose an estimator obtained by transforming the data set with a modification of the Champernowne cdf and then estimating the density of the transformed data by use of the classical kernel density estimator. We investigate the asymptotic bias and variance of the proposed estimator. In a simulation study, the proposed method shows a good performance. We also present two applications dealing with claims costs in insurance.     

Phi-Divergence Statistics for Testing Linear Hypotheses in Logistic Regression Models

In this paper we introduce and study two new families of statistics for the problem of testing linear combinations of the parameters in logistic regression models. These families are based on the phi-divergence measures. One of them includes the classical likelihood ratio statistic and the other the classical Pearson's statistic for this problem. It is interesting to note that the vector of unknown parameters, in the two new families of phi-divergence statistics considered in this paper, is estimated using the minimum phi-divergence estimator instead of the maximum likelihood estimator. Minimum phi-divergence estimators are a natural extension of the maximum likelihood estimator.     

Nonparametric likelihood and doubly robust estimating equations for marginal and nested structural models

This article considers Robins's marginal and nested structural models in the cross‐sectional setting and develops likelihood and regression estimators. First, a nonparametric likelihood method is proposed by retaining a finite subset of all inherent and modelling constraints on the joint distributions of potential outcomes and covariates under a correctly specified propensity score model. A profile likelihood is derived by maximizing the nonparametric likelihood over these joint distributions subject to the retained constraints. The maximum likelihood estimator is intrinsically efficient based on the retained constraints and weakly locally efficient. Second, two regression estimators, named hat and tilde, are derived as first‐order approximations to the likelihood estimator under the propensity score model. The tilde regression estimator is intrinsically and weakly locally efficient and doubly robust. The methods are illustrated by data analysis for an observational study on right heart catheterization. The Canadian Journal of Statistics 38: 609–632; 2010 © 2010 Statistical Society of Canada     

SB-Robust Estimators of the Parameters of the Wrapped Normal Distribution

In this article, we study the SB-robustness of various estimators of the mean direction (μ) and the concentration parameter (ρ) of the wrapped normal distribution. The functional corresponding to the sample mean direction is seen to be not SB-robust as an estimator of μ at the family of wrapped normal distributions with varying ρ, whereas the γ-trimmed mean direction is SB-robust at the same family of distributions for the different dispersion measures considered in this article. We also study the SB-robustness of the moment estimator of ρ and also that for a newly introduced trimmed estimator of ρ.     

On bias reduction estimators of skew-normal and skew-t distributions

A particular concerns of researchers in statistical inference is bias in parameters estimation. Maximum likelihood estimators are often biased and for small sample size, the first order bias of them can be large and so it may influence the efficiency of the estimator. There are different methods for reduction of this bias. In this paper, we proposed a modified maximum likelihood estimator for the shape parameter of two popular skew distributions, namely skew-normal and skew-t, by offering a new method. We show that this estimator has lower asymptotic bias than the maximum likelihood estimator and is more efficient than those based on the existing methods.