Asymptotic behavior of the grenander estimator at density flat regions

Over forty years ago, Grenander derived the MLE of a monotone decreasing density f with known mode. Prakasa Rao obtained the asymptotic distribution of this estimator at a fixed point x where f' (x) < 0. Here, we obtain the asymptotic distribution of this estimator at a fixed point x when f is constant and nonzero in some open neighborhood of x. This limiting distribution is expressible as the convolution of a closed-form density and a rescaled standard normal density. Groeneboom (1983) derived the aforementioned closed-form density and we provide an alternative, more direct derivation.     

Semiparametric Spatial Autoregressive Models With Endogenous Regressors: With an Application to Crime Data

This study considers semiparametric spatial autoregressive models that allow for endogenous regressors, as well as the heterogenous effects of these regressors across spatial units. For the model estimation, we propose a semiparametric series generalized method of moments estimator. We establish that the proposed estimator is both consistent and asymptotically normal. As an empirical illustration, we apply the proposed model and method to Tokyo crime data to estimate how the existence of a neighborhood police substation (NPS) affects the household burglary rate. The results indicate that the presence of an NPS helps reduce household burglaries, and that the effects of some variables are heterogenous with respect to residential distribution patterns. Furthermore, we show that using a model that does not adjust for the endogeneity of NPS does not allow us to observe the significant relationship between NPS and the household burglary rate. Supplementary materials for this article are available online.     

A Three-Parameter Asymmetric Laplace Distribution and Its Extension

ABSTRACT

In this article, a new three-parameter asymmetric Laplace distribution and its extension are introduced. This includes as special case the symmetric Laplace double-exponential distribution. The distribution has established a direct link to estimation of quantile and quantile regression. Properties of the new distribution are presented. Application is made to a flood data modeling example.     

Robust M- and L-Estimators of Scale Parameter

We consider first the class of M-estimators of scale that are location-scale equivariant and Fisher consistent at the error distribution of the shrinking contamination neighborhood and derive an expression for the maximal asymptotic mean-squared-error, for a suitably regular score function, followed by a lower bound on it. We next show that the minimax asymptotic mean-squzred-error is attained at an M-estimator of scale with the truncated MLE score function which, when specialized to the Standard Normal error distribution has the form of Huber's Proposal 2. The latter minimax property is also shown to hold for α-trimmed variance as an L-estimator of scale.     

The Minimum Density Power Divergence Estimation for the Lognormal Density

In this article, we implement the minimum density power divergence estimation for estimating the parameters of the lognormal density. We compare the minimum density power divergence estimator (MDPDE) and the maximum likelihood estimator (MLE) in terms of robustness and asymptotic distribution. The simulations and an example indicate that the MDPDE is less biased than MLE and is as good as MLE in terms of the mean square error under various distributional situations.     

A Method of Exploring the Mechanism of Inflationary Expectations Based on Qualitative Survey Data

The objective of this article is to propose a method of exploring the mechanism of expectation formation based on qualitative survey data. The survey data are regarded as a sample from a multinomial distribution whose parameters are time-variant functions of inflation expectations. The parameters are estimated using a Bayesian recursive approach, which is a generalization of the Kalman filtering technique. For illustrative purposes, the method is applied to Japanese data. One notable finding from the empirical analysis is that the expectation formation process of Japanese enterprises has varied greatly over time.     

Non-parametric Sampling Approximation via Voronoi Tessellations

In this article we propose a novel non-parametric sampling approach to estimate posterior distributions from parameters of interest. Starting from an initial sample over the parameter space, this method makes use of this initial information to form a geometrical structure known as Voronoi tessellation over the whole parameter space. This rough approximation to the posterior distribution provides a way to generate new points from the posterior distribution without any additional costly model evaluations. By using a traditional Markov Chain Monte Carlo (MCMC) over the non-parametric tessellation, the initial approximate distribution is refined sequentially. We applied this method to a couple of climate models to show that this hybrid scheme successfully approximates the posterior distribution of the model parameters.     

Generalized Inverse Gamma Distribution and its Application in Reliability

In this article, we introduce a new reliability model of inverse gamma distribution referred to as the generalized inverse gamma distribution (GIG). A generalization of inverse gamma distribution is defined based on the exact form of generalized gamma function of Kobayashi (1991). This function is useful in many problems of diffraction theory and corrosion problems in new machines. The new distribution has a number of lifetime special sub-models. For this model, some of its statistical properties are studied. The method of maximum likelihood is used for estimating the model parameters and the observed information matrix is derived. We also demonstrate the usefulness of this distribution on a real data set.     

A Modified Version of Logarithmic Series Distribution and Its Applications

In this article, we consider a modified version of logarithmic series distribution and study some of its properties. The maximum likelihood estimation of the parameters of the modified distribution is discussed and the distribution has been fitted to certain real life data sets. Tests are also carried out for justifying the significance of the additional parameter of the modified distribution.     

Transmuted Linear Exponential Distribution: A New Generalization of the Linear Exponential Distribution

In this article, a transmuted linear exponential distribution is developed that generalizes the linear exponential distribution with an additional parameter using the quadratic rank transmutation map which was studied by Shaw et al. Some statistical properties of the proposed distribution such as moments, quantiles, and the failure rate function are investigated. The maximum likelihood estimators of unknown parameters are also discussed and a real data analysis is carried out to illustrate the superiority of the proposed distribution.     

Entropy-based goodness-of-fit tests for the Pareto I distribution

In this article, having observed the generalized order statistics in a sample, we construct a test for the hypothesis that the underlying distribution is the Pareto I distribution. The Shannon entropy of generalized order statistics is used to test the null hypothesis.     

Maximum entropy estimation for survey sampling

Calibration methods have been widely studied in survey sampling over the last decades. Viewing calibration as an inverse problem, we extend the calibration technique by using a maximum entropy method. Finding the optimal weights is achieved by considering random weights and looking for a discrete distribution which maximizes an entropy under the calibration constraint. This method points a new frame for the computation of such estimates and the investigation of its statistical properties.     

Estimation of reliability of multicomponent stress–strength for a Kumaraswamy distribution

This article deals with the Bayesian and non Bayesian estimation of multicomponent stress–strength reliability by assuming the Kumaraswamy distribution. Both stress and strength are assumed to have a Kumaraswamy distribution with common and known shape parameter. The reliability of such a system is obtained by the methods of maximum likelihood and Bayesian approach and the results are compared using Markov Chain Monte Carlo (MCMC) technique for both small and large samples. Finally, two data sets are analyzed for illustrative purposes.     

A new quasi empirical bayes estimate in randomized response sampling

In this article, a new notion of “quasi-empirical” Bayes estimation is developed for estimating the proportion of a sensitive attribute in a population by making use of both a prior distribution of prevalence of the sensitive attribute in addition to the known prior distribution of an unrelated characteristic. The proposed quasi-empirical Bayes estimate is compared with those of the unrelated question model due to Greenberg et al. by means of a simulation study.     

Exact Interval Estimation for the Scale Family Under General Progressive Type-II Censoring

In this article, we consider a general progressively Type-II censored life test where the lifetime distribution of each test unit belongs to the scale family. We derive an exact confidence interval for the scale parameter. Using Monte Carlo simulation method, we assess the expected lower and upper limits of the proposed confidence interval for the exponential distribution. Finally, we present a numerical example to illustrate the proposed procedure.     

A Finite Mixture Three-Parameter Weibull Model for the Analysis of Wind Speed Data

This article presents a mixture three-parameter Weibull distribution to model wind speed data. The parameters are estimated by using maximum likelihood (ML) method in which the maximization problem is regarded as a nonlinear programming with only inequality constraints and is solved numerically by the interior-point method. By applying this model to four lattice-point wind speed sequences extracted from National Centers for Environmental Prediction (NCEP) reanalysis data, it is observed that the mixture three-parameter Weibull distribution model proposed in this paper provides a better fit than the existing Weibull models for the analysis of wind speed data under study.     

Semiparametric estimation of the cumulative incidence function under competing risks and left-truncated sampling

In this article, we propose semiparametric methods to estimate the cumulative incidence function of two dependent competing risks for left-truncated and right-censored data. The proposed method is based on work by Huang and Wang (1995). We extend previous model by allowing for a general parametric truncation distribution and a third competing risk before recruitment. Based on work by Vardi (1989), several iterative algorithms are proposed to obtain the semiparametric estimates of cumulative incidence functions. The asymptotic properties of the semiparametric estimators are derived. Simulation results show that a semiparametric approach assuming the parametric truncation distribution is correctly specified produces estimates with smaller mean squared error than those obtained in a fully nonparametric model.     

Asymptotic Distributions of Innovation Density Estimators in Linear Processes

In this article, we demonstrate that at a fixed point, the asymptotic distribution of the innovation density estimator is normal for stationary linear process. Also, we show that the asymptotic distribution of the global measure of the deviation of the density estimator from the expectation of the kernel innovation density (based on the true innovations) is the same as that in the case when we can observe the true innovations.     

The Kumaraswamy skew-t distribution and its related properties

Skew normal distribution is an alternative distribution to the normal distribution to accommodate asymmetry. Since then extensive studies have been done on applying Azzalini’s skewness mechanism to other well-known distributions, such as skew-t distribution, which is more flexible and can better accommodate long tailed data than the skew normal one. The Kumaraswamy generalized distribution (Kw ? F) is another new class of distribution which is capable of fitting skewed data that can not be fitted well by existing distributions. Such a distribution has been widely studied and various versions of generalization of this distribution family have been introduced. In this article, we introduce a new generalization of the skew-t distribution based on the Kumaraswamy generalized distribution. The new class of distribution, which we call the Kumaraswamy skew-t (KwST) has the ability of fitting skewed, long, and heavy-tailed data and is more flexible than the skew-t distribution as it contains the skew-t distribution as a special case. Related properties of this distribution family such as mathematical properties, moments, and order statistics are discussed. The proposed distribution is applied to a real dataset to illustrate the estimation procedure.