GMM estimation of a maximum entropy distribution with interval data

We develop a generalized method of moments (GMM) estimator for the distribution of a variable where summary statistics are available only for intervals of the random variable. Without individual data, one cannot calculate the weighting matrix for the GMM estimator. Instead, we propose a simulated weighting matrix based on a first-step consistent estimate. When the functional form of the underlying distribution is unknown, we estimate it using a simple yet flexible maximum entropy density. Our Monte Carlo simulations show that the proposed maximum entropy density is able to approximate various distributions extremely well. The two-step GMM estimator with a simulated weighting matrix improves the efficiency of the one-step GMM considerably. We use this method to estimate the U.S. income distribution and compare these results with those based on the underlying raw income data.     

Practical estimation of multivariate densities using wavelet methods

This paper describes a practical method for estimating multivariate densities using wavelets. As in kernel methods, wavelet methods depend on two types of parameters. On the one hand we have a functional parameter: the wavelet Ø (comparable to the kernel K ) and on the other hand we have a smoothing parameter: the resolution index (comparable to the bandwidth h ). Classically, we determine the resolution index with a cross-validation method. The advantage of wavelet methods compared to kernel methods is that we have a technique for choosing the wavelet Ø among a fixed family. Moreover, the wavelets method simplifies significantly both the theoretical and the practical computations.     

An EM algorithm for the estimation of parametric and nonparametric hierarchical nonlinear models

It is shown how to implement an EM algorithm for maximum likelihood estimation of hierarchical nonlinear models for data sets consisting of more than two levels of nesting. This upward–downward algorithm makes use of the conditional independence assumptions implied by the hierarchical model. It cannot only be used for the estimation of models with a parametric specification of the random effects, but also to extend the two-level nonparametric approach – sometimes referred to as latent class regression – to three or more levels. The proposed approach is illustrated with an empirical application.     

Percentage Points of the Multivariate t Distribution

The known methods for computing percentage points of multivariate t distributions are reviewed. We believe that this review will serve as an important reference and encourage further research activities in the area.     

Term structure estimation in the presence of autocorrelation

This paper assesses the effects of autocorrelation on parameter estimates of affine term structure models (ATSM) when principal components analysis is used to extract factors. In contrast to recent studies, we design and run a Monte Carlo experiment that relies on the construction of a simulation design that is consistent with the data, rather than theory or observation, and find that parameter estimation from ATSM is precise in the presence of serial correlation in the measurement error term. Our findings show that parameter estimation of ATSM with principal component based factors is robust to autocorrelation misspecification.     

Minimax estimation of a cumulative distribution function by converting to a parametric problem

Let X = (X ₁,...,X _n ) be a sample from an unknown cumulative distribution function F defined on the real line . The problem of estimating the cumulative distribution function F is considered using a decision theoretic approach. No assumptions are imposed on the unknown function F. A general method of finding a minimax estimator d(t;X) of F under the loss function of a general form is presented. The method of solution is based on converting the nonparametric problem of searching for minimax estimators of a distribution function to the parametric problem of searching for minimax estimators of the probability of success for a binomial distribution. The solution uses also the completeness property of the class of monotone decision procedures in a monotone decision problem. Some special cases of the underlying problem are considered in the situation when the loss function in the nonparametric problem is defined by a weighted squared, LINEX or a weighted absolute error.     

Multivariate survival functions characterized by constant product of mean remaining lives and hazard rates

Under the assumption that the products of multivariate mean remaining lives and hazard rates are the same constant, it is shown that the corresponding multivariate survival function belongs to one of three families: (1) multivariate Gumbel exponential distribution; (2) multivariate Lomax (Pareto type II) distribution; (3) multivariate rescaled Dirichlet distribution. This result is then used to derive another characterization of the latter two families based on the residual life distribution.     

Rough-and-ready assessment of the degree and importance of smoothing in functional estimation

In nonparametric estimation of functionals of a distribution, it may or may not be desirable, or indeed necessary, to introduce a degree of smoothing into this estimation. In this article, I describe a method for assessing, with just a little thought about the functional of interest, (i) whether smoothing is likely to prove worthwhile, and (ii) if so, roughly how much smoothing is appropriate (in order-of-magnitude terms). This rule-of-thumb is not guaranteed to be accurate nor does it give a complete answer to the smoothing problem. However, I have found it very useful over a number of years; many examples of its use, and limitations, are given.     

Nonparametric inference on multivariate versions of Blomqvist’s beta and related measures of tail dependence

We consider nonparametric estimation of multivariate versions of Blomqvist’s beta, also known as the medial correlation coefficient. For a two-dimensional population, the sample version of Blomqvist’s beta describes the proportion of data which fall into the first or third quadrant of a two-way contingency table with cutting points being the sample medians. Asymptotic normality and strong consistency of the estimators are established by means of the empirical copula process, imposing weak conditions on the copula. Though the asymptotic variance takes a complicated form, we are able to derive explicit formulas for large families of copulas. For the copulas of elliptically contoured distributions we obtain a variance stabilizing transformation which is similar to Fisher’s z-transformation. This allows for an explicit construction of asymptotic confidence bands used for hypothesis testing and eases the analysis of asymptotic efficiency. The computational complexity of estimating Blomqvist’s beta corresponds to the sample size n, which is lower than the complexity of most competing dependence measures.      

Tail index estimation based on linear combinations of intermediate order statistics

Based on linear combinations of intermediate order statistics, we introduce a new class of estimators for the exponent of a distribution function F with a regularly varying upper tail. We prove asymptotic normality and we make a comparison with existing proposals using the mean squared error as criterion.     

Improved estimation of extreme quantiles in the multivariate Lomax (Pareto II) distribution

Estimation of a quantile of the common marginal distribution in a multivariate Lomax (Pareto II) distribution with unknown location and scale parameters is considered. For quadratic loss and specified extreme quantiles, it is established that the best affine equivariant procedure is inadmissible by constructing a better estimator.     

温压战斗部爆炸冲击波毁伤小子样试验鉴定分析

针对温压战斗部爆炸冲击波毁伤的试验鉴定问题,提出冲击波毁伤的试验鉴定即为对毁伤参数所服从分布以及分布参数的统计推断问题,并进一步给出温压战斗部毁伤小子样试验鉴定的基本思路,重点对利用Bayes理论开展统计推断的方法进行了研究,介绍了两种验前分布的求取方法,并对Bayes估计理论进行了深入探讨,介绍了点估计和区间估计求取的具体方法。     

Nonparametric estimation in case of endogenous selection

This paper addresses the problem of estimation of a nonparametric regression function from selectively observed data when selection is endogenous. Our approach relies on independence between covariates and selection conditionally on potential outcomes. Endogeneity of regressors is also allowed for. In the exogenous and endogenous case, consistent two-step estimation procedures are proposed and their rates of convergence are derived. Pointwise asymptotic distribution of the estimators is established. In addition, bootstrap uniform confidence bands are obtained. Finite sample properties are illustrated in a Monte Carlo simulation study and an empirical illustration.     

The emperor's new clothes: a critique of the multivariate t regression model

Zellner (1976) proposed a regression model in which the data vector (or the error vector) is represented as a realization from the multivariate Student t distribution. This model has attracted considerable attention because it seems to broaden the usual Gaussian assumption to allow for heavier-tailed error distributions. A number of results in the literature indicate that the standard inference procedures for the Gaussian model remain appropriate under the broader distributional assumption, leading to claims of robustness of the standard methods. We show that, although mathematically the two models are different, for purposes of statistical inference they are indistinguishable. The empirical implications of the multivariate t model are precisely the same as those of the Gaussian model. Hence the suggestion of a broader distributional representation of the data is spurious, and the claims of robustness are misleading. These conclusions are reached from both frequentist and Bayesian perspectives.     

Interactive visualization of hierarchical clusters using MDS and MST

In this paper, we discuss interactively visualizing hierarchical clustering using multidimensional scaling (MDS) and the minimal spanning tree (MST). We can examine the sequential process leading to agglomerative or divisive hierarchical clustering, compare the different agglomerative methods, and detect influential observations better than is possible with dendrograms.     

基于分形理论谈安徽沿江环湖城市群的建立

简要介绍了分形理论的原理,运用分形理论,借助于罗特卡公式分析安徽省城市体系规模分布的分形特征,发现安徽省城市体系规模分布符合位序—规模率,存在城市体系规模分布过于均衡及首位城市、大城市规模过小的问题。提出建立安徽省沿江环湖城市群规模体系的发展对策。     

Minimax estimation under convex loss when the parameter interval is bounded

In the present paper families of truncated distributions with a Lebesgue density forx=(x ₁,...,x _n ) ε ℝ ⁿ are considered, wheref ₀:ℝ → (0, ∞) is a known continuous function andC _n (ϑ) denotes a normalization constant. The unknown truncation parameterϑ which is assumed to belong to a bounded parameter intervalΘ=[0,d] is to be estimated under a convex loss function. It is studied whether a two point prior and a corresponding Bayes estimator form a saddle point when the parameter interval is sufficiently small.     

An estimation model for replicating the rankings of the world competitiveness report

The World Competitiveness Report (WCR), a report annually produced by the Institute for Management Development, which is based in Switzerland, is a study that rates and ranks the competitiveness of a certain group of nations (OECD countries plus some newly emerging economies) and is a widely quoted report in the international media, especially by government and public leaders. Although some ideas as to the methodology used in the rating and ranking of countries are given, the details are not provided in the WCRs. Therefore, the methodology used in the WCRs is in large part unknown to the public.An intelligent use of the WCR requires a rather sound understanding of the methodology by its potential users; politicians, company executives, and public policy makers. The objective of this paper is to uncover and understand the methodology of the WCR through exact replications of its rankings at all levels of aggregation. An estimation model based on mathematical programming is used to replicate the WCR rankings.