Estimation of Common Location and Scale Parameters in Nonregular Cases Ahmad Razmpour Iowa State University
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Matrix Means and a Novel High-Dimensional Shrinkage Phenomenon
Matrix Means and a Novel High-Dimensional Shrinkage Phenomenon Asad Lodhia1, Keith Levin2, and Elizaveta Levina3 1Broad Institute of MIT and Harvard 2Department of Statistics, University of Wisconsin-Madison 3Department of Statistics, University of Michigan Abstract Many statistical settings call for estimating a population parameter, most typically the pop- ulation mean, based on a sample of matrices. The most natural estimate of the population mean is the arithmetic mean, but there are many other matrix means that may behave differently, especially in high dimensions. Here we consider the matrix harmonic mean as an alternative to the arithmetic matrix mean. We show that in certain high-dimensional regimes, the har- monic mean yields an improvement over the arithmetic mean in estimation error as measured by the operator norm. Counter-intuitively, studying the asymptotic behavior of these two matrix means in a spiked covariance estimation problem, we find that this improvement in operator norm error does not imply better recovery of the leading eigenvector. We also show that a Rao-Blackwellized version of the harmonic mean is equivalent to a linear shrinkage estimator studied previously in the high-dimensional covariance estimation literature, while applying a similar Rao-Blackwellization to regularized sample covariance matrices yields a novel nonlinear shrinkage estimator. Simulations complement the theoretical results, illustrating the conditions under which the harmonic matrix mean yields an empirically better estimate. 1 Introduction Matrix estimation problems arise in statistics in a number of areas, most prominently in covariance estimation, but also in network analysis, low-rank recovery and time series analysis. Typically, the focus is on estimating a matrix based on a single noisy realization of that matrix. -
A Comparison of Unbiased and Plottingposition Estimators of L
WATER RESOURCES RESEARCH, VOL. 31, NO. 8, PAGES 2019-2025, AUGUST 1995 A comparison of unbiased and plotting-position estimators of L moments J. R. M. Hosking and J. R. Wallis IBM ResearchDivision, T. J. Watson ResearchCenter, Yorktown Heights, New York Abstract. Plotting-positionestimators of L momentsand L moment ratios have several disadvantagescompared with the "unbiased"estimators. For generaluse, the "unbiased'? estimatorsshould be preferred. Plotting-positionestimators may still be usefulfor estimatingextreme upper tail quantilesin regional frequencyanalysis. Probability-Weighted Moments and L Moments •r+l-" (--1)r • P*r,k Olk '- E p *r,!•[J!•. Probability-weightedmoments of a randomvariable X with k=0 k=0 cumulativedistribution function F( ) and quantile function It is convenient to define dimensionless versions of L mo- x( ) were definedby Greenwoodet al. [1979]to be the quan- tities ments;this is achievedby dividingthe higher-orderL moments by the scale measure h2. The L moment ratios •'r, r = 3, Mp,ra= E[XP{F(X)}r{1- F(X)} s] 4, '", are definedby ßr-" •r/•2 ß {X(u)}PUr(1 -- U)s du. L momentratios measure the shapeof a distributionindepen- dently of its scaleof measurement.The ratios *3 ("L skew- ness")and *4 ("L kurtosis")are nowwidely used as measures Particularlyuseful specialcases are the probability-weighted of skewnessand kurtosis,respectively [e.g., Schaefer,1990; moments Pilon and Adamowski,1992; Royston,1992; Stedingeret al., 1992; Vogeland Fennessey,1993]. 12•r= M1,0, r = •01 (1 - u)rx(u) du, Estimators Given an ordered sample of size n, Xl: n • X2:n • ''' • urx(u) du. X.... there are two establishedways of estimatingthe proba- /3r--- Ml,r, 0 =f01 bility-weightedmoments and L moments of the distribution from whichthe samplewas drawn. -
The Asymmetric T-Copula with Individual Degrees of Freedom
The Asymmetric t-Copula with Individual Degrees of Freedom d-fine GmbH Christ Church University of Oxford A thesis submitted for the degree of MSc in Mathematical Finance Michaelmas 2012 Abstract This thesis investigates asymmetric dependence structures of multivariate asset returns. Evidence of such asymmetry for equity returns has been reported in the literature. In order to model the dependence structure, a new t-copula approach is proposed called the skewed t-copula with individ- ual degrees of freedom (SID t-copula). This copula provides the flexibility to assign an individual degree-of-freedom parameter and an individual skewness parameter to each asset in a multivariate setting. Applying this approach to GARCH residuals of bivariate equity index return data and using maximum likelihood estimation, we find significant asymmetry. By means of the Akaike information criterion, it is demonstrated that the SID t-copula provides the best model for the market data compared to other copula approaches without explicit asymmetry parameters. In addition, it yields a better fit than the conventional skewed t-copula with a single degree-of-freedom parameter. In a model impact study, we analyse the errors which can occur when mod- elling asymmetric multivariate SID-t returns with the symmetric multi- variate Gauss or standard t-distribution. The comparison is done in terms of the risk measures value-at-risk and expected shortfall. We find large deviations between the modelled and the true VaR/ES of a spread posi- tion composed of asymmetrically distributed risk factors. Going from the bivariate case to a larger number of risk factors, the model errors increase. -
A Bayesian Hierarchical Spatial Copula Model: an Application to Extreme Temperatures in Extremadura (Spain)
atmosphere Article A Bayesian Hierarchical Spatial Copula Model: An Application to Extreme Temperatures in Extremadura (Spain) J. Agustín García 1,† , Mario M. Pizarro 2,*,† , F. Javier Acero 1,† and M. Isabel Parra 2,† 1 Departamento de Física, Universidad de Extremadura, Avenida de Elvas, 06006 Badajoz, Spain; [email protected] (J.A.G.); [email protected] (F.J.A.) 2 Departamento de Matemáticas, Universidad de Extremadura, Avenida de Elvas, 06006 Badajoz, Spain; [email protected] * Correspondence: [email protected] † These authors contributed equally to this work. Abstract: A Bayesian hierarchical framework with a Gaussian copula and a generalized extreme value (GEV) marginal distribution is proposed for the description of spatial dependencies in data. This spatial copula model was applied to extreme summer temperatures over the Extremadura Region, in the southwest of Spain, during the period 1980–2015, and compared with the spatial noncopula model. The Bayesian hierarchical model was implemented with a Monte Carlo Markov Chain (MCMC) method that allows the distribution of the model’s parameters to be estimated. The results show the GEV distribution’s shape parameter to take constant negative values, the location parameter to be altitude dependent, and the scale parameter values to be concentrated around the same value throughout the region. Further, the spatial copula model chosen presents lower deviance information criterion (DIC) values when spatial distributions are assumed for the GEV distribution’s Citation: García, J.A.; Pizarro, M.M.; location and scale parameters than when the scale parameter is taken to be constant over the region. Acero, F.J.; Parra, M.I. A Bayesian Hierarchical Spatial Copula Model: Keywords: Bayesian hierarchical model; extreme temperature; Gaussian copula; generalized extreme An Application to Extreme value distribution Temperatures in Extremadura (Spain). -
Procedures for Estimation of Weibull Parameters James W
United States Department of Agriculture Procedures for Estimation of Weibull Parameters James W. Evans David E. Kretschmann David W. Green Forest Forest Products General Technical Report February Service Laboratory FPL–GTR–264 2019 Abstract Contents The primary purpose of this publication is to provide an 1 Introduction .................................................................. 1 overview of the information in the statistical literature on 2 Background .................................................................. 1 the different methods developed for fitting a Weibull distribution to an uncensored set of data and on any 3 Estimation Procedures .................................................. 1 comparisons between methods that have been studied in the 4 Historical Comparisons of Individual statistics literature. This should help the person using a Estimator Types ........................................................ 8 Weibull distribution to represent a data set realize some advantages and disadvantages of some basic methods. It 5 Other Methods of Estimating Parameters of should also help both in evaluating other studies using the Weibull Distribution .......................................... 11 different methods of Weibull parameter estimation and in 6 Discussion .................................................................. 12 discussions on American Society for Testing and Materials Standard D5457, which appears to allow a choice for the 7 Conclusion ................................................................ -
Location-Scale Distributions
Location–Scale Distributions Linear Estimation and Probability Plotting Using MATLAB Horst Rinne Copyright: Prof. em. Dr. Horst Rinne Department of Economics and Management Science Justus–Liebig–University, Giessen, Germany Contents Preface VII List of Figures IX List of Tables XII 1 The family of location–scale distributions 1 1.1 Properties of location–scale distributions . 1 1.2 Genuine location–scale distributions — A short listing . 5 1.3 Distributions transformable to location–scale type . 11 2 Order statistics 18 2.1 Distributional concepts . 18 2.2 Moments of order statistics . 21 2.2.1 Definitions and basic formulas . 21 2.2.2 Identities, recurrence relations and approximations . 26 2.3 Functions of order statistics . 32 3 Statistical graphics 36 3.1 Some historical remarks . 36 3.2 The role of graphical methods in statistics . 38 3.2.1 Graphical versus numerical techniques . 38 3.2.2 Manipulation with graphs and graphical perception . 39 3.2.3 Graphical displays in statistics . 41 3.3 Distribution assessment by graphs . 43 3.3.1 PP–plots and QQ–plots . 43 3.3.2 Probability paper and plotting positions . 47 3.3.3 Hazard plot . 54 3.3.4 TTT–plot . 56 4 Linear estimation — Theory and methods 59 4.1 Types of sampling data . 59 IV Contents 4.2 Estimators based on moments of order statistics . 63 4.2.1 GLS estimators . 64 4.2.1.1 GLS for a general location–scale distribution . 65 4.2.1.2 GLS for a symmetric location–scale distribution . 71 4.2.1.3 GLS and censored samples . -
Nearly Weighted Risk Minimal Unbiased Estimation✩ ∗ Ulrich K
Journal of Econometrics 209 (2019) 18–34 Contents lists available at ScienceDirect Journal of Econometrics journal homepage: www.elsevier.com/locate/jeconom Nearly weighted risk minimal unbiased estimationI ∗ Ulrich K. Müller a, , Yulong Wang b a Economics Department, Princeton University, United States b Economics Department, Syracuse University, United States article info a b s t r a c t Article history: Consider a small-sample parametric estimation problem, such as the estimation of the Received 26 July 2017 coefficient in a Gaussian AR(1). We develop a numerical algorithm that determines an Received in revised form 7 August 2018 estimator that is nearly (mean or median) unbiased, and among all such estimators, comes Accepted 27 November 2018 close to minimizing a weighted average risk criterion. We also apply our generic approach Available online 18 December 2018 to the median unbiased estimation of the degree of time variation in a Gaussian local-level JEL classification: model, and to a quantile unbiased point forecast for a Gaussian AR(1) process. C13, C22 ' 2018 Elsevier B.V. All rights reserved. Keywords: Mean bias Median bias Autoregression Quantile unbiased forecast 1. Introduction Competing estimators are typically evaluated by their bias and risk properties, such as their mean bias and mean-squared error, or their median bias. Often estimators have no known small sample optimality. What is more, if the estimation problem does not reduce to a Gaussian shift experiment even asymptotically, then in many cases, no optimality -
Lecture 22 - 11/19/2019 Lecture 22: Robust Location Estimation Lecturer: Jiantao Jiao Scribe: Vignesh Subramanian
EE290 Mathematics of Data Science Lecture 22 - 11/19/2019 Lecture 22: Robust Location Estimation Lecturer: Jiantao Jiao Scribe: Vignesh Subramanian In this lecture, we get a historical perspective into the robust estimation problem and discuss Huber's work [1] for robust estimation of a location parameter. The Huber loss function is given by, ( 1 2 2 t ; jtj ≤ k ρHuber(t) = 1 2 : (1) k jtj − 2 k ; jtj > k Here k is a parameter and the idea behind the loss function is to penalize outliers (beyond k) linearly instead of quadratically. Figure 1 shows the Huber loss function for k = 1. In this lecture we will get an intuitive 1 2 Figure 1: The green line plots the Huber-loss function for k = 1, and the blue line plots the quadratic function 2 t . understanding for the reasons behind the particular form of this function, quadratic in interior, linear in exterior and convex and will see that this loss function is optimal for one dimensional robust mean estimation for Gaussian location model. First we describe the problem setting. 1 Problem Setting Suppose we observe X1;X2;:::;Xn where Xi − µ ∼ F 2 F are i.i.d. Here, F = fF j F = (1 − )G + H; H 2 Mg; (2) where G 2 M is some fixed distribution function which is usually assumed to have zero mean, and M denotes the space of all probability measures. This describes the corruption model where the observed distribution is a convex combination of the true distribution G(x) and an arbitrary corruption distribution H. -
Robust Estimator of Location Parameter1)
The Korean Communications in Statistics Vol. 11 No. 1, 2004 pp. 153-160 Robust Estimator of Location Parameter1) Dongryeon Park2) Abstract In recent years, the size of data set which we usually handle is enormous, so a lot of outliers could be included in data set. Therefore the robust procedures that automatically handle outliers become very importance issue. We consider the robust estimation problem of location parameter in the univariate case. In this paper, we propose a new method for defining robustness weights for the weighted mean based on the median distance of observations and compare its performance with several existing robust estimators by a simulation study. It turns out that the proposed method is very competitive. Keywords : Location parameter, Median distance, Robust estimator, Robustness weight 1. Introduction It is often assumed in the social sciences that data conform to a normal distribution. When estimating the location of a normal distribution, a sample mean is well known to be the best estimator according to many criteria. However, numerous studies (Hample et al., 1986; Hoaglin et al., 1976; Rousseeuw and Leroy, 1987) have strongly questioned normal assumption in real world data sets. In fact, a few large errors might infect the data set so that the tails of the underlying distribution are heavier than those of the normal distribution. In this situation, the sample mean is no longer a good estimate for the center of symmetry because all the observations equally contribute to the value of the sample mean, so the estimators which are insensitive to extreme values should have better performance. -
Maximum Likelihood Characterization of Distributions
Bernoulli 20(2), 2014, 775–802 DOI: 10.3150/13-BEJ506 Maximum likelihood characterization of distributions MITIA DUERINCKX1,*, CHRISTOPHE LEY1,** and YVIK SWAN2 1D´epartement de Math´ematique, Universit´elibre de Bruxelles, Campus Plaine CP 210, Boule- vard du triomphe, B-1050 Bruxelles, Belgique. * ** E-mail: [email protected]; [email protected] 2 Universit´edu Luxembourg, Campus Kirchberg, UR en math´ematiques, 6, rue Richard Couden- hove-Kalergi, L-1359 Luxembourg, Grand-Duch´ede Luxembourg. E-mail: [email protected] A famous characterization theorem due to C.F. Gauss states that the maximum likelihood es- timator (MLE) of the parameter in a location family is the sample mean for all samples of all sample sizes if and only if the family is Gaussian. There exist many extensions of this re- sult in diverse directions, most of them focussing on location and scale families. In this paper, we propose a unified treatment of this literature by providing general MLE characterization theorems for one-parameter group families (with particular attention on location and scale pa- rameters). In doing so, we provide tools for determining whether or not a given such family is MLE-characterizable, and, in case it is, we define the fundamental concept of minimal necessary sample size at which a given characterization holds. Many of the cornerstone references on this topic are retrieved and discussed in the light of our findings, and several new characterization theorems are provided. Of particular interest is that one part of our work, namely the intro- duction of so-called equivalence classes for MLE characterizations, is a modernized version of Daniel Bernoulli’s viewpoint on maximum likelihood estimation. -
Parametric Forecasting of Value at Risk Using Heavy-Tailed Distribution
REVISTA INVESTIGACIÓN OPERACIONAL _____ Vol., 30, No.1, 32-39, 2009 DEPENDENCE BETWEEN VOLATILITY PERSISTENCE, KURTOSIS AND DEGREES OF FREEDOM Ante Rozga1 and Josip Arnerić, 2 Faculty of Economics, University of Split, Croatia ABSTRACT In this paper the dependence between volatility persistence, kurtosis and degrees of freedom from Student’s t-distribution will be presented in estimation alternative risk measures on simulated returns. As the most used measure of market risk is standard deviation of returns, i.e. volatility. However, based on volatility alternative risk measures can be estimated, for example Value-at-Risk (VaR). There are many methodologies for calculating VaR, but for simplicity they can be classified into parametric and nonparametric models. In category of parametric models the GARCH(p,q) model is used for modeling time-varying variance of returns. KEY WORDS: Value-at-Risk, GARCH(p,q), T-Student MSC 91B28 RESUMEN En este trabajo la dependencia de la persistencia de la volatilidad, kurtosis y grados de liberad de una Distribución T-Student será presentada como una alternativa para la estimación de medidas de riesgo en la simulación de los retornos. La medida más usada de riesgo de mercado es la desviación estándar de los retornos, i.e. volatilidad. Sin embargo, medidas alternativas de la volatilidad pueden ser estimadas, por ejemplo el Valor-al-Riesgo (Value-at-Risk, VaR). Existen muchas metodologías para calcular VaR, pero por simplicidad estas pueden ser clasificadas en modelos paramétricos y no paramétricos . En la categoría de modelos paramétricos el modelo GARCH(p,q) es usado para modelar la varianza de retornos que varían en el tiempo. -
Domination Over the Best Invariant Estimator Is, When Properly
DOCUMENT RESUME ED 394 999 TM 024 995 AUTHOR Brandwein, Ann Cohen; Strawderman, William E. TITLE James-Stein Estimation. Program Statistics Research, Technical Report No. 89-86. INSTITUTION Educational Testing Service, Princeton, N.J. REPORT NO ETS-RR-89-20 PUB DATE Apr 89 NOTE 46p. PUB TYPE Reports Evaluative/Feasibility (142) Statistical Data (110) EDRS PRICE MF01/PCO2 Plus Postage. DESCRIPTORS Equations (Mathematics); *Estimation (Mathematics); *Maximum Likelihood Statistics; *Statistical Distributions IDENTIFIERS *James Stein Estimation; Nonnormal Distributions; *Shrinkage ABSTRACT This paper presents an expository development of James-Stein estimation with substantial emphasis on exact results for nonnormal location models. The themes of the paper are: (1) the improvement possible over the best invariant estimator via shrinkage estimation is not surprising but expected from a variety of perspectives; (2) the amount of shrinkage allowable to preserve domination over the best invariant estimator is, when properly interpreted, relatively free from the assumption of normality; and (3) the potential savings in risk are substantial when accompanied by good quality prior information. Relatively, much less emphasis is placed on choosing a particular shrinkage estimator than on demonstrating that shrinkage should produce worthwhile gains in problems where the error distribution is spherically symmetric. In addition, such gains are relatively robust with respect to assumptions concerning distribution and loss. (Contains 1 figure and 53 references.)