Bias, Mean-Square Error, Relative Efficiency
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Lecture 12 Robust Estimation
Lecture 12 Robust Estimation Prof. Dr. Svetlozar Rachev Institute for Statistics and Mathematical Economics University of Karlsruhe Financial Econometrics, Summer Semester 2007 Prof. Dr. Svetlozar Rachev Institute for Statistics and MathematicalLecture Economics 12 Robust University Estimation of Karlsruhe Copyright These lecture-notes cannot be copied and/or distributed without permission. The material is based on the text-book: Financial Econometrics: From Basics to Advanced Modeling Techniques (Wiley-Finance, Frank J. Fabozzi Series) by Svetlozar T. Rachev, Stefan Mittnik, Frank Fabozzi, Sergio M. Focardi,Teo Jaˇsic`. Prof. Dr. Svetlozar Rachev Institute for Statistics and MathematicalLecture Economics 12 Robust University Estimation of Karlsruhe Outline I Robust statistics. I Robust estimators of regressions. I Illustration: robustness of the corporate bond yield spread model. Prof. Dr. Svetlozar Rachev Institute for Statistics and MathematicalLecture Economics 12 Robust University Estimation of Karlsruhe Robust Statistics I Robust statistics addresses the problem of making estimates that are insensitive to small changes in the basic assumptions of the statistical models employed. I The concepts and methods of robust statistics originated in the 1950s. However, the concepts of robust statistics had been used much earlier. I Robust statistics: 1. assesses the changes in estimates due to small changes in the basic assumptions; 2. creates new estimates that are insensitive to small changes in some of the assumptions. I Robust statistics is also useful to separate the contribution of the tails from the contribution of the body of the data. Prof. Dr. Svetlozar Rachev Institute for Statistics and MathematicalLecture Economics 12 Robust University Estimation of Karlsruhe Robust Statistics I Peter Huber observed, that robust, distribution-free, and nonparametrical actually are not closely related properties. -
Phase Transition Unbiased Estimation in High Dimensional Settings Arxiv
Phase Transition Unbiased Estimation in High Dimensional Settings St´ephaneGuerrier, Mucyo Karemera, Samuel Orso & Maria-Pia Victoria-Feser Research Center for Statistics, University of Geneva Abstract An important challenge in statistical analysis concerns the control of the finite sample bias of estimators. This problem is magnified in high dimensional settings where the number of variables p diverge with the sample size n. However, it is difficult to establish whether an estimator θ^ of θ0 is unbiased and the asymptotic ^ order of E[θ] − θ0 is commonly used instead. We introduce a new property to assess the bias, called phase transition unbiasedness, which is weaker than unbiasedness but stronger than asymptotic results. An estimator satisfying this property is such ^ ∗ that E[θ] − θ0 2 = 0, for all n greater than a finite sample size n . We propose a phase transition unbiased estimator by matching an initial estimator computed on the sample and on simulated data. It is computed using an algorithm which is shown to converge exponentially fast. The initial estimator is not required to be consistent and thus may be conveniently chosen for computational efficiency or for other properties. We demonstrate the consistency and the limiting distribution of the estimator in high dimension. Finally, we develop new estimators for logistic regression models, with and without random effects, that enjoy additional properties such as robustness to data contamination and to the problem of separability. arXiv:1907.11541v3 [math.ST] 1 Nov 2019 Keywords: Finite sample bias, Iterative bootstrap, Two-step estimators, Indirect inference, Robust estimation, Logistic regression. 1 1. Introduction An important challenge in statistical analysis concerns the control of the (finite sample) bias of estimators. -
Should We Think of a Different Median Estimator?
Comunicaciones en Estad´ıstica Junio 2014, Vol. 7, No. 1, pp. 11–17 Should we think of a different median estimator? ¿Debemos pensar en un estimator diferente para la mediana? Jorge Iv´an V´eleza Juan Carlos Correab [email protected] [email protected] Resumen La mediana, una de las medidas de tendencia central m´as populares y utilizadas en la pr´actica, es el valor num´erico que separa los datos en dos partes iguales. A pesar de su popularidad y aplicaciones, muchos desconocen la existencia de dife- rentes expresiones para calcular este par´ametro. A continuaci´on se presentan los resultados de un estudio de simulaci´on en el que se comparan el estimador cl´asi- co y el propuesto por Harrell & Davis (1982). Mostramos que, comparado con el estimador de Harrell–Davis, el estimador cl´asico no tiene un buen desempe˜no pa- ra tama˜nos de muestra peque˜nos. Basados en los resultados obtenidos, se sugiere promover la utilizaci´on de un mejor estimador para la mediana. Palabras clave: mediana, cuantiles, estimador Harrell-Davis, simulaci´on estad´ısti- ca. Abstract The median, one of the most popular measures of central tendency widely-used in the statistical practice, is often described as the numerical value separating the higher half of the sample from the lower half. Despite its popularity and applica- tions, many people are not aware of the existence of several formulas to estimate this parameter. We present the results of a simulation study comparing the classic and the Harrell-Davis (Harrell & Davis 1982) estimators of the median for eight continuous statistical distributions. -
A Joint Central Limit Theorem for the Sample Mean and Regenerative Variance Estimator*
Annals of Operations Research 8(1987)41-55 41 A JOINT CENTRAL LIMIT THEOREM FOR THE SAMPLE MEAN AND REGENERATIVE VARIANCE ESTIMATOR* P.W. GLYNN Department of Industrial Engineering, University of Wisconsin, Madison, W1 53706, USA and D.L. IGLEHART Department of Operations Research, Stanford University, Stanford, CA 94305, USA Abstract Let { V(k) : k t> 1 } be a sequence of independent, identically distributed random vectors in R d with mean vector ~. The mapping g is a twice differentiable mapping from R d to R 1. Set r = g(~). A bivariate central limit theorem is proved involving a point estimator for r and the asymptotic variance of this point estimate. This result can be applied immediately to the ratio estimation problem that arises in regenerative simulation. Numerical examples show that the variance of the regenerative variance estimator is not necessarily minimized by using the "return state" with the smallest expected cycle length. Keywords and phrases Bivariate central limit theorem,j oint limit distribution, ratio estimation, regenerative simulation, simulation output analysis. 1. Introduction Let X = {X(t) : t I> 0 } be a (possibly) delayed regenerative process with regeneration times 0 = T(- 1) ~< T(0) < T(1) < T(2) < .... To incorporate regenerative sequences {Xn: n I> 0 }, we pass to the continuous time process X = {X(t) : t/> 0}, where X(0 = X[t ] and [t] is the greatest integer less than or equal to t. Under quite general conditions (see Smith [7] ), *This research was supported by Army Research Office Contract DAAG29-84-K-0030. The first author was also supported by National Science Foundation Grant ECS-8404809 and the second author by National Science Foundation Grant MCS-8203483. -
Weak Instruments and Finite-Sample Bias
7 Weak instruments and finite-sample bias In this chapter, we consider the effect of weak instruments on instrumen- tal variable (IV) analyses. Weak instruments, which were introduced in Sec- tion 4.5.2, are those that do not explain a large proportion of the variation in the exposure, and so the statistical association between the IV and the expo- sure is not strong. This is of particular relevance in Mendelian randomization studies since the associations of genetic variants with exposures of interest are often weak. This chapter focuses on the impact of weak instruments on the bias and coverage of IV estimates. 7.1 Introduction Although IV techniques can be used to give asymptotically unbiased estimates of causal effects in the presence of confounding, these estimates suffer from bias when evaluated in finite samples [Nelson and Startz, 1990]. A weak instrument (or a weak IV) is still a valid IV, in that it satisfies the IV assumptions, and es- timates using the IV with an infinite sample size will be unbiased; but for any finite sample size, the average value of the IV estimator will be biased. This bias, known as weak instrument bias, is towards the observational confounded estimate. Its magnitude depends on the strength of association between the IV and the exposure, which is measured by the F statistic in the regression of the exposure on the IV [Bound et al., 1995]. In this chapter, we assume the context of ‘one-sample’ Mendelian randomization, in which evidence on the genetic variant, exposure, and outcome are taken on the same set of indi- viduals, rather than subsample (Section 8.5.2) or two-sample (Section 9.8.2) Mendelian randomization, in which genetic associations with the exposure and outcome are estimated in different sets of individuals (overlapping sets in subsample, non-overlapping sets in two-sample Mendelian randomization). -
STAT 830 the Basics of Nonparametric Models The
STAT 830 The basics of nonparametric models The Empirical Distribution Function { EDF The most common interpretation of probability is that the probability of an event is the long run relative frequency of that event when the basic experiment is repeated over and over independently. So, for instance, if X is a random variable then P (X ≤ x) should be the fraction of X values which turn out to be no more than x in a long sequence of trials. In general an empirical probability or expected value is just such a fraction or average computed from the data. To make this precise, suppose we have a sample X1;:::;Xn of iid real valued random variables. Then we make the following definitions: Definition: The empirical distribution function, or EDF, is n 1 X F^ (x) = 1(X ≤ x): n n i i=1 This is a cumulative distribution function. It is an estimate of F , the cdf of the Xs. People also speak of the empirical distribution of the sample: n 1 X P^(A) = 1(X 2 A) n i i=1 ^ This is the probability distribution whose cdf is Fn. ^ Now we consider the qualities of Fn as an estimate, the standard error of the estimate, the estimated standard error, confidence intervals, simultaneous confidence intervals and so on. To begin with we describe the best known summaries of the quality of an estimator: bias, variance, mean squared error and root mean squared error. Bias, variance, MSE and RMSE There are many ways to judge the quality of estimates of a parameter φ; all of them focus on the distribution of the estimation error φ^−φ. -
1 Estimation and Beyond in the Bayes Universe
ISyE8843A, Brani Vidakovic Handout 7 1 Estimation and Beyond in the Bayes Universe. 1.1 Estimation No Bayes estimate can be unbiased but Bayesians are not upset! No Bayes estimate with respect to the squared error loss can be unbiased, except in a trivial case when its Bayes’ risk is 0. Suppose that for a proper prior ¼ the Bayes estimator ±¼(X) is unbiased, Xjθ (8θ)E ±¼(X) = θ: This implies that the Bayes risk is 0. The Bayes risk of ±¼(X) can be calculated as repeated expectation in two ways, θ Xjθ 2 X θjX 2 r(¼; ±¼) = E E (θ ¡ ±¼(X)) = E E (θ ¡ ±¼(X)) : Thus, conveniently choosing either EθEXjθ or EX EθjX and using the properties of conditional expectation we have, θ Xjθ 2 θ Xjθ X θjX X θjX 2 r(¼; ±¼) = E E θ ¡ E E θ±¼(X) ¡ E E θ±¼(X) + E E ±¼(X) θ Xjθ 2 θ Xjθ X θjX X θjX 2 = E E θ ¡ E θ[E ±¼(X)] ¡ E ±¼(X)E θ + E E ±¼(X) θ Xjθ 2 θ X X θjX 2 = E E θ ¡ E θ ¢ θ ¡ E ±¼(X)±¼(X) + E E ±¼(X) = 0: Bayesians are not upset. To check for its unbiasedness, the Bayes estimator is averaged with respect to the model measure (Xjθ), and one of the Bayesian commandments is: Thou shall not average with respect to sample space, unless you have Bayesian design in mind. Even frequentist agree that insisting on unbiasedness can lead to bad estimators, and that in their quest to minimize the risk by trading off between variance and bias-squared a small dosage of bias can help. -
11. Parameter Estimation
11. Parameter Estimation Chris Piech and Mehran Sahami May 2017 We have learned many different distributions for random variables and all of those distributions had parame- ters: the numbers that you provide as input when you define a random variable. So far when we were working with random variables, we either were explicitly told the values of the parameters, or, we could divine the values by understanding the process that was generating the random variables. What if we don’t know the values of the parameters and we can’t estimate them from our own expert knowl- edge? What if instead of knowing the random variables, we have a lot of examples of data generated with the same underlying distribution? In this chapter we are going to learn formal ways of estimating parameters from data. These ideas are critical for artificial intelligence. Almost all modern machine learning algorithms work like this: (1) specify a probabilistic model that has parameters. (2) Learn the value of those parameters from data. Parameters Before we dive into parameter estimation, first let’s revisit the concept of parameters. Given a model, the parameters are the numbers that yield the actual distribution. In the case of a Bernoulli random variable, the single parameter was the value p. In the case of a Uniform random variable, the parameters are the a and b values that define the min and max value. Here is a list of random variables and the corresponding parameters. From now on, we are going to use the notation q to be a vector of all the parameters: Distribution Parameters Bernoulli(p) q = p Poisson(l) q = l Uniform(a,b) q = (a;b) Normal(m;s 2) q = (m;s 2) Y = mX + b q = (m;b) In the real world often you don’t know the “true” parameters, but you get to observe data. -
Bias in Parametric Estimation: Reduction and Useful Side-Effects
Bias in parametric estimation: reduction and useful side-effects Ioannis Kosmidis Department of Statistical Science, University College London August 30, 2018 Abstract The bias of an estimator is defined as the difference of its expected value from the parameter to be estimated, where the expectation is with respect to the model. Loosely speaking, small bias reflects the desire that if an experiment is repeated in- definitely then the average of all the resultant estimates will be close to the parameter value that is estimated. The current paper is a review of the still-expanding repository of methods that have been developed to reduce bias in the estimation of parametric models. The review provides a unifying framework where all those methods are seen as attempts to approximate the solution of a simple estimating equation. Of partic- ular focus is the maximum likelihood estimator, which despite being asymptotically unbiased under the usual regularity conditions, has finite-sample bias that can re- sult in significant loss of performance of standard inferential procedures. An informal comparison of the methods is made revealing some useful practical side-effects in the estimation of popular models in practice including: i) shrinkage of the estimators in binomial and multinomial regression models that guarantees finiteness even in cases of data separation where the maximum likelihood estimator is infinite, and ii) inferential benefits for models that require the estimation of dispersion or precision parameters. Keywords: jackknife/bootstrap, indirect inference, penalized likelihood, infinite es- timates, separation in models with categorical responses 1 Impact of bias in estimation By its definition, bias necessarily depends on how the model is written in terms of its parameters and this dependence makes it not a strong statistical principle in terms of evaluating the performance of estimators; for example, unbiasedness of the familiar sample variance S2 as an estimator of σ2 does not deliver an unbiased estimator of σ itself. -
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 -
Bayes Estimator Recap - Example
Recap Bayes Risk Consistency Summary Recap Bayes Risk Consistency Summary . Last Lecture . Biostatistics 602 - Statistical Inference Lecture 16 • What is a Bayes Estimator? Evaluation of Bayes Estimator • Is a Bayes Estimator the best unbiased estimator? . • Compared to other estimators, what are advantages of Bayes Estimator? Hyun Min Kang • What is conjugate family? • What are the conjugate families of Binomial, Poisson, and Normal distribution? March 14th, 2013 Hyun Min Kang Biostatistics 602 - Lecture 16 March 14th, 2013 1 / 28 Hyun Min Kang Biostatistics 602 - Lecture 16 March 14th, 2013 2 / 28 Recap Bayes Risk Consistency Summary Recap Bayes Risk Consistency Summary . Recap - Bayes Estimator Recap - Example • θ : parameter • π(θ) : prior distribution i.i.d. • X1, , Xn Bernoulli(p) • X θ fX(x θ) : sampling distribution ··· ∼ | ∼ | • π(p) Beta(α, β) • Posterior distribution of θ x ∼ | • α Prior guess : pˆ = α+β . Joint fX(x θ)π(θ) π(θ x) = = | • Posterior distribution : π(p x) Beta( xi + α, n xi + β) | Marginal m(x) | ∼ − • Bayes estimator ∑ ∑ m(x) = f(x θ)π(θ)dθ (Bayes’ rule) | α + x x n α α + β ∫ pˆ = i = i + α + β + n n α + β + n α + β α + β + n • Bayes Estimator of θ is ∑ ∑ E(θ x) = θπ(θ x)dθ | θ Ω | ∫ ∈ Hyun Min Kang Biostatistics 602 - Lecture 16 March 14th, 2013 3 / 28 Hyun Min Kang Biostatistics 602 - Lecture 16 March 14th, 2013 4 / 28 Recap Bayes Risk Consistency Summary Recap Bayes Risk Consistency Summary . Loss Function Optimality Loss Function Let L(θ, θˆ) be a function of θ and θˆ. -
Chapter 4 Parameter Estimation
Chapter 4 Parameter Estimation Thus far we have concerned ourselves primarily with probability theory: what events may occur with what probabilities, given a model family and choices for the parameters. This is useful only in the case where we know the precise model family and parameter values for the situation of interest. But this is the exception, not the rule, for both scientific inquiry and human learning & inference. Most of the time, we are in the situation of processing data whose generative source we are uncertain about. In Chapter 2 we briefly covered elemen- tary density estimation, using relative-frequency estimation, histograms and kernel density estimation. In this chapter we delve more deeply into the theory of probability density es- timation, focusing on inference within parametric families of probability distributions (see discussion in Section 2.11.2). We start with some important properties of estimators, then turn to basic frequentist parameter estimation (maximum-likelihood estimation and correc- tions for bias), and finally basic Bayesian parameter estimation. 4.1 Introduction Consider the situation of the first exposure of a native speaker of American English to an English variety with which she has no experience (e.g., Singaporean English), and the problem of inferring the probability of use of active versus passive voice in this variety with a simple transitive verb such as hit: (1) The ball hit the window. (Active) (2) The window was hit by the ball. (Passive) There is ample evidence that this probability is contingent