Point Estimation Decision Theory
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Multiple-Change-Point Detection for Auto-Regressive Conditional Heteroscedastic Processes
J. R. Statist. Soc. B (2014) Multiple-change-point detection for auto-regressive conditional heteroscedastic processes P.Fryzlewicz London School of Economics and Political Science, UK and S. Subba Rao Texas A&M University, College Station, USA [Received March 2010. Final revision August 2013] Summary.The emergence of the recent financial crisis, during which markets frequently under- went changes in their statistical structure over a short period of time, illustrates the importance of non-stationary modelling in financial time series. Motivated by this observation, we propose a fast, well performing and theoretically tractable method for detecting multiple change points in the structure of an auto-regressive conditional heteroscedastic model for financial returns with piecewise constant parameter values. Our method, termed BASTA (binary segmentation for transformed auto-regressive conditional heteroscedasticity), proceeds in two stages: process transformation and binary segmentation. The process transformation decorrelates the original process and lightens its tails; the binary segmentation consistently estimates the change points. We propose and justify two particular transformations and use simulation to fine-tune their parameters as well as the threshold parameter for the binary segmentation stage. A compara- tive simulation study illustrates good performance in comparison with the state of the art, and the analysis of the Financial Times Stock Exchange FTSE 100 index reveals an interesting correspondence between the estimated change points -
Chapter 6 Point Estimation
Chapter 6 Point estimation This chapter deals with one of the central problems of statistics, that of using a sample of data to estimate the parameters for a hypothesized population model from which the data are assumed to arise. There are many approaches to estimation, and several of these are mentioned; but one method has very wide acceptance, the method of maximum likelihood. So far in the course, a major distinction has been drawn between sample quan- tities on the one hand-values calculated from data, such as the sample mean and the sample standard deviation-and corresponding population quantities on the other. The latter arise when a statistical model is assumed to be an ad- equate representation of the underlying variation in the population. Usually the model family is specified (binomial, Poisson, normal, . ) but the index- ing parameters (the binomial probability p, the Poisson mean p, the normal variance u2, . ) might be unknown-indeed, they usually will be unknown. Often one of the main reasons for collecting data is to estimate, from a sample, the value of the model parameter or parameters. Here are two examples. Example 6.1 Counts of females in queues In a paper about graphical methods for testing model quality, an experiment Jinkinson, R.A. and Slater, M. is described to count the number of females in each of 100 queues, all of length (1981) Critical discussion of a ten, at a London underground train station. graphical method for identifying discrete distributions. The The number of females in each queue could theoretically take any value be- Statistician, 30, 239-248. -
THE THEORY of POINT ESTIMATION a Point Estimator Uses the Information Available in a Sample to Obtain a Single Number That Estimates a Population Parameter
THE THEORY OF POINT ESTIMATION A point estimator uses the information available in a sample to obtain a single number that estimates a population parameter. There can be a variety of estimators of the same parameter that derive from different principles of estimation, and it is necessary to choose amongst them. Therefore, criteria are required that will indicate which are the acceptable estimators and which of these is the best in given circumstances. Often, the choice of an estimate is governed by practical considerations such as the ease of computation or the ready availability of a computer program. In what follows, we shall ignore such considerations in order to concentrate on the criteria that, ideally, we would wish to adopt. The circumstances that affect the choice are more complicated than one might imagine at first. Consider an estimator θˆn based on a sample of size n that purports to estimate a population parameter θ, and let θ˜n be any other estimator based on the same sample. If P (θ − c1 ≤ θˆn ≤ θ + c2) ≥ P (θ − c1 ≤ θ˜n ≤ θ + c2) for all values of c1,c2 > 0, then θˆn would be unequivocally the best estimator. However, it is possible to show that an estimator has such a property only in very restricted circumstances. Instead, we must evaluate an estimator against a list of partial criteria, of which we shall itemise the principal ones. The first is the criterion of unbiasedness. (a) θˆ is an unbiased estimator of the population parameter θ if E(θˆ)=θ. On it own, this is an insufficient criterion. -
Chapter 7 Point Estimation Method of Moments
Introduction Method of Moments Procedure Mark and Recapture Monsoon Rains Chapter 7 Point Estimation Method of Moments 1 / 23 Introduction Method of Moments Procedure Mark and Recapture Monsoon Rains Outline Introduction Classical Statistics Method of Moments Procedure Mark and Recapture Monsoon Rains 2 / 23 Introduction Method of Moments Procedure Mark and Recapture Monsoon Rains Parameter Estimation For parameter estimation, we consider X = (X1;:::; Xn), independent random variables chosen according to one of a family of probabilities Pθ where θ is element from the parameter spaceΘ. Based on our analysis, we choose an estimator θ^(X ). If the data x takes on the values x1; x2;:::; xn, then θ^(x1; x2;:::; xn) is called the estimate of θ. Thus we have three closely related objects. 1. θ - the parameter, an element of the parameter space, is a number or a vector. 2. θ^(x1; x2;:::; xn) - the estimate, is a number or a vector obtained by evaluating the estimator on the data x = (x1; x2;:::; xn). 3. θ^(X1;:::; Xn) - the estimator, is a random variable. We will analyze the distribution of this random variable to decide how well it performs in estimating θ. 3 / 23 Introduction Method of Moments Procedure Mark and Recapture Monsoon Rains Classical Statistics In classical statistics, the state of nature is assumed to be fixed, but unknown to us. Thus, one goal of estimation is to determine which of the Pθ is the source of the data. The estimate is a statistic θ^ : data ! Θ: For estimation procedures, the classical approach to statistics is based -
Point Estimation
6.1 Point Estimation What is an estimate? • Want to study a population • Ideally the knowledge of the distribution, • Some parameters of the population may be the first consideration: θ • Use the information from a sample to estimate • Estimator θˆ : defines the procedure (the formula) • Different estimators, each is a rv. Examples of estimators • want to estimate the mean µ of a normal dist • Possibilities of estimators: X sample mean X¯ = i • n ! • sample median • average of the smallest and the largest • 10% trimmed mean Which one is best (better)? • Error in estimation θˆ = θ + error of estimation • Error is random, depending on the sample • We would like to control the error • mean 0 (unbiased) • smallest variance Unbiased Estimator • estimator: a particular statistic • different estimators: some tend to over (under) estimate • unbiased: E(θˆ) = θ • difference is called the bias • examples: sample mean (yes), sample proportion (yes) if X is a binomial rv. • If possible, we should always choose an unbiased Sample Variance • Need an estimator for the variance • Natural candidate: sample variance • Why divide by n-1, not n? • Make sure the estimator is unbiased • We can not say the same thing for sample standard deviation Unbiased Estimators for Population Mean • several choices (make things complicated) • sample mean • if distribution is continuous and symmetric, sample median and any trimmed mean Minimum Variance • suppose we have two estimators, both unbiased, which one do we prefer? • pick the one with smaller variance • minimum variance unbiased estimator (MVUE) • for normal distribution, the sample mean X ¯ is the MVUE for the population mean. -
Statistical Inference a Work in Progress
Ronald Christensen Department of Mathematics and Statistics University of New Mexico Copyright c 2019 Statistical Inference A Work in Progress Springer v Seymour and Wes. Preface “But to us, probability is the very guide of life.” Joseph Butler (1736). The Analogy of Religion, Natural and Revealed, to the Constitution and Course of Nature, Introduction. https://www.loc.gov/resource/dcmsiabooks. analogyofreligio00butl_1/?sp=41 Normally, I wouldn’t put anything this incomplete on the internet but I wanted to make parts of it available to my Advanced Inference Class, and once it is up, you have lost control. Seymour Geisser was a mentor to Wes Johnson and me. He was Wes’s Ph.D. advisor. Near the end of his life, Seymour was finishing his 2005 book Modes of Parametric Statistical Inference and needed some help. Seymour asked Wes and Wes asked me. I had quite a few ideas for the book but then I discovered that Sey- mour hated anyone changing his prose. That was the end of my direct involvement. The first idea for this book was to revisit Seymour’s. (So far, that seems only to occur in Chapter 1.) Thinking about what Seymour was doing was the inspiration for me to think about what I had to say about statistical inference. And much of what I have to say is inspired by Seymour’s work as well as the work of my other professors at Min- nesota, notably Christopher Bingham, R. Dennis Cook, Somesh Das Gupta, Mor- ris L. Eaton, Stephen E. Fienberg, Narish Jain, F. Kinley Larntz, Frank B. -
An Economic Theory of Statistical Testing∗
An Economic Theory of Statistical Testing∗ Aleksey Tetenovy August 21, 2016 Abstract This paper models the use of statistical hypothesis testing in regulatory ap- proval. A privately informed agent proposes an innovation. Its approval is beneficial to the proponent, but potentially detrimental to the regulator. The proponent can conduct a costly clinical trial to persuade the regulator. I show that the regulator can screen out all ex-ante undesirable proponents by committing to use a simple statistical test. Its level is the ratio of the trial cost to the proponent's benefit from approval. In application to new drug approval, this level is around 15% for an average Phase III clinical trial. The practice of statistical hypothesis testing has been widely criticized across the many fields that use it. Examples of such criticism are Cohen (1994), Johnson (1999), Ziliak and McCloskey (2008), and Wasserstein and Lazar (2016). While conventional test levels of 5% and 1% are widely agreed upon, these numbers lack any substantive motivation. In spite of their arbitrary choice, they affect thousands of influential decisions. The hypothesis testing criterion has an unusual lexicographic structure: first, ensure that the ∗Thanks to Larbi Alaoui, V. Bhaskar, Ivan Canay, Toomas Hinnosaar, Chuck Manski, Jos´eLuis Montiel Olea, Ignacio Monz´on, Denis Nekipelov, Mario Pagliero, Amedeo Piolatto, Adam Rosen and Joel Sobel for helpful comments and suggestions. I have benefited from presenting this paper at the Cornell Conference on Partial Identification and Statistical Decisions, ASSET 2013, Econometric Society NAWM 2014, SCOCER 2015, Oxford, UCL, Pittsburgh, Northwestern, Michigan, Virginia, and Western. Part of this research was conducted at UCL/CeMMAP with financial support from the European Research Council grant ERC-2009-StG-240910-ROMETA. -
6 X 10.5 Long Title.P65
Cambridge University Press 978-0-521-68567-2 - Principles of Statistical Inference D. R. Cox Index More information Author index Aitchison, J., 131, 175, 199 Daniels, H.E., 132, 201 Akahira, M., 132, 199 Darmois, G., 28 Amari, S., 131, 199 Davies, R.B., 159, 201 Andersen, P.K., 159, 199 Davis, R.A., 16, 200 Anderson, T.W., 29, 132, 199 Davison, A.C., 14, 132, 200, 201 Anscombe, F.J., 94, 199 Dawid, A.P., 131, 201 Azzalini, A., 14, 160, 199 Day, N.E., 160, 200 Dempster, A.P., 132, 201 de Finetti, B., 196 Baddeley, A., 192, 199 de Groot, M., 196 Barnard, G.A., 28, 29, 63, 195, 199 Dunsmore, I.R., 175, 199 Barndorff-Nielsen, O.E., 28, 94, 131, 132, 199 Barnett, V., 14, 200 Edwards, A.W.F., 28, 195, 202 Barnett, V.D., 14, 200 Efron, B., 132, 202 Bartlett, M.S., 131, 132, 159, 200 Eguchi, S., 94, 201 Berger, J., 94, 200 Berger, R.L., 14, 200 Bernardo, J.M., 62, 94, 196, 200 Farewell, V., 160, 202 Besag, J.E., 16, 160, 200 Fisher, R.A., 27, 28, 40, 43, 44, 53, 55, Birnbaum, A., 62, 200 62, 63, 66, 93, 95, 132, 176, 190, Blackwell, D., 176 192, 194, 195, 202 Boole, G., 194 Fraser, D.A.S., 63, 202 Borgan, Ø., 199 Fridette, M., 175, 203 Box, G.E.P., 14, 62, 200 Brazzale, A.R., 132, 200 Breslow, N.E., 160, 200 Brockwell, P.J., 16, 200 Garthwaite, P.H., 93, 202 Butler, R.W., 132, 175, 200 Gauss, C.F., 15, 194 Geisser, S., 175, 202 Gill, R.D., 199 Carnap, R., 195 Godambe, V.P., 176, 202 Casella, G.C., 14, 29, 132, 200, 203 Good, I.J., 196 Christensen, R., 93, 200 Green, P.J., 160, 202 Cochran, W.G., 176, 192, 200 Greenland, S., 94, 202 Copas, J., 94, 201 Cox, D.R., 14, 16, 43, 63, 94, 131, 132, 159, 160, 192, 199, 201, 204 Hacking, I., 28, 202 Creasy, M.A., 44, 201 Hald, A., 194, 202 209 © Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-68567-2 - Principles of Statistical Inference D. -
Principles of Statistical Inference
Principles of Statistical Inference In this important book, D. R. Cox develops the key concepts of the theory of statistical inference, in particular describing and comparing the main ideas and controversies over foundational issues that have rumbled on for more than 200 years. Continuing a 60-year career of contribution to statistical thought, Professor Cox is ideally placed to give the comprehensive, balanced account of the field that is now needed. The careful comparison of frequentist and Bayesian approaches to inference allows readers to form their own opinion of the advantages and disadvantages. Two appendices give a brief historical overview and the author’s more personal assessment of the merits of different ideas. The content ranges from the traditional to the contemporary. While specific applications are not treated, the book is strongly motivated by applications across the sciences and associated technologies. The underlying mathematics is kept as elementary as feasible, though some previous knowledge of statistics is assumed. This book is for every serious user or student of statistics – in particular, for anyone wanting to understand the uncertainty inherent in conclusions from statistical analyses. Principles of Statistical Inference D.R. COX Nuffield College, Oxford CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521866736 © D. R. Cox 2006 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. -
Point Estimators Math 218, Mathematical Statistics
that has \central tendencies." Which of these is best, and why? Desirable criteria for point estimators. The MSE. Well, of course, we want our point estima- ^ Point estimators tor θ of the parameter θ to be close to θ. We can't Math 218, Mathematical Statistics expect them to be equal, of course, because of sam- D Joyce, Spring 2016 pling error. How should we measure how far off the random variable θ^ is from the unknown constant θ? The main job of statistics is to make inferences, One standard measure is what is called the mean specifically inferences bout parameters based on squared error, abbreviated MSE and defined by data from a sample. MSE(θ^) = E((θ^ − θ)2) We assume that a sample X1;:::;Xn comes from a particular distribution, called the population dis- the expected square of the error. If we have two dif- tribution, and although that particular distribution ferent estimators of θ, the one that has the smaller is not known, it is assumed that it is one of a fam- MSE is closer to the actual parameter (in some ily of distributions parametrized by one or more sense of \closer"). parameters. For example, if there are only two possible out- Variance and bias. The MSE of an estimator θ^ comes, then the distribution is a Bernoulli distri- can be split into two parts, the estimator's variance bution parametrized by one parameter p, the prob- and a \bias" term. We're familiar with the variance ability of success. of a random variable X; it's For another example, many measurements are as- sumed to be normally distributed, and for a normal Var(X) = σ2 = E((X − µ )2) = E(X2) − µ2 : distribution, there are two parameters µ and σ2. -
1 Introduction
Optimal Decision Rules for Weak GMM By Isaiah Andrews1 and Anna Mikusheva2 Abstract This paper studies optimal decision rules, including estimators and tests, for weakly identi- fied GMM models. We derive the limit experiment for weakly identified GMM, and propose a theoretically-motivated class of priors which give rise to quasi-Bayes decision rules as a limiting case. Together with results in the previous literature, this establishes desirable properties for the quasi-Bayes approach regardless of model identification status, and we recommend quasi-Bayes for settings where identification is a concern. We further propose weighted average power- optimal identification-robust frequentist tests and confidence sets, and prove a Bernstein-von Mises-type result for the quasi-Bayes posterior under weak identification. Keywords: Limit Experiment, Quasi Bayes, Weak Identification, Nonlinear GMM JEL Codes: C11, C12, C20 First draft: July 2020. This draft: July 2021. 1 Introduction Weak identification arises in a wide range of empirical settings. Weakly identified non- linear models have objective functions which are near-flat in certain directions, or have multiple near-optima. Standard asymptotic approximations break down when identifi- cation is weak, resulting in biased and non-normal estimates, as well as invalid standard errors and confidence sets. Further, existing optimality results do not apply in weakly 1Harvard Department of Economics, Littauer Center M18, Cambridge, MA 02138. Email ian- [email protected]. Support from the National Science Foundation under grant number 1654234, and from the Sloan Research Fellowship is gratefully acknowledged. 2Department of Economics, M.I.T., 50 Memorial Drive, E52-526, Cambridge, MA, 02142. Email: [email protected]. -
Multiple Hypothesis Testing and Multiple Outlier Identification Methods
Multiple Hypothesis Testing and Multiple Outlier Identification Methods A Thesis Submitted to the College of Graduate Studies and Research in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in the Department of Mathematics and Statistics University of Saskatchewan Saskatoon, Saskatchewan By Yaling Yin March 2010 c Yaling Yin , March 2010. All rights reserved. Permission to Use In presenting this thesis in partial fulfilment of the requirements for a Postgraduate degree from the University of Saskatchewan, I agree that the Libraries of this University may make it freely available for inspection. I further agree that permission for copying of this thesis in any manner, in whole or in part, for scholarly purposes may be granted by the professor or professors who supervised my thesis work or, in their absence, by the Head of the Department or the Dean of the College in which my thesis work was done. It is understood that any copying or publication or use of this thesis or parts thereof for financial gain shall not be allowed without my written permission. It is also understood that due recognition shall be given to me and to the University of Saskatchewan in any scholarly use which may be made of any material in my thesis. Requests for permission to copy or to make other use of material in this thesis in whole or part should be addressed to: Head of the Department of Mathematics and Statistics University of Saskatchewan Saskatoon, Saskatchewan Canada S7N 5E6 i Abstract Traditional multiple hypothesis testing procedures, such as that of Benjamini and Hochberg, fix an error rate and determine the corresponding rejection region.