Exact Statistical Inferences for Functions of Parameters of the Log-Gamma Distribution
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Ordinary Least Squares 1 Ordinary Least Squares
Ordinary least squares 1 Ordinary least squares In statistics, ordinary least squares (OLS) or linear least squares is a method for estimating the unknown parameters in a linear regression model. This method minimizes the sum of squared vertical distances between the observed responses in the dataset and the responses predicted by the linear approximation. The resulting estimator can be expressed by a simple formula, especially in the case of a single regressor on the right-hand side. The OLS estimator is consistent when the regressors are exogenous and there is no Okun's law in macroeconomics states that in an economy the GDP growth should multicollinearity, and optimal in the class of depend linearly on the changes in the unemployment rate. Here the ordinary least squares method is used to construct the regression line describing this law. linear unbiased estimators when the errors are homoscedastic and serially uncorrelated. Under these conditions, the method of OLS provides minimum-variance mean-unbiased estimation when the errors have finite variances. Under the additional assumption that the errors be normally distributed, OLS is the maximum likelihood estimator. OLS is used in economics (econometrics) and electrical engineering (control theory and signal processing), among many areas of application. Linear model Suppose the data consists of n observations { y , x } . Each observation includes a scalar response y and a i i i vector of predictors (or regressors) x . In a linear regression model the response variable is a linear function of the i regressors: where β is a p×1 vector of unknown parameters; ε 's are unobserved scalar random variables (errors) which account i for the discrepancy between the actually observed responses y and the "predicted outcomes" x′ β; and ′ denotes i i matrix transpose, so that x′ β is the dot product between the vectors x and β. -
Quantum State Estimation with Nuisance Parameters 2
Quantum state estimation with nuisance parameters Jun Suzuki E-mail: [email protected] Graduate School of Informatics and Engineering, The University of Electro-Communications, Tokyo, 182-8585 Japan Yuxiang Yang E-mail: [email protected] Institute for Theoretical Physics, ETH Z¨urich, 8093 Z¨urich, Switzerland Masahito Hayashi E-mail: [email protected] Graduate School of Mathematics, Nagoya University, Nagoya, 464-8602, Japan Shenzhen Institute for Quantum Science and Engineering, Southern University of Science and Technology, Shenzhen, 518055, China Center for Quantum Computing, Peng Cheng Laboratory, Shenzhen 518000, China Centre for Quantum Technologies, National University of Singapore, 117542, Singapore Abstract. In parameter estimation, nuisance parameters refer to parameters that are not of interest but nevertheless affect the precision of estimating other parameters of interest. For instance, the strength of noises in a probe can be regarded as a nuisance parameter. Despite its long history in classical statistics, the nuisance parameter problem in quantum estimation remains largely unexplored. The goal of this article is to provide a systematic review of quantum estimation in the presence of nuisance parameters, and to supply those who work in quantum tomography and quantum arXiv:1911.02790v3 [quant-ph] 29 Jun 2020 metrology with tools to tackle relevant problems. After an introduction to the nuisance parameter and quantum estimation theory, we explicitly formulate the problem of quantum state estimation with nuisance parameters. We extend quantum Cram´er-Rao bounds to the nuisance parameter case and provide a parameter orthogonalization tool to separate the nuisance parameters from the parameters of interest. In particular, we put more focus on the case of one-parameter estimation in the presence of nuisance parameters, as it is most frequently encountered in practice. -
Nearly Optimal Tests When a Nuisance Parameter Is Present
Nearly Optimal Tests when a Nuisance Parameter is Present Under the Null Hypothesis Graham Elliott Ulrich K. Müller Mark W. Watson UCSD Princeton University Princeton University and NBER January 2012 (Revised July 2013) Abstract This paper considers nonstandard hypothesis testing problems that involve a nui- sance parameter. We establish an upper bound on the weighted average power of all valid tests, and develop a numerical algorithm that determines a feasible test with power close to the bound. The approach is illustrated in six applications: inference about a linear regression coe¢ cient when the sign of a control coe¢ cient is known; small sample inference about the di¤erence in means from two independent Gaussian sam- ples from populations with potentially di¤erent variances; inference about the break date in structural break models with moderate break magnitude; predictability tests when the regressor is highly persistent; inference about an interval identi…ed parame- ter; and inference about a linear regression coe¢ cient when the necessity of a control is in doubt. JEL classi…cation: C12; C21; C22 Keywords: Least favorable distribution, composite hypothesis, maximin tests This research was funded in part by NSF grant SES-0751056 (Müller). The paper supersedes the corresponding sections of the previous working papers "Low-Frequency Robust Cointegration Testing" and "Pre and Post Break Parameter Inference" by the same set of authors. We thank Lars Hansen, Andriy Norets, Andres Santos, two referees, the participants of the AMES 2011 meeting at Korea University, of the 2011 Greater New York Metropolitan Econometrics Colloquium, and of a workshop at USC for helpful comments. -
Composite Hypothesis, Nuisance Parameters’
Practical Statistics – part II ‘Composite hypothesis, Nuisance Parameters’ W. Verkerke (NIKHEF) Wouter Verkerke, NIKHEF Summary of yesterday, plan for today • Start with basics, gradually build up to complexity of Statistical tests with simple hypotheses for counting data “What do we mean with probabilities” Statistical tests with simple hypotheses for distributions “p-values” “Optimal event selection & Hypothesis testing as basis for event selection machine learning” Composite hypotheses (with parameters) for distributions “Confidence intervals, Maximum Likelihood” Statistical inference with nuisance parameters “Fitting the background” Response functions and subsidiary measurements “Profile Likelihood fits” Introduce concept of composite hypotheses • In most cases in physics, a hypothesis is not “simple”, but “composite” • Composite hypothesis = Any hypothesis which does not specify the population distribution completely • Example: counting experiment with signal and background, that leaves signal expectation unspecified Simple hypothesis ~ s=0 With b=5 s=5 s=10 s=15 Composite hypothesis (My) notation convention: all symbols with ~ are constants Wouter Verkerke, NIKHEF A common convention in the meaning of model parameters • A common convention is to recast signal rate parameters into a normalized form (e.g. w.r.t the Standard Model rate) Simple hypothesis ~ s=0 With b=5 s=5 s=10 s=15 Composite hypothesis ‘Universal’ parameter interpretation makes it easier to work with your models μ=0 no signal Composite hypothesis μ=1 expected signal -
Likelihood Inference in the Presence of Nuisance Parameters N
PHYSTAT2003, SLAC, Stanford, California, September 8-11, 2003 Likelihood Inference in the Presence of Nuisance Parameters N. Reid, D.A.S. Fraser Department of Statistics, University of Toronto, Toronto Canada M5S 3G3 We describe some recent approaches to likelihood based inference in the presence of nuisance parameters. Our approach is based on plotting the likelihood function and the p-value function, using recently developed third order approximations. Orthogonal parameters and adjustments to profile likelihood are also discussed. Connections to classical approaches of conditional and marginal inference are outlined. 1. INTRODUCTION it is defined only up to arbitrary multiples which may depend on y but not on θ. This ensures in particu- We take the view that the most effective form of lar that the likelihood function is invariant to one-to- inference is provided by the observed likelihood func- one transformations of the measurement(s) y. In the tion along with the associated p-value function. In context of independent, identically distributed sam- the case of a scalar parameter the likelihood func- pling, where y = (y1; : : : ; yn) and each yi follows the tion is simply proportional to the density function. model f(y; θ) the likelihood function is proportional The p-value function can be obtained exactly if there to Πf(yi; θ) and the log-likelihood function becomes a is a one-dimensional statistic that measures the pa- sum of independent and identically distributed com- rameter. If not, the p-value can be obtained to a ponents: high order of approximation using recently developed methods of likelihood asymptotics. In the presence `(θ) = `(θ; y) = Σ log f(yi; θ) + a(y): (2) of nuisance parameters, the likelihood function for a ^ (one-dimensional) parameter of interest is obtained The maximum likelihood estimate θ is the value of via an adjustment to the profile likelihood function. -
Systems Manual 1I Overview
System Overview General The Statistical Solutions System for Missing Data Analysis incorporates the latest advanced techniques for imputing missing values. Researchers are regularly confronted with the problem of missing data in data sets. According to the type of missing data in your data set, the Statistical Solutions System allows you to choose the most appropriate technique to apply to a subset of your data. In addition to resolving the problem of missing data, the system incorporates a wide range of statistical tests and techniques. This manual discusses the main statistical tests and techniques available as output from the system. It is not a complete reference guide to the system. About this Manual Chapter 1 Data Management Describes how to import a data file, specifying the attributes of a variable, transformations that can be performed on a variable, and defining design and longitudinal variables. Information about the Link Manager and its comprehensive set of tools for screening data is also given. Chapter 2 Descriptive Statistics Discusses Descriptive Statistics such as: Univariate Summary Statistics, Sample Size, Frequency and Proportion. Chapter 3 Regression Introduces Regression, discusses Simple Linear, and Multiple Regression techniques, and describes the available Output Options. Chapter 4 Tables (Frequency Analysis) Introduces Frequency Analysis and describes the Tables, Measures, and Tests output options, which are available to the user when performing an analysis. Chapter 5 t- Tests and Nonparametric Tests Describes Two-group, Paired, and One-Group t-Tests, and the Output Options available for each test type. Systems Manual 1i Overview Chapter 6 Analysis (ANOVA) Introduces the Analysis of Variance (ANOVA) method, describes the One-way ANOVA and Two-way ANOVA techniques, and describes the available Output Options. -
Learning to Pivot with Adversarial Networks
Learning to Pivot with Adversarial Networks Gilles Louppe Michael Kagan Kyle Cranmer New York University SLAC National Accelerator Laboratory New York University [email protected] [email protected] [email protected] Abstract Several techniques for domain adaptation have been proposed to account for differences in the distribution of the data used for training and testing. The majority of this work focuses on a binary domain label. Similar problems occur in a scientific context where there may be a continuous family of plausible data generation processes associated to the presence of systematic uncertainties. Robust inference is possible if it is based on a pivot – a quantity whose distribution does not depend on the unknown values of the nuisance parameters that parametrize this family of data generation processes. In this work, we introduce and derive theoretical results for a training procedure based on adversarial networks for enforcing the pivotal property (or, equivalently, fairness with respect to continuous attributes) on a predictive model. The method includes a hyperparameter to control the trade- off between accuracy and robustness. We demonstrate the effectiveness of this approach with a toy example and examples from particle physics. 1 Introduction Machine learning techniques have been used to enhance a number of scientific disciplines, and they have the potential to transform even more of the scientific process. One of the challenges of applying machine learning to scientific problems is the need to incorporate systematic uncertainties, which affect both the robustness of inference and the metrics used to evaluate a particular analysis strategy. In this work, we focus on supervised learning techniques where systematic uncertainties can be associated to a data generation process that is not uniquely specified. -
Confidence Intervals and Nuisance Parameters Common Example
Confidence Intervals and Nuisance Parameters Common Example Interval Estimation in Poisson Sampling with Scale Factor and Background Subtraction The Problem (eg): A “Cut and Count” analysis for a branching fraction B finds n events. ˆ – The background estimate is b ± σb events. – The efficiency and parent sample are estimated to give a scaling factor ˆ f ± σf . How do we determine a (frequency) Confidence Interval? – Assume n is sampled from Poisson, µ = hni = fB + b. ˆ – Assume b is sampled from normal N(b, σb). ˆ – Assume f is sampled from normal N(f, σf ). 1 Frank Porter, March 22, 2005, CITBaBar Example, continued The likelihood function is: 2 2 n −µ 1ˆb−b 1fˆ−f µ e 1 −2 σ −2 σ L(n,ˆb, ˆf; B, b, f) = e b f . n! 2πσbσf µ = hni = fB + b We are interested in the branching fraction B. In particular, would like to summarize data relevant to B, for example, in the form of a confidence interval, without dependence on the uninteresting quantities b and f. b and f are “nuisance parameters”. 2 Frank Porter, March 22, 2005, CITBaBar Interval Estimation in Poisson Sampling (continued) Variety of Approaches – Dealing With the Nuisance Parameters ˆ ˆ Just give n, b ± σb, and f ± σf . – Provides “complete” summary. – Should be done anyway. – But it isn’t a confidence interval. ˆ ˆ Integrate over N(f, σf ) “PDF” for f, N(b, σb) “PDF” for b. (variant: normal assumption in 1/f). – Quasi-Bayesian (uniform prior for f, b (or, eg, for 1/f)). -
Reducing the Sensitivity to Nuisance Parameters in Pseudo-Likelihood Functions
544 The Canadian Journal of Statistics Vol. 42, No. 4, 2014, Pages 544–562 La revue canadienne de statistique Reducing the sensitivity to nuisance parameters in pseudo-likelihood functions Yang NING1*, Kung-Yee LIANG2 and Nancy REID3 1Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Canada 2Department of Life Sciences and Institute of Genome Sciences, National Yang-Ming University, Taiwan, ROC 3Department of Statistical Science, University of Toronto, Toronto, Canada Key words and phrases: Composite likelihood; higher order inference; invariance; nuisance parameters; pseudo-likelihood. MSC 2010: Primary 62F03; secondary 62F10 Abstract: In a parametric model, parameters are often partitioned into parameters of interest and nuisance parameters. However, as the data structure becomes more complex, inference based on the full likeli- hood may be computationally intractable or sensitive to potential model misspecification. Alternative likelihood-based methods proposed in these settings include pseudo-likelihood and composite likelihood. We propose a simple adjustment to these likelihood functions to reduce the impact of nuisance parameters. The advantages of the modification are illustrated through examples and reinforced through simulations. The adjustment is still novel even if attention is restricted to the profile likelihood. The Canadian Journal of Statistics 42: 544–562; 2014 © 2014 Statistical Society of Canada Resum´ e:´ Les parametres` d’un modele` sont souvent categoris´ es´ comme nuisibles ou d’inter´ et.ˆ A` mesure que la structure des donnees´ devient plus complexe, la vraisemblance peut devenir incalculable ou sensible a` des erreurs de specification.´ La pseudo-vraisemblance et la vraisemblance composite ont et´ epr´ esent´ ees´ comme des solutions dans ces situations. -
IBM SPSS Exact Tests
IBM SPSS Exact Tests Cyrus R. Mehta and Nitin R. Patel Note: Before using this information and the product it supports, read the general information under Notices on page 213. This edition applies to IBM® SPSS® Exact Tests 21 and to all subsequent releases and modifications until otherwise indicated in new editions. Microsoft product screenshots reproduced with permission from Microsoft Corporation. Licensed Materials - Property of IBM © Copyright IBM Corp. 1989, 2012. U.S. Government Users Restricted Rights - Use, duplication or disclosure restricted by GSA ADP Schedule Contract with IBM Corp. Preface Exact Tests™ is a statistical package for analyzing continuous or categorical data by exact methods. The goal in Exact Tests is to enable you to make reliable inferences when your data are small, sparse, heavily tied, or unbalanced and the validity of the corresponding large sample theory is in doubt. This is achieved by computing exact p values for a very wide class of hypothesis tests, including one-, two-, and K- sample tests, tests for unordered and ordered categorical data, and tests for measures of asso- ciation. The statistical methodology underlying these exact tests is well established in the statistical literature and may be regarded as a natural generalization of Fisher’s ex- act test for the single 22× contingency table. It is fully explained in this user manual. The real challenge has been to make this methodology operational through software development. Historically, this has been a difficult task because the computational de- mands imposed by the exact methods are rather severe. We and our colleagues at the Harvard School of Public Health have worked on these computational problems for over a decade and have developed exact and Monte Carlo algorithms to solve them. -
The NPAR1WAY Procedure This Document Is an Individual Chapter from SAS/STAT® 14.1 User’S Guide
SAS/STAT® 14.1 User’s Guide The NPAR1WAY Procedure This document is an individual chapter from SAS/STAT® 14.1 User’s Guide. The correct bibliographic citation for this manual is as follows: SAS Institute Inc. 2015. SAS/STAT® 14.1 User’s Guide. Cary, NC: SAS Institute Inc. SAS/STAT® 14.1 User’s Guide Copyright © 2015, SAS Institute Inc., Cary, NC, USA All Rights Reserved. Produced in the United States of America. For a hard-copy book: No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, or otherwise, without the prior written permission of the publisher, SAS Institute Inc. For a web download or e-book: Your use of this publication shall be governed by the terms established by the vendor at the time you acquire this publication. The scanning, uploading, and distribution of this book via the Internet or any other means without the permission of the publisher is illegal and punishable by law. Please purchase only authorized electronic editions and do not participate in or encourage electronic piracy of copyrighted materials. Your support of others’ rights is appreciated. U.S. Government License Rights; Restricted Rights: The Software and its documentation is commercial computer software developed at private expense and is provided with RESTRICTED RIGHTS to the United States Government. Use, duplication, or disclosure of the Software by the United States Government is subject to the license terms of this Agreement pursuant to, as applicable, FAR 12.212, DFAR 227.7202-1(a), DFAR 227.7202-3(a), and DFAR 227.7202-4, and, to the extent required under U.S. -
Arxiv:1807.05996V2 [Physics.Data-An] 4 Aug 2018
Lectures on Statistics in Theory: Prelude to Statistics in Practice Robert D. Cousins∗ Dept. of Physics and Astronomy University of California, Los Angeles Los Angeles, California 90095 August 4, 2018 Abstract This is a writeup of lectures on \statistics" that have evolved from the 2009 Hadron Collider Physics Summer School at CERN to the forthcoming 2018 school at Fermilab. The emphasis is on foundations, using simple examples to illustrate the points that are still debated in the professional statistics literature. The three main approaches to interval estimation (Neyman confidence, Bayesian, likelihood ratio) are discussed and compared in detail, with and without nuisance parameters. Hypothesis testing is discussed mainly from the frequentist point of view, with pointers to the Bayesian literature. Various foundational issues are emphasized, including the conditionality principle and the likelihood principle. arXiv:1807.05996v2 [physics.data-an] 4 Aug 2018 ∗[email protected] 1 Contents 1 Introduction 5 2 Preliminaries 6 2.1 Why foundations matter . 6 2.2 Definitions are important . 6 2.3 Key tasks: Important to distinguish . 7 3 Probability 8 3.1 Definitions, Bayes's theorem . 8 3.2 Example of Bayes's theorem using frequentist P . 10 3.3 Example of Bayes's theorem using Bayesian P . 10 3.4 A note re Decisions ................................ 12 3.5 Aside: What is the \whole space"? . 12 4 Probability, probability density, likelihood 12 4.1 Change of observable variable (\metric") x in pdf p(xjµ) . 13 4.2 Change of parameter µ in pdf p(xjµ) ...................... 14 4.3 Probability integral transform . 15 4.4 Bayes's theorem generalized to probability densities .