DOCSLIB.ORG

Does Robust Estimation Change Conclusions from Economic Data?

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Does Robust Estimation Change Conclusions from Economic Data? DOES ROBUST ESTIMATION CHANGE CONCLUSIONS FROM ECONOMIC DATA? Richard i-lndrew Simson B.Comm. Honours (Eco~lornics)University of Natal M.Cornm. University of Natal A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOROF PHILOSOPHY in the Department of Economics @ ltichard Andrew Simson 1993 SIMON FRASER UNIVERSITY July 1993 All rights reserved. This work may not be reproduced in whole or in part, by photocopy or other means, without the permission of the author. APPROVAL , Name: Richard A. Simson Degree: Ph.D. (Economics) Title of Thesis: Does Robust Estimation Change Conclusions From Economic Data? Examining Committee Chairman: Dr. Lawrence A. Boland Dr. pet& E. ~enne&y Senior Supervisor i Sr. Stephen T. Easton Supervisor Dr. R. A. ~onr Department Economics Internal/External Examiner Dr. Lonnie J. Magee Associate Professor Department of Economics McMaster University External Examiner Date Approved: &L\/ lZ 1993 PARTIAL COPYRIGHT LICENSE I hereby grant to Simon Fraser University the right to lend my thesis, project or extended essay (the title of which is shown below) to users of the Simon Fraser University Library, and to make partial or single copies only for such users or in response to a request from the library of any other university, or other educational institution, on its own behalf or for one of its users. I hrther agree that permission for multiple copying of this work for scholarly purposes may be granted by me or the Dean of Graduate Studies. It is understood that copying or publication of this work for financial gain shall not be allowed without my written permission. Title of Thesis/Project/Extended Essay Does Robust Estimation Change Conclusions From Economic Data? Author: (signature) Richard A. Simson (name) July 12, 1993 (date) Abstract Using ninety economic data sets from forty-four publications, robust estimates are compared with the original ordinary least squares' results. Robust methods, insen- sitive to large errors, have recei~reda great deal of attention by statisticians and econorrietricians; economists have been exhorted to use them. Do these robust meth- ods make a difference with economic data, in terms of the estimated coefficients. the results of hypothesis tests, or the quality of forecasts? The results of this thesis sug- gest that on the whole robust met hods do make a difference, albeit more so on some criteria than on others. Forecasting differences are not great, for example, but hy- pothesis testing differences are large enough to be of special concern. This result led to an extension of the thesis, examining the quality of the standard error estimates produced by popular econornetrics packages. This extension included the use of ran- domization tests and Monte Carlo work. Results indicate some methods in common use are markedly inferior to others and should be avoided. Acknowledgements This thesis would not have been possible without the support of Kimberley Sinlson. Contents ... Abstract ..................................... 111 hcknow ledgements ............................... iv ... List of Tables .................................. v111 List of Figures .................................. X 1 Introduction ................................ 1 1.1 The Purpose of the Thesis .................... l 1.2 A View of the Literature ..................... 2 1.3 New Avenues of Research .................... -I 1.4 Why be Concerned with Robust Regression? .......... 9 1.4.1 Introduction ...................... 9 1.4.2 Some Quotations Supporting R.obust Methods .... 10 1.4.3 Summary ........................ 15 1.5 Some Prior Work .......................... 16 1.5.1 Introduction ...................... 16 1.5.2 'Two important studies ................. 17 1.5.3 The Other Studies ................... 20 1.5.4 Least Median Squares ................. 25 1.5.5 Surrlrnary ........................ 25 1.5.6 What; is to be done? .................. 26 2 Robust Estimators ............................. 27 2.1 Introduction ............................. 27 2.2 Empirical Work In Economics .................. 25 2.3 Properties of Ordinary Least Squares .............. 28 2.4 Some Robust Estimators ..................... 30 2.5 M-Estimators ........................... 31 2.5.1 M-Estimators ...................... 31 2.5.2 Three M-Estimators .................. 33 2.5.3 The Estirnation Process ................ 35 2.5.4 Hypothesis Testing ................... 38 2.5.5 Monte Carlo ...................... 43 2.5.6 So Which M-Estimator? ................ 44 2.6 L-Estimators ........................... 45 2.6.1 Regression Quantiles .................. 46 2.7 Trimmed Least Squares ..................... 51 2.8 Summary and conclusion .................... 52 3 Criteria to Compare Robust to Least Squares ............. 53 3.1 Criteria for Comparing Estimates ................ 53 3.1 .1 Introduction ...................... 53 3.1.2 The Fourteen Criteria ................. 55 3.1.3 Sample Sizes ...................... 69 4 Results ................................... 71 4.1 Introduction ............................ 71 4.2 The Data ............................. 72 4.3 Results for the thirteen criteria ................. 75 4.4 Criterion I ............................. 75 4.5 Criterion II ............................ 77 4.6 Criterion 111 ............................ 77 4.7 Criterion IV ............................ 79 4.8 Criterion V ............................ 81 4.9 Criterion V1 ............................ 84 4.10 Criterion VII and VIII ...................... 88 4.1 1 Criterion IX ............................. 92 4.12 Criterion X ............................ 94 4.13 Criterion XI ............................ 95 4.14 Criterion XI1 ........................... 99 4.15 Criterion XI11 ........................... 100 4.16 Criterion XIV ........................... 101 4.17 Outliers in the Studies' Data Sets ................ 105 4.17.1 The Extent of Outliers ................. 107 4.17.2 R LOGIT Analysis of Outliers ............ 108 4.18 Conclusions ............................ 112 4.19 Studies ............................... 115 5 Afraid of Heights: A Monte Carlo Study ................ 121 5.1 Introduction ............................ 121 5.2 A Proposed Monte Carlo Study ................. 122 5.3 Steps of the i'vlonte Carlo Study ................. 123 5.4 The SHAZAM Programme .................... 125 5.5 Monte Carlo Results ....................... 127 5.6 More Monte Carlo Results .................... 131 6 Summary arid Conclusions ........................ 136 Bibliography .................................. 151 vii List of Tables 2.1 An Example of Raw and Winsorized Data ............... 37 3.1 Sample Size-Coefficients ......................... 70 3.2 Sample Size-Regressions ......................... 70 3.3 Equations Forecast ............................ 70 4.1 Rounding ................................. 73 4.2 Types of Replication ........................... 75 4.3 Percentage Changes of Robust Slope Coefficients Changing By More than Stated ................................ 76 4.4 Percentage Changes More Than Thirty and Fifty ........... 78 4.5 Changes More Than OLS Standard Deviation ............. 79 4.6 Changes more than OLS standard deviations .............. 80 4.7 Standard Deviation ............................ 53 4.8 ,y vest Results .............................. 84 4.9 Test Results: random coef ...................... 85 4.10 Wilcoxon Signed Ranks Test ....................... 87 4.11 Siegel-Tukey Alternating Ranks Test .................. 87 4.12 P~rcentageof robust estimators with a lower estimated variance ... 89 4.13 Average Percentage Change when robust estimated variance lower . 90 4.14 Average Percentage Change when least square's cstirnated variance lower 90 4.15 Percentage when robust forecast beats least squares (me(di)an percent- age error criterion) ............................ 93 ... Vlll 4.16 Average improvement in MAPE whe11 robust (OLS) forecast beats OLS (robust) .................................. 93 4.17 Average improvement in MedianAPE when robust (OLS) forecast beats OLS (robust) ............................... 94 4.18 Percentage Becoming S (1ns)ignificant .................. 97 4.10 Percentage Becoming Significant and Insignificant ........... 98 4.20 PercentageWithBiggerP-Value ..................... 99 4.21 Average and Median Percentage Change When Bigger P-Value .... 100 4.22 Average and Median Percentage Change When Smaller P-Value ... 4.23 Average and Median Percentage Point Change When Bigger P-Value 4.24 Average and Median Percentage Point Change When Smaller P-Vdue 4.25 Percentage Bigger P-Value ........................ 4.26 Percentage Smaller P-Value ....................... 4.27 Percentage Bigger P-Value ........................ 4.28 Percentage Smaller P-Value ....................... 4.29 Outliers in Data Sets: Part One ..................... 4.30 Outliers in Data Sets: Part Two ..................... 4.31 Results from the Logit Analysis ..................... 5.1 Monte Carlo Results-Benderley and Zwick ............. 127 5.2 Monte Carlo Results-Barkley ...................... 130 5.3 "Robilst" Hypothesisrkst Results .................... 131 5.4 Monte Carlo Results-Benderley and Zwick-Normal ......... 104 5.5 Monte Carlo Results-Barkley-Nor~nal ................. 134 5.6 Monte Carlo. Results-Benderley and Zwick-Mixed (.75) ...... 134 5.7 Monte Carlo Results-Barkley-Mixed (0.75) .............. 135 5.8 Monte Carlo R.esults.Benderley and Zwick-Slash .......... 135 5.9 Monte Carlo R.esults.Barkley.Slash
Load more

Recommended publications
  • Efficient Inflation Estimation
    1 Efficient Inflation Estimclticn by Michael F. Bryan, Stephen G. Cecchetti, and Rodney L. Wiggins II Working Paper 9707 EFFICIENT JIVFLATION ESTIMATION by Michael F. Bryan, Stephen G. Cecchetti, and Rodney L. Wiggins I1 Michael F. Bryan is assistant vice president and economist at the Federal Reserve Bank of Cleveland. Stephen G. Cecchetti is executive vice president and director of research at the Federal Reserve Bank of New York and a research associate at the National Bureau of Econonlic Research. Rodney L. Wiggins I1 is a member of the Research Department of the Federal Reserve Bank of Cleveland. The authors gratefully acknowledge the comments and assistance of Todd Clark, Ben Craig, and Scott Roger. Working papers of the Federal Reserve Bank of Cleveland are preliminary materials circulated to stimulate discussion and critical comment. The views stated herein are those of the authors and are not necessarily those of the Federal Reserve Bank of Cleveland or of the Board of Governors of the Federal Reserve System. Federal Reserve Bank of Cleveland working papers are distributed for the purpose of promoting discussion of research in progress. These papers may not have been subject to the formal editorial review accorded official Federal Reserve Bank of Cleveland publications. Working papers are now available electronically through the Cleveland Fed's home page on the World Wide Web: http:Ilwww.clev.frb.org. August 1997 Abstract This paper investigates the use of trimmed means as high-frequency estimators of inflation. The known characteristics of price change distributions, specifically the . observation that they generally exhibit high levels of kurtosis, imply that simple averages of price data are unlikely to produce efficient estimates of inflation.
    [Show full text]
  • A Globally Optimal Solution to 3D ICP Point-Set Registration
    1 Go-ICP: A Globally Optimal Solution to 3D ICP Point-Set Registration Jiaolong Yang, Hongdong Li, Dylan Campbell, and Yunde Jia Abstract—The Iterative Closest Point (ICP) algorithm is one of the most widely used methods for point-set registration. However, being based on local iterative optimization, ICP is known to be susceptible to local minima. Its performance critically relies on the quality of the initialization and only local optimality is guaranteed. This paper presents the first globally optimal algorithm, named Go-ICP, for Euclidean (rigid) registration of two 3D point-sets under the L2 error metric defined in ICP. The Go-ICP method is based on a branch-and-bound (BnB) scheme that searches the entire 3D motion space SE(3). By exploiting the special structure of SE(3) geometry, we derive novel upper and lower bounds for the registration error function. Local ICP is integrated into the BnB scheme, which speeds up the new method while guaranteeing global optimality. We also discuss extensions, addressing the issue of outlier robustness. The evaluation demonstrates that the proposed method is able to produce reliable registration results regardless of the initialization. Go-ICP can be applied in scenarios where an optimal solution is desirable or where a good initialization is not always available. Index Terms—3D point-set registration, global optimization, branch-and-bound, SE(3) space search, iterative closest point F 1 INTRODUCTION given an initial transformation (rotation and transla- Point-set registration is a fundamental problem in tion), it alternates between building closest-point cor- computer and robot vision.
    [Show full text]
  • Trimmed Mean Group Estimation
    Trimmed Mean Group Estimation Yoonseok Lee∗ Donggyu Sul† Syracuse University University of Texas at Dallas May 2020 Abstract This paper develops robust panel estimation in the form of trimmed mean group estimation. It trims outlying individuals of which the sample variances of regressors are either extremely small or large. As long as the regressors are statistically independent of the regression coefficients, the limiting distribution of the trimmed estimator can be obtained in a similar way to the standard mean group estimator. We consider two trimming methods. The first one is based on the order statistic of the sample variance of each regressor. The second one is based on the Mahalanobis depth of the sample variances of regressors. We apply them to the mean group estimation of the two-way fixed effect model with potentially heterogeneous slope parameters and to the commonly correlated regression, and we derive limiting distribution of each estimator. As an empirical illustration, we consider the effect of police on property crime rates using the U.S. state-level panel data. Keywords: Trimmed mean group estimator, Robust estimator, Heterogeneous panel, Two- way fixed effect, Commonly correlated estimator JEL Classifications: C23, C33 ∗Address: Department of Economics and Center for Policy Research, Syracuse University, 426 Eggers Hall, Syracuse, NY 13244. E-mail: [email protected] †Address: Department of Economics, University of Texas at Dallas, 800 W. Campbell Road, Richardson, TX 75080. E-mail: [email protected] 1Introduction Though it is popular in practice to impose homogeneity in panel data regression, the homogeneity restriction is often rejected (e.g., Baltagi, Bresson, and Pirotte, 2008).
    [Show full text]
  • Programs Tramo (Time Series Regression with Arima Noise
    PROGRAMS TRAMO (TIME SERIES REGRESSION WITH ARIMA NOISE, MISSING OBSERVATIONS AND OUTLIERS) AND SEATS (SIGNALEXTRACTIONS AREMA TIME SERIES) INSTRUCTIONS FOR THE USER (BETA VERSION: JUNIO 1997) Víctor Gómez * Agustín Maravall ** SGAPE-97001 Julio 1997 * Ministerio de Economía y Hacienda ** Banco de España This software is available at the following Internet address: http://www.bde.es The Working Papers of the Dirección General de Análisis y Programación Presupuestaria are not official statements of the Ministerio de Economía y Hacienda Abstract Brief summaries and user Instructions are presented for the programs-* TRAMO ("Time 'Series Regression with ARTMA Noise, Missing Observations and Outliers") and SEATS ("Signal Extraction in ARIMA Time Series"). TRAMO is a program for estimation and forecasting of regression models with possibly nonstationary (ARIMA) errors and any sequence of missing val- ues. The program interpolates these values, identifies and corrects for several types of outliers, and estimates special effects such as Trading Day and Easter and, in general, intervention variable type of effects. Fully automatic model identification and outlier correction procedures are available. SEATS is a program for estimation of unobserved components in time se- ries following the so-called ARlMA-model-based method. The trend, seasonal, irregular, and cyclical components are estimated and forecasted with signal extraction techniques applied to ARIMA models. The standard errors of the es- timates and forecasts are obtained and the model-based structure is exploited to answer questions of interest in short-term analysis of the data. The two programs are structured so as to be used together both for in- depth analysis of a few series (as presently done at the Bank of Spain) or for automatic routine applications to a large number of series (as presently done at Eurostat).
    [Show full text]
  • Learning Entangled Single-Sample Distributions Via Iterative Trimming
    Learning Entangled Single-Sample Distributions via Iterative Trimming Hui Yuan Yingyu Liang Department of Statistics and Finance Department of Computer Sciences University of Science and Technology of China University of Wisconsin-Madison Abstract have n data points from n different distributions with a common mean but different variances (the mean and all the variances are unknown), and our goal is to es- In the setting of entangled single-sample dis- timate the mean. tributions, the goal is to estimate some com- mon parameter shared by a family of distri- This setting is motivated for both theoretical and prac- butions, given one single sample from each tical reasons. From the theoretical perspective, it goes distribution. We study mean estimation and beyond the typical i.i.d. setting and raises many inter- linear regression under general conditions, esting open questions, even on basic topics like mean and analyze a simple and computationally ef- estimation for Gaussians. It can also be viewed as ficient method based on iteratively trimming a generalization of the traditional mixture modeling, samples and re-estimating the parameter on since the number of distinct mixture components can the trimmed sample set. We show that the grow with the number of samples. From the practical method in logarithmic iterations outputs an perspective, many modern applications have various estimation whose error only depends on the forms of heterogeneity, for which the i.i.d. assumption noise level of the dαne-th noisiest data point can lead to bad modeling of their data. The entan- where α is a constant and n is the sample gled single-sample setting provides potentially better size.
    [Show full text]
  • Power-Shrinkage and Trimming: Two Ways to Mitigate Excessive Weights
    ASA Section on Survey Research Methods Power-Shrinkage and Trimming: Two Ways to Mitigate Excessive Weights Chihnan Chen1, Nanhua Duan2, Xiao-Li Meng3, Margarita Alegria4 Boston University1 UCLA2 Harvard University3 Cambridge Health Alliance and Harvard Medical School4 Abstract 2 Power-Shrinkage and Trimming Large-scale surveys often produce raw weights with very The survey weights are constructed as the reciprocals of large variations. A standard approach is to perform the selection probabilities. Let w be the survey weight, n some form of trimming, as a way to reduce potentially be the sample size, and y be the variable of interest. Let large variances in various survey estimators. The amount p 2 [0; 1] be the power-shrinkage parameter and T 2 [0; 1] of trimming is usually determined by considerations of be the trimming threshold de¯ned in terms of percentile. bias-variance trade-o®. While bias-variance trade-o® is The original weighted estimator, power-shrinkage estima- a sound principle, the trimming method itself is popular tor and the trimmed estimator for the expectation, E(y), only because of its simplicity, not because it has statisti- are given by cally desirable properties. In this paper we investigate a Xn more principled method by shrinking the variability of the wiyi log of the weights. We use the same mean squared error ² Original weighted estimator :y ¹ = i=1 (MSE) criterion to determine the amount of shrinking. w Xn The shrinking on the log scale implies shrinking weights wi by a power parameter p between [0,1]. Our investiga- i=1 tion suggests an empirical way to predict the optimal Xn choice.
    [Show full text]
  • 26 Sep 2019 the Trimmed Mean in Non-Parametric Regression
    The Trimmed Mean in Non-parametric Regression Function Estimation Subhra Sankar Dhar1, Prashant Jha2, and Prabrisha Rakhshit3 1Department of Mathematics and Statistics, IIT Kanpur, India 2Department of Mathematics and Statistics, IIT Kanpur, India 3Department of Statistics, Rutgers University, USA September 27, 2019 Abstract This article studies a trimmed version of the Nadaraya-Watson estimator to estimate the unknown non-parametric regression function. The characterization of the estimator through minimization problem is established, and its pointwise asymptotic distribution is derived. The robustness property of the proposed estimator is also studied through breakdown point. Moreover, as the trimmed mean in the location model, here also for a wide range of trim- ming proportion, the proposed estimator poses good efficiency and high breakdown point for various cases, which is out of the ordinary property for any estimator. Furthermore, the usefulness of the proposed estimator is shown for three benchmark real data and various simulated data. Keywords: Heavy-tailed distribution; Kernel density estimator; L-estimator; Nadaraya-Watson estimator; Robust estimator. arXiv:1909.10734v2 [math.ST] 26 Sep 2019 1 Introduction We have a random sample (X1,Y1),..., (Xn,Yn), which are i.i.d. copies of (X,Y ), and the regres- sion of Y on X is defined as Yi = g(Xi)+ ei, i =1, . , n, (1) where g( ) is unknown, and e are independent copies of error random variable e with E(e X)=0 · i | and var(e X = x)= σ2(x) < for all x. Note that the condition E(e X) = 0 is essentially the | ∞ | 2000 AMS Subject Classification: 62G08 1 identifiable condition for mean regression, and it varies over the different procedures of regression.
    [Show full text]
  • Measuring Core Inflation in India: an Application of Asymmetric-Trimmed Mean Approach
    Measuring Core Inflation in India: An Application of Asymmetric-Trimmed Mean Approach Motilal Bicchal* Naresh Kumar Sharma ** Abstract The paper seek to obtain an optimal asymmetric trimmed means-core inflation measure in the class of trimmed means measures when the distribution of price changes is leptokurtic and skewed to the right for any given period. Several estimators based on asymmetric trimmed mean approach are constructed, and evaluated by the conditions set out in Marques et al. (2000). The data used in the study is the monthly 69 individual price indices which are constituent components of Wholesale Price Index (WPI) and covers the period, April 1994 to April 2009, with 1993-94 as the base year. Results of the study indicate that an optimally trimmed estimator is found when we trim 29.5 per cent from the left-hand tail and 20.5 per cent from the right-hand tail of the distribution of price changes. Key words: Core inflation, underlying inflation, asymmetric trimmed mean JEL Classification: C43, E31, E52 * Ph.D Research scholar and ** Professor at Department of Economics, University of Hyderabad, P.O Central University, Hyderabad-500046, India. Corresponding author: [email protected] We are very grateful to Carlos Robalo Marques of Banco de Portugal for providing the program for computing the asymmetric trimmed-mean as well as for his valuable comments and suggestions. We also wish to thank Bandi Kamaiah for going through previous draft and for offering comments, which improved the presentation of this paper. 1. Introduction Various approaches to measuring core inflation have been discussed in the literature.
    [Show full text]
  • Robust Statistics
    Robust statistics Stéphane Paltani Why robust statistics? Removal of data Robust statistics L-estimators and some non-parametric statistics R-estimators M-estimators Stéphane Paltani ISDC Data Center for Astrophysics Astronomical Observatory of the University of Geneva Statistics Course for Astrophysicists, 2010–2011 Outline Robust statistics Stéphane Paltani Why robust statistics? Removal of data Why robust statistics? L-estimators R-estimators Removal of data M-estimators L-estimators R-estimators M-estimators Caveat Robust statistics Stéphane Paltani Why robust statistics? Removal of data The presentation aims at being user-focused and at L-estimators presenting usable recipes R-estimators Do not expect a fully mathematically rigorous description! M-estimators This has been prepared in the hope to be useful even in the stand-alone version Please provide me feedback with any misconception or error you may find, and with any topic not addressed here but which should be included Outline Robust statistics Stéphane Paltani Why robust statistics? Why robust statistics? First example General concepts First example Data representation General concepts Removal of data Data representation L-estimators R-estimators M-estimators Removal of data L-estimators R-estimators M-estimators Example: The average weight of opossums Robust statistics Stéphane Paltani Why robust statistics? First example General concepts Data representation Removal of data L-estimators R-estimators M-estimators I Find a group of N opossums I Weight them: wi ; i = 1;:::; N
    [Show full text]
  • Configurable FPGA-Based Outlier Detection for Time Series Data
    Configurable FPGA-Based Outlier Detection for Time Series Data by Ghazaleh Vazhbakht, B.Sc. A thesis submitted to the Faculty of Graduate and Postdoctoral Affairs in partial fulfillment of the requirements for the degree of Master of Applied Science in Electrical and Computer Engineering Ottawa-Carleton Institute for Electrical and Computer Engineering Carleton University Ottawa, Ontario c 2017 Ghazaleh Vazhbakht TABLE OF CONTENTS Page LIST OF FIGURES iv LIST OF TABLES v LIST OF ACRONYMS vi ACKNOWLEDGMENTS vii ABSTRACT viii 1 Introduction 1 1.1 Outlier Detection in Time Series . 2 1.2 Thesis Objective and Contributions . 3 2 Background 5 2.1 Definition of Outlier . 5 2.2 Outlier Detection . 6 2.3 Scope of Work . 8 2.3.1 Types of Outliers . 10 2.3.2 Techniques Based on Outlier Aware Prediction Models . 13 2.3.3 Managing Detected Outliers . 14 2.4 FPGA-based Outlier Detection Techniques . 15 3 Theory and Reduction to Practice 20 3.1 ARMA Modeling . 21 3.1.1 Model Selection . 23 3.1.2 Parameter Estimation . 26 3.1.3 Model Checking . 35 3.2 Test Statistics . 35 3.2.1 Critical Value . 36 3.3 Outlier Detection Algorithm . 37 3.3.1 Estimating Outliers Effects . 39 3.3.2 Locating and Identifying Outliers . 41 3.3.3 Residual Standard Deviation . 42 ii 3.3.4 The Procedure of Joint Estimation of Model Parameters and Outlier Effects . 43 4 Design and Implementation 48 4.1 Profiling the Outlier Detection Algorithm . 49 4.2 MATLAB Implementation of the Simplified Algorithm . 50 4.3 Hardware Implementation of the Outlier Detection Algorithm .
    [Show full text]
  • Outlier Detection and Trimmed Estimation for General Functional Data
    Outlier detection and trimmed estimation for general functional data Daniel Gervini Department of Mathematical Sciences University of Wisconsin–Milwaukee P.O. Box 413, Milwaukee, WI 53201 [email protected] December 3, 2012 arXiv:1001.1014v2 [stat.ME] 30 Nov 2012 Abstract This article introduces trimmed estimators for the mean and covariance function of general functional data. The estimators are based on a new measure of “outlying- ness” or data depth that is well defined on any metric space, although this paper focuses on Euclidean spaces. We compute the breakdown point of the estimators and show that the optimal breakdown point is attainable for the appropriate choice of tuning parameters. The small-sample behavior of the estimators is studied by simulation, and we show that they have better outlier-resistance properties than alternative estimators. This is confirmed by two real-data applications, that also show that the outlyingness measure can be used as a graphical outlier-detection tool in functional spaces where visual screening of the data is difficult. Key Words: Breakdown Point; Data Depth; Robust Statistics; Stochastic Pro- cesses. 1 Introduction Many statistical applications today involve data that does not fit into the classi- cal univariate or multivariate frameworks; for example, growth curves, spectral curves, and time-dependent gene expression profiles. These are samples of func- tions, rather than numbers or vectors. We can think of them as realizations of a stochastic process with sample paths in L 2(R), the space of square-integrable functions. The statistical analysis of function-valued data has received a lot of at- tention in recent years (see e.g.
    [Show full text]
  • Robust Mean Estimation for Real-Time Blanking in Radioastronomy
    ROBUST MEAN ESTIMATION FOR REAL-TIME BLANKING IN RADIOASTRONOMY Philippe Ravier1, Cedric Dumez-Viou1−2 1Laboratory of Electronics, Signals, Images, Polytech'Orleans´ - University of Orleans 12, rue de Blois, BP 6744 Cedex 2, F-45067 Orleans,´ France 2Observatoire de Paris/(CNRS No.704), Station de radioastronomie, F-18330 Nanc¸ay, France phone: + (33).2.38.49.48.63, fax: + (33).2.38.41.72.45, email: [email protected] web: www.obs-nancay.fr/rfi ABSTRACT assumption does not hold anymore and the exact PDF must be taken into account for precise threshold derivation. Radio astronomy observations suffer from strong interfer- ences that need to be blanked both in the time and frequency domains. In order to achieve real-time computations, in- 2. PROCEDURE USED FOR REAL-TIME terference detection is made by simple thresholding. The In order to achieve real-time computation, a simple thresh- threshold value is linked to the mean estimation of the power olding drives the time and/or frequency blanking decision. 2 of clean observed data that follows a χ distribution. Spu- The problem is formulated as the following binary hy- rious values insensitivity is obtained by replacing mean by pothesis test: robust mean. A theoretical study of the variance of three es- timators of the mean is presented. The study leads to a prac- tical trimmed percentage that is chosen for the robust mean. H0 : the SOI only is present, H1 : the SOI is polluted with additive interference. 1. INTRODUCTION The SOI is supposed to be Gaussian. On the other hand, Radioastronomy benefits from protected bands for the study no a priori information about the interference PDF is avail- of radio-sources.
    [Show full text]