DOCSLIB.ORG

Robust Mean-Squared Error Estimation in the Presence of Model Uncertainties Yonina C

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Robust Mean-Squared Error Estimation in the Presence of Model Uncertainties Yonina C 168 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 53, NO. 1, JANUARY 2005 Robust Mean-Squared Error Estimation in the Presence of Model Uncertainties Yonina C. Eldar, Member, IEEE, Aharon Ben-Tal, and Arkadi Nemirovski Abstract—We consider the problem of estimating an unknown so that is chosen to minimize the Euclidian norm of the data parameter vector x in a linear model that may be subject to error . However, in an estimation context, the objective x uncertainties, where the vector is known to satisfy a weighted typically is to minimize the size of the estimation error , norm constraint. We first assume that the model is known ex- actly and seek the linear estimator that minimizes the worst-case rather than that of the data error . mean-squared error (MSE) across all possible values of x. We show To develop an estimation method that is based directly on the that for an arbitrary choice of weighting, the optimal minimax estimation error, we may seek the estimator that minimizes MSE estimator can be formulated as a solution to a semidefinite the mean-squared error (MSE), where the MSE of an estimate programming problem (SDP), which can be solved very efficiently. of is the expected value of the squared norm of the esti- We then develop a closed form expression for the minimax MSE mation error and is equal to the sum of the variance and the estimator for a broad class of weighting matrices and show that it coincides with the shrunken estimator of Mayer and Willke, with squared norm of the bias. Since the bias generally depends on a specific choice of shrinkage factor that explicitly takes the prior the unknown parameters , we cannot choose an estimator to information into account. directly minimize the MSE. A common approach is to restrict Next, we consider the case in which the model matrix is subject to the estimator to be linear and unbiased and then seek the esti- uncertainties and seek the robust linear estimator that minimizes mator of this form that minimizes the variance or the MSE. It is the worst-case MSE across all possible values of x and all possible values of the model matrix. As we show, the robust minimax MSE well known that the LS estimator minimizes the variance in the estimator can also be formulated as a solution to an SDP. estimate among all unbiased linear estimators. However, this Finally, we demonstrate through several examples that the min- does not imply that the LS estimator leads to a small variance or imax MSE estimator can significantly increase the performance a small mean-squared error (MSE). A difficulty often encoun- over the conventional least-squares estimator, and when the model tered in this estimation problem is that the resulting variance can matrix is subject to uncertainties, the robust minimax MSE esti- be very large, particularly in nonorthogonal and ill-conditioned mator can lead to a considerable improvement in performance over the minimax MSE estimator. problems. Various modifications of the LS estimator for the case in Index Terms—Data uncertainty, linear estimation, mean squared error estimation, minimax estimation, robust estimation. which the data model holds i.e., with and known exactly, have been proposed. Among the alternatives are Tikhonov regularization [5], which is also known in the I. INTRODUCTION statistical literature as the ridge estimator [6], the shrunken HE problem of estimating a set of unknown deterministic estimator [7], and the covariance shaping LS estimator [8], [9]. T parameters observed through a linear transformation In general, these LS alternatives attempt to reduce the MSE , and corrupted by additive noise , arises in a large variety in estimating by allowing for a bias. However, each of the of areas in science and engineering, e.g., communication, estimators above is designed to optimize an objective which economics, signal processing, seismology, and control. does not depend directly on the MSE, but rather depends on the Owing to the lack of statistical information about the pa- data error . rameters , often, the estimated parameters are chosen to opti- In many engineering applications, the model matrix is also mize a criterion based on the observed signal . The celebrated subject to uncertainties. For example, the matrix may be es- least-squares (LS) estimator [1]–[3], which was first used by timated from noisy data, in which case, may not be known Gauss to predict movements of planets [4], seeks the linear esti- exactly. If the actual data matrix deviates from the one assumed, mate of that results in an estimated data vector that then the performance of an estimator designed based on alone is closest, in a LS sense, to the given data vector may deteriorate considerably. Various methods have been pro- posed to account for uncertainties in . The Total LS method [10], [11] seeks the parameters and the minimum perturbation Manuscript received January 30, 2003; revised January 29, 2004. The work of Y. C. Eldar was supported in part by the Taub Foundation and by a Technion to the model matrix that minimize the data error. Although V.P.R. Fund. The associate editor coordinating the review of this manuscript and the total LS method allows for uncertainties in , in many cases, approving it for publication was Dr. Jean Pierre Delmas. it results in correction terms that are unnecessarily large. In par- Y. C. Eldar is with the Department of Electrical Engineering, Technion—Is- ticular, when the model matrix is square, the total LS method rael Institute of Technology, Haifa 32000, Israel (e-mail: [email protected] nion.ac.il). recuse to the conventional LS method, which does not take the A. Ben-Tal and A. Nemirovski are with the MINERVA Optimization uncertainties into account. Recently, several methods [12]–[14] Center, Department of Industrial Engineering, Technion—Israel Institute of have been developed to treat the case in which the perturba- Technology, Haifa 32000, Israel (e-mail: [email protected]; nemirovs@ ie.technion.ac.il). tion to the model matrix is bounded. These methods seek Digital Object Identifier 10.1109/TSP.2004.838933 the parameters that minimize the worst-case data error across 1053-587X/$20.00 © 2005 IEEE Authorized licensed use limited to: Georgia Institute of Technology. Downloaded on February 16,2010 at 17:11:22 EST from IEEE Xplore. Restrictions apply. ELDAR et al.: ROBUST MEAN-SQUARED ERROR ESTIMATION IN THE PRESENCE OF MODEL UNCERTAINTIES 169 all bounded perturbations of and possibly bounded perturba- have the same eigenvector matrix. In Section IV, we tions of the data vector. In [15], the authors seek the estimator consider the case in which both and the model matrix are that minimizes the best possible data error over all possible per- subject to uncertainties and show that the minimax MSE esti- turbations of . Here again, the above objectives depend on the mator that minimizes the worst-case MSE in the region of un- data error and not on the estimation error or the MSE. certainty can again be formulated as an SDP. We then consider, In this paper, we consider the case in which the (possibly in Section V, the special case in which and have weighted) norm of the unknown vector is bounded and de- the same eigenvector matrix and and have the same velop robust estimators of , whose performance is reason- eigenvector matrix. Examples illustrating the performance ad- ably good across all possible values of in the region of uncer- vantage of the minimax MSE estimator over the LS estimator, tainty, by minimizing objectives that depend explicitly on the and the advantage of the robust minimax MSE estimator over MSE. the minimax MSE estimator in the presence of model uncer- We first consider the case in which is known exactly and tainties, are discussed in Section VI. develop a minimax linear robust estimator that minimizes the worst-case MSE across all possible bounded values of , i.e., II. MINIMAX MSE ESTIMATION WITH KNOWN over all values of such that for some constant We denote vectors in by boldface lowercase letters and and weighting matrix . The minimax MSE estimator for the matrices in by boldface uppercase letters. denotes the special case in which and the covariance matrix of identity matrix of appropriate dimension, denotes the Her- the noise vector is given by for some mitian conjugate of the corresponding matrix, and denotes has been developed in [16]. Here, we extend the results to ar- an estimated vector or matrix. The notation means that bitrary and and show that the minimax MSE estimator can be formulated as the solution to a semidefinite programming is positive semidefinite. problem (SDP) [17]–[19], which is a tractable convex optimiza- Consider the problem of estimating the unknown determin- tion problem that can be solved efficiently, e.g., using interior istic parameters in the linear model point methods [19], [20]. We then develop a closed-form solu- (1) tion to the minimax estimation problem for the case in which the weighting and have the same eigenvector matrix. where is a known matrix with full rank , and is a In particular, when , we show that the optimal estimator is zero-mean random vector with covariance . We assume that a shrunken estimator proposed by Mayer and Willke [7], with a is known to satisfy the weighted norm constraint specific choice of shrinkage factor, that explicitly takes the prior for some positive definite matrix and scalar , where information into account. We demonstrate through simulations, . that the minimax MSE estimator can increase the performance We estimate using a linear estimator so that for over the conventional LS approach. some matrix . The MSE of the estimator is We then consider the case in which the model matrix is given by not known exactly, but is rather given by , where is known, and is a bounded perturbation matrix.
Load more

Recommended publications
  • Lecture 13: Simple Linear Regression in Matrix Format
    11:55 Wednesday 14th October, 2015 See updates and corrections at http://www.stat.cmu.edu/~cshalizi/mreg/ Lecture 13: Simple Linear Regression in Matrix Format 36-401, Section B, Fall 2015 13 October 2015 Contents 1 Least Squares in Matrix Form 2 1.1 The Basic Matrices . .2 1.2 Mean Squared Error . .3 1.3 Minimizing the MSE . .4 2 Fitted Values and Residuals 5 2.1 Residuals . .7 2.2 Expectations and Covariances . .7 3 Sampling Distribution of Estimators 8 4 Derivatives with Respect to Vectors 9 4.1 Second Derivatives . 11 4.2 Maxima and Minima . 11 5 Expectations and Variances with Vectors and Matrices 12 6 Further Reading 13 1 2 So far, we have not used any notions, or notation, that goes beyond basic algebra and calculus (and probability). This has forced us to do a fair amount of book-keeping, as it were by hand. This is just about tolerable for the simple linear model, with one predictor variable. It will get intolerable if we have multiple predictor variables. Fortunately, a little application of linear algebra will let us abstract away from a lot of the book-keeping details, and make multiple linear regression hardly more complicated than the simple version1. These notes will not remind you of how matrix algebra works. However, they will review some results about calculus with matrices, and about expectations and variances with vectors and matrices. Throughout, bold-faced letters will denote matrices, as a as opposed to a scalar a. 1 Least Squares in Matrix Form Our data consists of n paired observations of the predictor variable X and the response variable Y , i.e., (x1; y1);::: (xn; yn).
    [Show full text]
  • In This Segment, We Discuss a Little More the Mean Squared Error
    MITOCW | MITRES6_012S18_L20-04_300k In this segment, we discuss a little more the mean squared error. Consider some estimator. It can be any estimator, not just the sample mean. We can decompose the mean squared error as a sum of two terms. Where does this formula come from? Well, we know that for any random variable Z, this formula is valid. And if we let Z be equal to the difference between the estimator and the value that we're trying to estimate, then we obtain this formula here. The expected value of our random variable Z squared is equal to the variance of that random variable plus the square of its mean. Let us now rewrite these two terms in a more suggestive way. We first notice that theta is a constant. When you add or subtract the constant from a random variable, the variance does not change. So this term is the same as the variance of theta hat. This quantity here, we will call it the bias of the estimator. It tells us whether theta hat is systematically above or below than the unknown parameter theta that we're trying to estimate. And using this terminology, this term here is just equal to the square of the bias. So the mean squared error consists of two components, and these capture different aspects of an estimator's performance. Let us see what they are in a concrete setting. Suppose that we're estimating the unknown mean of some distribution, and that our estimator is the sample mean. In this case, the mean squared error is the variance, which we know to be sigma squared over n, plus the bias term.
    [Show full text]
  • Minimum Mean Squared Error Model Averaging in Likelihood Models
    Statistica Sinica 26 (2016), 809-840 doi:http://dx.doi.org/10.5705/ss.202014.0067 MINIMUM MEAN SQUARED ERROR MODEL AVERAGING IN LIKELIHOOD MODELS Ali Charkhi1, Gerda Claeskens1 and Bruce E. Hansen2 1KU Leuven and 2University of Wisconsin, Madison Abstract: A data-driven method for frequentist model averaging weight choice is developed for general likelihood models. We propose to estimate the weights which minimize an estimator of the mean squared error of a weighted estimator in a local misspecification framework. We find that in general there is not a unique set of such weights, meaning that predictions from multiple model averaging estimators might not be identical. This holds in both the univariate and multivariate case. However, we show that a unique set of empirical weights is obtained if the candidate models are appropriately restricted. In particular a suitable class of models are the so-called singleton models where each model only includes one parameter from the candidate set. This restriction results in a drastic reduction in the computational cost of model averaging weight selection relative to methods which include weights for all possible parameter subsets. We investigate the performance of our methods in both linear models and generalized linear models, and illustrate the methods in two empirical applications. Key words and phrases: Frequentist model averaging, likelihood regression, local misspecification, mean squared error, weight choice. 1. Introduction We study a focused version of frequentist model averaging where the mean squared error plays a central role. Suppose we have a collection of models S 2 S to estimate a population quantity µ, this is the focus, leading to a set of estimators fµ^S : S 2 Sg.
    [Show full text]
  • Calibration: Calibrate Your Model
    DUE TODAYCOMPUTER FILES AND QUESTIONS for Assgn#6 Assignment # 6 Steady State Model Calibration: Calibrate your model. If you want to conduct a transient calibration, talk with me first. Perform calibration using UCODE. Be sure your report addresses global, graphical, and spatial measures of error as well as common sense. Consider more than one conceptual model and compare the results. Remember to make a prediction with your calibrated models and evaluate confidence in your prediction. Be sure to save your files because you will want to use them later in the semester. Suggested Calibration Report Outline Title Introduction describe the system to be calibrated (use portions of your previous report as appropriate) Observations to be matched in calibration type of observations locations of observations observed values uncertainty associated with observations explain specifically what the observation will be matched to in the model Calibration Procedure Evaluation of calibration residuals parameter values quality of calibrated model Calibrated model results Predictions Uncertainty associated with predictions Problems encountered, if any Comparison with uncalibrated model results Assessment of future work needed, if appropriate Summary/Conclusions Summary/Conclusions References submit the paper as hard copy and include it in your zip file of model input and output submit the model files (input and output for both simulations) in a zip file labeled: ASSGN6_LASTNAME.ZIP Calibration (Parameter Estimation, Optimization, Inversion, Regression) adjusting parameter values, boundary conditions, model conceptualization, and/or model construction until the model simulation matches field observations We calibrate because 1. the field measurements are not accurate reflecti ons of the model scale properties, and 2.
    [Show full text]
  • 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.
    [Show full text]
  • An Analysis of Random Design Linear Regression
    An Analysis of Random Design Linear Regression Daniel Hsu1,2, Sham M. Kakade2, and Tong Zhang1 1Department of Statistics, Rutgers University 2Department of Statistics, Wharton School, University of Pennsylvania Abstract The random design setting for linear regression concerns estimators based on a random sam- ple of covariate/response pairs. This work gives explicit bounds on the prediction error for the ordinary least squares estimator and the ridge regression estimator under mild assumptions on the covariate/response distributions. In particular, this work provides sharp results on the \out-of-sample" prediction error, as opposed to the \in-sample" (fixed design) error. Our anal- ysis also explicitly reveals the effect of noise vs. modeling errors. The approach reveals a close connection to the more traditional fixed design setting, and our methods make use of recent ad- vances in concentration inequalities (for vectors and matrices). We also describe an application of our results to fast least squares computations. 1 Introduction In the random design setting for linear regression, one is given pairs (X1;Y1);:::; (Xn;Yn) of co- variates and responses, sampled from a population, where each Xi are random vectors and Yi 2 R. These pairs are hypothesized to have the linear relationship > Yi = Xi β + i for some linear map β, where the i are noise terms. The goal of estimation in this setting is to find coefficients β^ based on these (Xi;Yi) pairs such that the expected prediction error on a new > 2 draw (X; Y ) from the population, measured as E[(X β^ − Y ) ], is as small as possible.
    [Show full text]
  • Review of Basic Statistics and the Mean Model for Forecasting
    Review of basic statistics and the simplest forecasting model: the sample mean Robert Nau Fuqua School of Business, Duke University August 2014 Most of what you need to remember about basic statistics Consider a random variable called X that is a time series (a set of observations ordered in time) consisting of the following 20 observations: 114, 126, 123, 112, 68, 116, 50, 108, 163, 79, 67, 98, 131, 83, 56, 109, 81, 61, 90, 92. 180 160 140 120 100 80 60 40 20 0 0 5 10 15 20 25 How should we forecast what will happen next? The simplest forecasting model that we might consider is the mean model,1 which assumes that the time series consists of independently and identically distributed (“i.i.d.”) values, as if each observation is randomly drawn from the same population. Under this assumption, the next value should be predicted to be equal to the historical sample mean if the goal is to minimize mean squared error. This might sound trivial, but it isn’t. If you understand the details of how this works, you are halfway to understanding linear regression. (No kidding: see section 3 of the regression notes handout.) To set the stage for using the mean model for forecasting, let’s review some of the most basic concepts of statistics. Let: X = a random variable, with its individual values denoted by x1, x2, etc. N = size of the entire population of values of X (possibly infinite)2 n = size of a finite sample of X 1 This might also be called a “constant model” or an “intercept-only regression.” 2 The term “population” does not refer to the number of distinct values of X.
    [Show full text]
  • Linear Regression
    Linear regression Fitting a line to a bunch of points. Linear regression Example: college GPAs Better predictions with more information We also have SAT scores of all students. Distribution of GPAs of Mean squared error students at a certain Ivy (MSE) drops to 0.43. League university. What GPA to predict for a random student from this group? • Without further information, predict the mean, 2.47. • What is the average squared error of this prediction? This is a regression problem with: 2 That is, E[((student's GPA) − (predicted GPA)) ]? • Predictor variable: SAT score The variance of the distribution, 0.55. • Response variable: College GPA Parametrizing a line The line fitting problem (1) (1) (n) (n) Pick a line (a; b) based on (x ; y );:::; (x ; y ) 2 R × R A line can be parameterized as y = ax + b (a: slope, b: intercept). • x(i); y (i) are predictor and response variables. E.g. SAT score, GPA of ith student. • Minimize the mean squared error, n 1 X MSE(a; b) = (y (i) − (ax(i) + b))2: n i=1 This is the loss function. Minimizing the loss function Multivariate regression: diabetes study Given (x(1); y (1));:::; (x(n); y (n)), minimize n Data from n = 442 diabetes patients. X L(a; b) = (y (i) − (ax(i) + b))2: i=1 For each patient: • 10 features x = (x1;:::; x10) age, sex, body mass index, average blood pressure, and six blood serum measurements. • A real value y: the progression of the disease a year later. Regression problem: • response y 2 R 10 • predictor variables x 2 R Least-squares regression Back to the diabetes data 10 Linear function of 10 variables: for x 2 R , • No predictor variables: mean squared error (MSE) = 5930 f (x) = w1x1 + w2x2 + ··· + w10x10 + b = w · x + b • One predictor ('bmi'): MSE = 3890 where w = (w1; w2;:::; w10).
    [Show full text]
  • Lecture 9 Estimators
    Lecture 9: Estimators 1 of 13 Course: Mathematical Statistics Term: Fall 2018 Instructor: Gordan Žitkovi´c Lecture 9 Estimators We defined an estimator as any function of the data which is not allowed to depend on the value of the unknown parameters. Such a broad definition allows for very silly examples. To rule out bad estimators and to find the ones that will provide us with the most “bang for the buck”, we need to discuss, first, what it means for an estimator to be “good”. There is no one answer to this question, and a large part of entire discipline (decision theory) tries to answer it. Some desirable properties of estimators are, however, hard to argue with, so we start from those. 9.1 Unbiased estimators Definition 9.1.1. We say that an estimator qˆ is unbiased for the pa- rameter q if E[qˆ] = q. We also define the bias E[qˆ] − q of qˆ, and denote it by bias(qˆ). In words, qˆ is unbiased if we do not expect qˆ to be systematically above or systematically below q. Clearly, in order to talk about the bias of an estimator, we need to specify what that estimator is trying to estimate. We will see below that same estimator can be unbiased as an estimator for one parameter, but biased when used to estimate another parameter. Another important question that needs to be asked about Definition 9.1.1 above is: what does it mean to take the expected value of qˆ, when we do not know what its distribution is.
    [Show full text]
  • MS&E 226: Fundamentals of Data Science
    MS&E 226: Fundamentals of Data Science Lecture 6: Bias and variance Ramesh Johari 1 / 49 Our plan today In general, when creating predictive models, we are trading off two different goals: I On one hand, we want models to fit the training data well. I On the other hand, we want to avoid models that become so finely tuned to the training data that they perform poorly on new data. In this lecture we develop a systematic vocabulary for talking about this kind of trade off, through the notions of bias and variance. 2 / 49 Conditional expectation 3 / 49 Conditional expectation Given the population model for X~ and Y , suppose we are allowed to choose any predictive model f^ we want. What is the best one? ^ ~ 2 minimize EX;Y~ [(Y − f(X)) ]: (Here expectation is over (X;Y~ ).) Theorem The predictive model that minimizes squared error is ^ ~ ~ f(X) = EY [Y jX]. 4 / 49 Conditional expectation Proof: ^ ~ 2 EX;Y~ [(Y − f(X)) ] ~ ~ ^ ~ 2 = EX;Y~ [(Y − EY [Y jX] + EY [Y jX] − f(X)) ] ~ 2 ~ ^ ~ 2 = EX;Y~ [(Y − EY [Y jX]) ] + EY [(E[Y jX] − f(X)) ] ~ ~ ^ ~ + 2EX;Y~ [(Y − EY [Y jX])(EY [Y jX] − f(X))]: The first two terms are positive, and minimized if ^ ~ ~ f(X) = EY [Y jX]. For the third term, using the tower property of conditional expectation: ~ ~ ^ ~ EX;Y~ [(Y − EY [Y jX])(EY [Y jX] − f(X))] h ~ ~ ~ ^ ~ i = EX~ EY [Y − EY [Y jX]jX](EY [Y jX] − f(X)) = 0: So the squared error minimizing solution is to choose ^ ~ ~ f(X) = EY [Y jX].
    [Show full text]
  • R2 Is Rescaled Mean Squared Error
    R2 is rescaled mean squared error Brendan O’Connor September 3, 2009 R2 – “the coefficient of determination” – is a rescaling of MSE (relative to the dataset in question). Alternative definitions are (1) the regression’s proportion of total sum of squares, or (2) the squared correlation between predictions and responses. Setup: items xi and we’re targeting real-valued responses yi by fitting a function f(x). Let’s be vague on training vs. test sets; all that matters is we want to evaluate the prediction function’s accuracy on these items. Then 2 MSE = X(f(xi) − yi) /N i Let’s use definition #1 of (1−R2), that it’s the “sum of squared error divided by total sum of squares”. These terms are 2 • SStot = (yi −E[y]) : total sum of squares, which is a rescaling of response variance 2 • SSerr = (yi−f(xi)) : sum of squared errors, a.k.a. “residual sum of squares”, a rescaling of the model’s predictions’ MSE So we have: 2 SSerr 1 − R = SStot − 2 P(yi f(xi)) /N = − 2 P(yi E[y]) /N MSE = V ar(y) (Mean)SqErr of predictions = (Mean)SqErr of guessing the mean R2 can be thought of as a rescaling of MSE, comparing it to the variance of the outcome response. It’s nice to interpret because it’s bounded between 0 and 1. Higher is better. If MSE=0, then R2 = 100%: you have perfect predictions. 1 If MSE is as bad as just guessing the mean for everything, then R2 = 0%: about as bad as possible.
    [Show full text]
  • A Simulation Study of the Robustness of the Least Median of Squares Estimator of Slope in a Regression Through the Origin Model
    A SIMULATION STUDY OF THE ROBUSTNESS OF THE LEAST MEDIAN OF SQUARES ESTIMATOR OF SLOPE IN A REGRESSION THROUGH THE ORIGIN MODEL by THILANKA DILRUWANI PARANAGAMA B.Sc., University of Colombo, Sri Lanka, 2005 A REPORT submitted in partial fulfillment of the requirements for the degree MASTER OF SCIENCE Department of Statistics College of Arts and Sciences KANSAS STATE UNIVERSITY Manhattan, Kansas 2010 Approved by: Major Professor Dr. Paul Nelson Abstract The principle of least squares applied to regression models estimates parameters by minimizing the mean of squared residuals. Least squares estimators are optimal under normality but can perform poorly in the presence of outliers. This well known lack of robustness motivated the development of alternatives, such as least median of squares estimators obtained by minimizing the median of squared residuals. This report uses simulation to examine and compare the robustness of least median of squares estimators and least squares estimators of the slope of a regression line through the origin in terms of bias and mean squared error in a variety of conditions containing outliers created by using mixtures of normal and heavy tailed distributions. It is found that least median of squares estimation is almost as good as least squares estimation under normality and can be much better in the presence of outliers. Table of Contents List of Figures ................................................................................................................................ iv List of Tables .................................................................................................................................
    [Show full text]