DOCSLIB.ORG

Statistical Inference Parametric Inference Maximum Likelihood

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Statistical Inference Parametric Inference Maximum Likelihood Statistical Inference Parametric Inference Maximum Likelihood Inference Exponential Families Expectation Maximization (EM) Bayesian Inference Statistical Decison Theory IP, José Bioucas Dias, IST, 2007 1 Statistical Inference Statistics aims at retriving the “causes” (e.g., parameters of a pdf) from the observations (effects) Probability Statistics Statistical inference problems can thus be seen as Inverse Problems As a result of this perpective, at the eighteenth century (at the time of Bayes and Laplace) Statistics was often called Inverse Probability IP, José Bioucas Dias, IST, 2007 2 Parametric Inference Consider the parametric model where is the parameter space and is the parameter The problem of inference reduces to the estimation of from ; i.e, Parameters of interest and nuisance parameters Let Sometimes we are only interested in some function that depends only on - parameter of interest; - nuisance parameter Example: IP, José Bioucas Dias, IST, 2007 3 Parametric Inference (theoretical limits) The Cramer Rao Lower Bound (CRLB) Under under appropriate regularity conditions, the covariance matrix of any Unbiased estimator , satisfies where is the Fisher information matrix given by An unbiased estimator that attains the CRLB may be found iif For some function h. The estimator is IP, José Bioucas Dias, IST, 2007 4 CRLB for the general Gaussian case Example: Parameter of a signal in white noise If Example: Known signal in unknown white noise IP, José Bioucas Dias, IST, 2007 5 Maximum Likelihood Method is the likelihood function If for all f we can use the log-likelihood Example (Bernoulli) IP, José Bioucas Dias, IST, 2007 6 Maximum Likelihood Example (Uniform) 1 1 IP, José Bioucas Dias, IST, 2007 7 Maximum Likelihood Example (Gaussian) IID Sample mean Sample variace IP, José Bioucas Dias, IST, 2007 8 Maximum Likelihood Example (Multivariate Gaussian) IID Sample mean Sample covariance IP, José Bioucas Dias, IST, 2007 9 Maximum Likelihood (linear observation model) Example: Linear observation in Gaussian noise A is full rank IP, José Bioucas Dias, IST, 2007 10 Example: Linear observation in Gaussian noise (cont.) • MLE is equivalent to the LSE using the norm • If , , is given by the Moore-Penrose Pseudo-Inverse • is a projection matrix (SVD) • If the noise is zero-mean but not Gaussian, the Best Linear Unbiased estimator (BLUE) is still given by IP, José Bioucas Dias, IST, 2007 11 Maximum likelihood Linear observation in Gaussian noise MLE Properties (MLE is optimal for the linear model) • Is the Minimum Variance Unbiased (MVU) estimator [ and is the minimum among all unbiased estimators] • Is efficient (it attains the Camer Rao Lower Bound (CRLB)) • Its PDF is IP, José Bioucas Dias, IST, 2007 12 Maximum likelihood (characterization) Appealing properties of MLE Let A sequence of IID vectors in and 1. The MLE is consistent: ( denotes the true parameter) 2. The MLE is equivariant: if is the MLE estimate of , then is the MLE estimate of 3. The MLE (under appropriate regularity conditions) is asymptotically Normal and optimal or efficient: Fisher information matrix IP, José Bioucas Dias, IST, 2007 13 The exponential Family Definition: the set an exponential family of dimension k if there there are functions such that is a sufficient statistic for f , i.e, Theorem: (Neyman-Fisher Factorization) is a sufficient statistic for f iif can be factored as IP, José Bioucas Dias, IST, 2007 14 The exponential family Natural (or canonical) form Given an exponential family, it is always possible to introduce the change of variables and the reparemeterization such that Since is a PDF, it must integrate to one IP, José Bioucas Dias, IST, 2007 15 The exponential family (The partition function) Computing moments from the derivatives of the partition function After some calculus IP, José Bioucas Dias, IST, 2007 16 The exponential family (IID sequences) Let a member of an exponential family defined by The density of the IID sequence is belongs exponential family defined by IP, José Bioucas Dias, IST, 2007 17 Examples of exponential families Many of the most common probabilistic models belong to exponential families; e.g., Gaussian, Poisson, Bernoulli, binomial, exponential, gamma, and beta. Example: Canonical form IP, José Bioucas Dias, IST, 2007 18 Examples of exponential families (Gaussian) Example: Canonical form IP, José Bioucas Dias, IST, 2007 19 Computing maximum likelihood estimates Very often the MLE can not be found analytically. Commonly used numerical methods: 1. Newton-Raphson 2. Scoring 3. Expectation Maximization (EM) Newton-Raphson method Scoring method Can be computed off-line IP, José Bioucas Dias, IST, 2007 20 Computing maximum likelihood estimates (EM) Expectation Maximization (EM) [Dempster, Laird, and Rubin, 1977] Suppose that is hard to maximize But we can find a vector z such that is easy to maximze and Idea: iterate between two steps: E-step: “Fill in z” in Terminology Observed data M-step: Maximize Missing data Complete data IP, José Bioucas Dias, IST, 2007 21 Expectation maximization The EM algorithm 1. Pick up a starting vector : repeat steps 2. and 3. 2. E-step: Calculate 3. M-step Alternatively (GEM) IP, José Bioucas Dias, IST, 2007 22 Expectation maximization The EM (GEM) algorithm always increases the likelihood. Define 1. 2. Kulback Leibler distance 3. 4. KL distance maximization IP, José Bioucas Dias, IST, 2007 23 Expectation maximization (why does it work?) IP, José Bioucas Dias, IST, 2007 24 EM: Mixtures of densities Let be the random variable that selects the active mode: where and IP, José Bioucas Dias, IST, 2007 25 EM: Mixtures of densities Consider now that is a sequence of IID random variables Let be IID random variables, where selects the active mode in the sample : IP, José Bioucas Dias, IST, 2007 26 EM: Mixtures of densities Equivalent Q Where is the sample mean of x, i.e., IP, José Bioucas Dias, IST, 2007 27 EM: Mixtures of densities E-step: M-step: IP, José Bioucas Dias, IST, 2007 28 EM: Mixtures of densities E-step: M-step: IP, José Bioucas Dias, IST, 2007 29 EM: Mixtures of Gaussian densities (MOGs) E-step: M-step: Weighted sample mean Weighted sample covariance IP, José Bioucas Dias, IST, 2007 30 EM: Mixtures of Gaussian densities. 1D Example 0 1 0.6316 -0.0288 1.0287 0.6258 3 3 0.3158 2.8952 2.5649 0.3107 6 10 0.0526 6.1687 7.3980 0.0635 p = 3 loglikelihood L(fk) N = 1900 -3800 -4000 -4200 -4400 -4600 -4800 -5000 -5200 0 5 10 15 20 25 30 IP, José Bioucas Dias, IST, 2007 31 EM: Mixtures of Gaussian Densities (MOGs) 0 1 0.6316 p = 3 Example – 1D 3 3 0.3158 N = 1900 6 10 0.0526 0.35 0.35 hist hist est MOG est modes 0.3 0.3 true MOG true modes 0.25 0.25 0.2 0.2 0.15 0.15 0.1 0.1 0.05 0.05 0 0 -5 0 5 10 15 -6 -4 -2 0 2 4 6 8 10 12 IP, José Bioucas Dias, IST, 2007 32 EM: Mixtures of Gaussian Densities: 2D Example MOG with determination of the number of modes [M. Figueiredo, 2002] k=3 2 0 -2 -2 0 2 4 IP, José Bioucas Dias, IST, 2007 33 Bayesian Estimation IP, José Bioucas Dias, IST, 2007 34 The Bayesian Philosophy ([Wasserman, 2004]) Bayesian Inference B1 – Probabilities describe degrees of belief, not limiting relative frequency B2 – We can make probability statements about parameters, even though they are fixed parameters B3 – We make inferences about a parameter by producing a probalility distribution for Frequencist or Classical Inference F1 – Probabilities refer to limiting relative frequencies and are objective properties of the real world F2 – Parameters are fixed unknown parameters F3 – The criteria for obtaining statistical procedures are based on long run frequency properties. IP, José Bioucas Dias, IST, 2007 35 The Bayesian Philosophy Observation model unknown observation Prior knowledge Bayesian Inference Classical Inference describes degrees of belief (subjective), not limiting frequency IP, José Bioucas Dias, IST, 2007 36 The Bayesian method 1. Choose a prior density , called the prior (or a priori) distribution that expresses our beliefs about f, before we see any data 2. Choose the observation model that reflects our belief about g given f 3. Calculate the posterior (or a posteriori) distribution using the Bayes law: where is the marginal on g (other names: evidence, unconditional, predictive) 4. Any inference should be based on the posterior IP, José Bioucas Dias, IST, 2007 37 The Bayesian method Example: Let IID and 4 ==0.5 for = >1, pulls 3.5 ==1 ==2 towards 1/2 3 ==10 2.5 2 1.5 1 0.5 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 IP, José Bioucas Dias, IST, 2007 38 Example (cont.): (Bernoulli observations, Beta prior) Observation model Prior Posterior Thus, IP, José Bioucas Dias, IST, 2007 39 Example (cont.): (Bernoulli observations, Beta prior) Maximum a posteriori estimate (MAP) • Total ignorance: flat prior = =1 • Note that for large values of n The von Mises Theorem If the prior is continuous and not zero at the location of the MLestimate, then, IP, José Bioucas Dias, IST, 2007 40 Conjugate priors In the previous example, the prior and the posterior are both Beta distributed. We say that the prior is conjugate with respect to the model • Formally, let and be two parametrized families of priors and observation models, respectively • is a conjugate family for if for some • Very often, prior information about f is very small, allowing to select conjugate priors • Conjugate priors why? Computing the posterior density simply consists in updating the parameters of the prior IP, José Bioucas Dias, IST, 2007 41 Conjugate priors (Gaussian observation, Gaussian prior) • Gaussian observations • Gaussian prior • The posterior distribution is Gaussian 1.
Load more

Recommended publications
  • Parametric Models
    An Introduction to Event History Analysis Oxford Spring School June 18-20, 2007 Day Two: Regression Models for Survival Data Parametric Models We’ll spend the morning introducing regression-like models for survival data, starting with fully parametric (distribution-based) models. These tend to be very widely used in social sciences, although they receive almost no use outside of that (e.g., in biostatistics). A General Framework (That Is, Some Math) Parametric models are continuous-time models, in that they assume a continuous parametric distribution for the probability of failure over time. A general parametric duration model takes as its starting point the hazard: f(t) h(t) = (1) S(t) As we discussed yesterday, the density is the probability of the event (at T ) occurring within some differentiable time-span: Pr(t ≤ T < t + ∆t) f(t) = lim . (2) ∆t→0 ∆t The survival function is equal to one minus the CDF of the density: S(t) = Pr(T ≥ t) Z t = 1 − f(t) dt 0 = 1 − F (t). (3) This means that another way of thinking of the hazard in continuous time is as a conditional limit: Pr(t ≤ T < t + ∆t|T ≥ t) h(t) = lim (4) ∆t→0 ∆t i.e., the conditional probability of the event as ∆t gets arbitrarily small. 1 A General Parametric Likelihood For a set of observations indexed by i, we can distinguish between those which are censored and those which aren’t... • Uncensored observations (Ci = 1) tell us both about the hazard of the event, and the survival of individuals prior to that event.
    [Show full text]
  • Robust Estimation on a Parametric Model Via Testing 3 Estimators
    Bernoulli 22(3), 2016, 1617–1670 DOI: 10.3150/15-BEJ706 Robust estimation on a parametric model via testing MATHIEU SART 1Universit´ede Lyon, Universit´eJean Monnet, CNRS UMR 5208 and Institut Camille Jordan, Maison de l’Universit´e, 10 rue Tr´efilerie, CS 82301, 42023 Saint-Etienne Cedex 2, France. E-mail: [email protected] We are interested in the problem of robust parametric estimation of a density from n i.i.d. observations. By using a practice-oriented procedure based on robust tests, we build an estimator for which we establish non-asymptotic risk bounds with respect to the Hellinger distance under mild assumptions on the parametric model. We show that the estimator is robust even for models for which the maximum likelihood method is bound to fail. A numerical simulation illustrates its robustness properties. When the model is true and regular enough, we prove that the estimator is very close to the maximum likelihood one, at least when the number of observations n is large. In particular, it inherits its efficiency. Simulations show that these two estimators are almost equal with large probability, even for small values of n when the model is regular enough and contains the true density. Keywords: parametric estimation; robust estimation; robust tests; T-estimator 1. Introduction We consider n independent and identically distributed random variables X1,...,Xn de- fined on an abstract probability space (Ω, , P) with values in the measure space (X, ,µ). E F We suppose that the distribution of Xi admits a density s with respect to µ and aim at estimating s by using a parametric approach.
    [Show full text]
  • A PARTIALLY PARAMETRIC MODEL 1. Introduction a Notable
    A PARTIALLY PARAMETRIC MODEL DANIEL J. HENDERSON AND CHRISTOPHER F. PARMETER Abstract. In this paper we propose a model which includes both a known (potentially) nonlinear parametric component and an unknown nonparametric component. This approach is feasible given that we estimate the finite sample parameter vector and the bandwidths simultaneously. We show that our objective function is asymptotically equivalent to the individual objective criteria for the parametric parameter vector and the nonparametric function. In the special case where the parametric component is linear in parameters, our single-step method is asymptotically equivalent to the two-step partially linear model esti- mator in Robinson (1988). Monte Carlo simulations support the asymptotic developments and show impressive finite sample performance. We apply our method to the case of a partially constant elasticity of substitution production function for an unbalanced sample of 134 countries from 1955-2011 and find that the parametric parameters are relatively stable for different nonparametric control variables in the full sample. However, we find substan- tial parameter heterogeneity between developed and developing countries which results in important differences when estimating the elasticity of substitution. 1. Introduction A notable development in applied economic research over the last twenty years is the use of the partially linear regression model (Robinson 1988) to study a variety of phenomena. The enthusiasm for the partially linear model (PLM) is not confined to
    [Show full text]
  • Options for Development of Parametric Probability Distributions for Exposure Factors
    EPA/600/R-00/058 July 2000 www.epa.gov/ncea Options for Development of Parametric Probability Distributions for Exposure Factors National Center for Environmental Assessment-Washington Office Office of Research and Development U.S. Environmental Protection Agency Washington, DC 1 Introduction The EPA Exposure Factors Handbook (EFH) was published in August 1997 by the National Center for Environmental Assessment of the Office of Research and Development (EPA/600/P-95/Fa, Fb, and Fc) (U.S. EPA, 1997a). Users of the Handbook have commented on the need to fit distributions to the data in the Handbook to assist them when applying probabilistic methods to exposure assessments. This document summarizes a system of procedures to fit distributions to selected data from the EFH. It is nearly impossible to provide a single distribution that would serve all purposes. It is the responsibility of the assessor to determine if the data used to derive the distributions presented in this report are representative of the population to be assessed. The system is based on EPA’s Guiding Principles for Monte Carlo Analysis (U.S. EPA, 1997b). Three factors—drinking water, population mobility, and inhalation rates—are used as test cases. A plan for fitting distributions to other factors is currently under development. EFH data summaries are taken from many different journal publications, technical reports, and databases. Only EFH tabulated data summaries were analyzed, and no attempt was made to obtain raw data from investigators. Since a variety of summaries are found in the EFH, it is somewhat of a challenge to define a comprehensive data analysis strategy that will cover all cases.
    [Show full text]
  • Exponential Families and Theoretical Inference
    EXPONENTIAL FAMILIES AND THEORETICAL INFERENCE Bent Jørgensen Rodrigo Labouriau August, 2012 ii Contents Preface vii Preface to the Portuguese edition ix 1 Exponential families 1 1.1 Definitions . 1 1.2 Analytical properties of the Laplace transform . 11 1.3 Estimation in regular exponential families . 14 1.4 Marginal and conditional distributions . 17 1.5 Parametrizations . 20 1.6 The multivariate normal distribution . 22 1.7 Asymptotic theory . 23 1.7.1 Estimation . 25 1.7.2 Hypothesis testing . 30 1.8 Problems . 36 2 Sufficiency and ancillarity 47 2.1 Sufficiency . 47 2.1.1 Three lemmas . 48 2.1.2 Definitions . 49 2.1.3 The case of equivalent measures . 50 2.1.4 The general case . 53 2.1.5 Completeness . 56 2.1.6 A result on separable σ-algebras . 59 2.1.7 Sufficiency of the likelihood function . 60 2.1.8 Sufficiency and exponential families . 62 2.2 Ancillarity . 63 2.2.1 Definitions . 63 2.2.2 Basu's Theorem . 65 2.3 First-order ancillarity . 67 2.3.1 Examples . 67 2.3.2 Main results . 69 iii iv CONTENTS 2.4 Problems . 71 3 Inferential separation 77 3.1 Introduction . 77 3.1.1 S-ancillarity . 81 3.1.2 The nonformation principle . 83 3.1.3 Discussion . 86 3.2 S-nonformation . 91 3.2.1 Definition . 91 3.2.2 S-nonformation in exponential families . 96 3.3 G-nonformation . 99 3.3.1 Transformation models . 99 3.3.2 Definition of G-nonformation . 103 3.3.3 Cox's proportional risks model .
    [Show full text]
  • A Primer on the Exponential Family of Distributions
    A Primer on the Exponential Family of Distributions David R. Clark, FCAS, MAAA, and Charles A. Thayer 117 A PRIMER ON THE EXPONENTIAL FAMILY OF DISTRIBUTIONS David R. Clark and Charles A. Thayer 2004 Call Paper Program on Generalized Linear Models Abstract Generahzed Linear Model (GLM) theory represents a significant advance beyond linear regression theor,], specifically in expanding the choice of probability distributions from the Normal to the Natural Exponential Famdy. This Primer is intended for GLM users seeking a hand)' reference on the model's d]smbutional assumptions. The Exponential Faintly of D,smbutions is introduced, with an emphasis on variance structures that may be suitable for aggregate loss models m property casualty insurance. 118 A PRIMER ON THE EXPONENTIAL FAMILY OF DISTRIBUTIONS INTRODUCTION Generalized Linear Model (GLM) theory is a signtficant advance beyond linear regression theory. A major part of this advance comes from allowmg a broader famdy of distributions to be used for the error term, rather than just the Normal (Gausstan) distributton as required m hnear regression. More specifically, GLM allows the user to select a distribution from the Exponentzal Family, which gives much greater flexibility in specifying the vanance structure of the variable being forecast (the "'response variable"). For insurance apphcations, this is a big step towards more realistic modeling of loss distributions, while preserving the advantages of regresston theory such as the ability to calculate standard errors for estimated parameters. The Exponentml family also includes several d~screte distributions that are attractive candtdates for modehng clatm counts and other events, but such models will not be considered here The purpose of this Primer is to give the practicmg actuary a basic introduction to the Exponential Family of distributions, so that GLM models can be designed to best approximate the behavior of the insurance phenomenon.
    [Show full text]
  • 2 May 2021 Parametric Bootstrapping in a Generalized Extreme Value
    Parametric bootstrapping in a generalized extreme value regression model for binary response DIOP Aba∗ E-mail: [email protected] Equipe de Recherche en Statistique et Mod`eles Al´eatoires Universit´eAlioune Diop, Bambey, S´en´egal DEME El Hadji E-mail: [email protected] Laboratoire d’Etude et de Recherche en Statistique et D´eveloppement Universit´eGaston Berger, Saint-Louis, S´en´egal Abstract Generalized extreme value (GEV) regression is often more adapted when we in- vestigate a relationship between a binary response variable Y which represents a rare event and potentiel predictors X. In particular, we use the quantile function of the GEV distribution as link function. Bootstrapping assigns measures of accuracy (bias, variance, confidence intervals, prediction error, test of hypothesis) to sample es- timates. This technique allows estimation of the sampling distribution of almost any statistic using random sampling methods. Bootstrapping estimates the properties of an estimator by measuring those properties when sampling from an approximating distribution. In this paper, we fitted the generalized extreme value regression model, then we performed parametric bootstrap method for testing hupthesis, estimating confidence interval of parameters for generalized extreme value regression model and a real data application. arXiv:2105.00489v1 [stat.ME] 2 May 2021 Keywords: generalized extreme value, parametric bootstrap, confidence interval, test of hypothesis, Dengue. ∗Corresponding author 1 1 Introduction Classical methods of statistical inference do not provide correct answers to all the concrete problems the user may have. They are only valid under some specific application conditions (e.g. normal distribution of populations, independence of samples etc.).
    [Show full text]
  • Parametric and Nonparametric Bayesian Models for Ecological Inference in 2 × 2 Tables∗
    Parametric and Nonparametric Bayesian Models for Ecological Inference in 2 × 2 Tables∗ Kosuke Imai† Department of Politics, Princeton University Ying Lu‡ Office of Population Research, Princeton University November 18, 2004 Abstract The ecological inference problem arises when making inferences about individual behavior from aggregate data. Such a situation is frequently encountered in the social sciences and epi- demiology. In this article, we propose a Bayesian approach based on data augmentation. We formulate ecological inference in 2×2 tables as a missing data problem where only the weighted average of two unknown variables is observed. This framework directly incorporates the deter- ministic bounds, which contain all information available from the data, and allow researchers to incorporate the individual-level data whenever available. Within this general framework, we first develop a parametric model. We show that through the use of an EM algorithm, the model can formally quantify the effect of missing information on parameter estimation. This is an important diagnostic for evaluating the degree of aggregation effects. Next, we introduce a nonparametric model using a Dirichlet process prior to relax the distributional assumption of the parametric model. Through simulations and an empirical application, we evaluate the rela- tive performance of our models in various situations. We demonstrate that in typical ecological inference problems, the fraction of missing information often exceeds 50 percent. We also find that the nonparametric model generally outperforms the parametric model, although the latter gives reasonable in-sample predictions when the bounds are informative. C-code, along with an R interface, is publicly available for implementing our Markov chain Monte Carlo algorithms to fit the proposed models.
    [Show full text]
  • Parametric Versus Semi and Nonparametric Regression Models
    Parametric versus Semi and Nonparametric Regression Models Hamdy F. F. Mahmoud1;2 1 Department of Statistics, Virginia Tech, Blacksburg VA, USA. 2 Department of Statistics, Mathematics and Insurance, Assiut University, Egypt. Abstract Three types of regression models researchers need to be familiar with and know the requirements of each: parametric, semiparametric and nonparametric regression models. The type of modeling used is based on how much information are available about the form of the relationship between response variable and explanatory variables, and the random error distribution. In this article, differences between models, common methods of estimation, robust estimation, and applications are introduced. The R code for all the graphs and analyses presented here, in this article, is available in the Appendix. Keywords: Parametric, semiparametric, nonparametric, single index model, robust estimation, kernel regression, spline smoothing. 1 Introduction The aim of this paper is to answer many questions researchers may have when they fit their data, such as what are the differences between parametric, semi/nonparamertic models? which one should be used to model a real data set? what estimation method should be used?, and which type of modeling is better? These questions and others are addressed by examples. arXiv:1906.10221v1 [stat.ME] 24 Jun 2019 Assume that a researcher collected data about Y, a response variable, and explanatory variables, X = (x1; x2; : : : ; xk). The model that describes the relationship between Y and X can be written as Y = f(X; β) + , (1) where β is a vector of k parameters, is an error term whose distribution may or may not be normal, f(·) is some function that describe the relationship between Y and X.
    [Show full text]
  • Basics of Statistical Machine Learning 1 Parametric Vs. Nonparametric
    CS761 Spring 2013 Advanced Machine Learning Basics of Statistical Machine Learning Lecturer: Xiaojin Zhu [email protected] Modern machine learning is rooted in statistics. You will find many familiar concepts here with a different name. 1 Parametric vs. Nonparametric Statistical Models A statistical model H is a set of distributions. F In machine learning, we call H the hypothesis space. A parametric model is one that can be parametrized by a finite number of parameters. We write the PDF f(x) = f(x; θ) to emphasize the parameter θ 2 Rd. In general, d H = f(x; θ): θ 2 Θ ⊂ R (1) where Θ is the parameter space. We will often use the notation Z Eθ(g) = g(x)f(x; θ) dx (2) x to denote the expectation of a function g with respect to f(x; θ). Note the subscript in Eθ does not mean integrating over all θ. F This notation is standard but unfortunately confusing. It means \w.r.t. this fixed θ." In machine learning terms, this means w.r.t. different training sets all sampled from this θ. We will see integration over all θ when discussing Bayesian methods. Example 1 Consider the parametric model H = fN(µ, 1) : µ 2 Rg. Given iid data x1; : : : ; xn, the optimal 1 P estimator of the mean is µb = n xi. F All (parametric) models are wrong. Some are more useful than others. A nonparametric model is one which cannot be parametrized by a fixed number of parameters. Example 2 Consider the nonparametric model H = fP : V arP (X) < 1g.
    [Show full text]
  • Survival Techniques, Parametric Models (Weibull, Exponential and Log-Logistic), Displaced Proportion, Retrospective Data, Lord Resistance Army (LRA)
    American Journal of Mathematics and Statistics 2014, 4(5): 205-213 DOI: 10.5923/j.ajms.20140405.01 Parametric Models and Future Event Prediction Base on Right Censored Data Joseph. O. Okello1,*, D. Abdou Ka2 1Pan Africa University Institute of Basic Sciences, Technology and Innovation (PAUISTI), Nairobi, Kenya 2Department of statistics, University of Gaston Berger, Senegal and a visiting lecturer to (PAUISTI), Nairobi, Kenya Abstract In this paper, we developed a parametric displacement models base on the time that the Internally Displaced Persons (IDPs) took to return from IDPs Camps to their ancestral homes in Northern Uganda. The objective is to analyze the displaced proportion of the IDPs using suitable time-to-event parametric models. The accelerated failure time (AFT) models (Weibull, Exponential and log-logistic) were considered. A retrospective data of seven years study of 590 subjects is considered. Maximum likelihood method together with the Davidon-Fletcher- Powell optimization algorithm in MATLAB is used in the estimation of the parameters of the models. The estimated displaced proportions of these AFT models are used in predicting the displaced proportion of the IDPs at a time t. Weibull and exponential models provided better estimates of the displaced proportion of the IDPs due to their good convergence power to four decimal points and predicted the 2027 and 2044 respectively as the year when the displaced proportion can be approximated to be zero. Keywords Survival Techniques, Parametric Models (Weibull, Exponential and Log-logistic), Displaced Proportion, Retrospective Data, Lord Resistance Army (LRA) in which Weibull regression model showed a superior fit. 1. Introduction There is already a very wide literature on parametric distribution (Weibull, Exponential and Log-logistic) in The underlying foundation of most inferential statistical analyzing the time-to-event data, for instance [11], [13], analysis is the concept of probability distribution.
    [Show full text]
  • Econstor Wirtschaft Leibniz Information Centre Make Your Publications Visible
    A Service of Leibniz-Informationszentrum econstor Wirtschaft Leibniz Information Centre Make Your Publications Visible. zbw for Economics Schwiebert, Jörg Working Paper Semiparametric Estimation of a Binary Choice Model with Sample Selection Diskussionsbeitrag, No. 505 Provided in Cooperation with: School of Economics and Management, University of Hannover Suggested Citation: Schwiebert, Jörg (2012) : Semiparametric Estimation of a Binary Choice Model with Sample Selection, Diskussionsbeitrag, No. 505, Leibniz Universität Hannover, Wirtschaftswissenschaftliche Fakultät, Hannover This Version is available at: http://hdl.handle.net/10419/73118 Standard-Nutzungsbedingungen: Terms of use: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Documents in EconStor may be saved and copied for your Zwecken und zum Privatgebrauch gespeichert und kopiert werden. personal and scholarly purposes. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle You are not to copy documents for public or commercial Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich purposes, to exhibit the documents publicly, to make them machen, vertreiben oder anderweitig nutzen. publicly available on the internet, or to distribute or otherwise use the documents in public. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, If the documents have been made available under an Open gelten abweichend von diesen Nutzungsbedingungen die in der dort Content Licence (especially Creative Commons Licences), you genannten Lizenz gewährten Nutzungsrechte. may exercise further usage rights as specified in the indicated licence. www.econstor.eu Semiparametric Estimation of a Binary Choice Model with Sample Selection J¨orgSchwiebert∗ Abstract In this paper we provide semiparametric estimation strategies for a sample selection model with a binary dependent variable.
    [Show full text]