Chapter 2 the Maximum Likelihood Estimator
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Concentration and Consistency Results for Canonical and Curved Exponential-Family Models of Random Graphs
CONCENTRATION AND CONSISTENCY RESULTS FOR CANONICAL AND CURVED EXPONENTIAL-FAMILY MODELS OF RANDOM GRAPHS BY MICHAEL SCHWEINBERGER AND JONATHAN STEWART Rice University Statistical inference for exponential-family models of random graphs with dependent edges is challenging. We stress the importance of additional structure and show that additional structure facilitates statistical inference. A simple example of a random graph with additional structure is a random graph with neighborhoods and local dependence within neighborhoods. We develop the first concentration and consistency results for maximum likeli- hood and M-estimators of a wide range of canonical and curved exponential- family models of random graphs with local dependence. All results are non- asymptotic and applicable to random graphs with finite populations of nodes, although asymptotic consistency results can be obtained as well. In addition, we show that additional structure can facilitate subgraph-to-graph estimation, and present concentration results for subgraph-to-graph estimators. As an ap- plication, we consider popular curved exponential-family models of random graphs, with local dependence induced by transitivity and parameter vectors whose dimensions depend on the number of nodes. 1. Introduction. Models of network data have witnessed a surge of interest in statistics and related areas [e.g., 31]. Such data arise in the study of, e.g., social networks, epidemics, insurgencies, and terrorist networks. Since the work of Holland and Leinhardt in the 1970s [e.g., 21], it is known that network data exhibit a wide range of dependencies induced by transitivity and other interesting network phenomena [e.g., 39]. Transitivity is a form of triadic closure in the sense that, when a node k is connected to two distinct nodes i and j, then i and j are likely to be connected as well, which suggests that edges are dependent [e.g., 39]. -
Lecture 12 Robust Estimation
Lecture 12 Robust Estimation Prof. Dr. Svetlozar Rachev Institute for Statistics and Mathematical Economics University of Karlsruhe Financial Econometrics, Summer Semester 2007 Prof. Dr. Svetlozar Rachev Institute for Statistics and MathematicalLecture Economics 12 Robust University Estimation of Karlsruhe Copyright These lecture-notes cannot be copied and/or distributed without permission. The material is based on the text-book: Financial Econometrics: From Basics to Advanced Modeling Techniques (Wiley-Finance, Frank J. Fabozzi Series) by Svetlozar T. Rachev, Stefan Mittnik, Frank Fabozzi, Sergio M. Focardi,Teo Jaˇsic`. Prof. Dr. Svetlozar Rachev Institute for Statistics and MathematicalLecture Economics 12 Robust University Estimation of Karlsruhe Outline I Robust statistics. I Robust estimators of regressions. I Illustration: robustness of the corporate bond yield spread model. Prof. Dr. Svetlozar Rachev Institute for Statistics and MathematicalLecture Economics 12 Robust University Estimation of Karlsruhe Robust Statistics I Robust statistics addresses the problem of making estimates that are insensitive to small changes in the basic assumptions of the statistical models employed. I The concepts and methods of robust statistics originated in the 1950s. However, the concepts of robust statistics had been used much earlier. I Robust statistics: 1. assesses the changes in estimates due to small changes in the basic assumptions; 2. creates new estimates that are insensitive to small changes in some of the assumptions. I Robust statistics is also useful to separate the contribution of the tails from the contribution of the body of the data. Prof. Dr. Svetlozar Rachev Institute for Statistics and MathematicalLecture Economics 12 Robust University Estimation of Karlsruhe Robust Statistics I Peter Huber observed, that robust, distribution-free, and nonparametrical actually are not closely related properties. -
Use of Statistical Tables
TUTORIAL | SCOPE USE OF STATISTICAL TABLES Lucy Radford, Jenny V Freeman and Stephen J Walters introduce three important statistical distributions: the standard Normal, t and Chi-squared distributions PREVIOUS TUTORIALS HAVE LOOKED at hypothesis testing1 and basic statistical tests.2–4 As part of the process of statistical hypothesis testing, a test statistic is calculated and compared to a hypothesised critical value and this is used to obtain a P- value. This P-value is then used to decide whether the study results are statistically significant or not. It will explain how statistical tables are used to link test statistics to P-values. This tutorial introduces tables for three important statistical distributions (the TABLE 1. Extract from two-tailed standard Normal, t and Chi-squared standard Normal table. Values distributions) and explains how to use tabulated are P-values corresponding them with the help of some simple to particular cut-offs and are for z examples. values calculated to two decimal places. STANDARD NORMAL DISTRIBUTION TABLE 1 The Normal distribution is widely used in statistics and has been discussed in z 0.00 0.01 0.02 0.03 0.050.04 0.05 0.06 0.07 0.08 0.09 detail previously.5 As the mean of a Normally distributed variable can take 0.00 1.0000 0.9920 0.9840 0.9761 0.9681 0.9601 0.9522 0.9442 0.9362 0.9283 any value (−∞ to ∞) and the standard 0.10 0.9203 0.9124 0.9045 0.8966 0.8887 0.8808 0.8729 0.8650 0.8572 0.8493 deviation any positive value (0 to ∞), 0.20 0.8415 0.8337 0.8259 0.8181 0.8103 0.8206 0.7949 0.7872 0.7795 0.7718 there are an infinite number of possible 0.30 0.7642 0.7566 0.7490 0.7414 0.7339 0.7263 0.7188 0.7114 0.7039 0.6965 Normal distributions. -
Should We Think of a Different Median Estimator?
Comunicaciones en Estad´ıstica Junio 2014, Vol. 7, No. 1, pp. 11–17 Should we think of a different median estimator? ¿Debemos pensar en un estimator diferente para la mediana? Jorge Iv´an V´eleza Juan Carlos Correab [email protected] [email protected] Resumen La mediana, una de las medidas de tendencia central m´as populares y utilizadas en la pr´actica, es el valor num´erico que separa los datos en dos partes iguales. A pesar de su popularidad y aplicaciones, muchos desconocen la existencia de dife- rentes expresiones para calcular este par´ametro. A continuaci´on se presentan los resultados de un estudio de simulaci´on en el que se comparan el estimador cl´asi- co y el propuesto por Harrell & Davis (1982). Mostramos que, comparado con el estimador de Harrell–Davis, el estimador cl´asico no tiene un buen desempe˜no pa- ra tama˜nos de muestra peque˜nos. Basados en los resultados obtenidos, se sugiere promover la utilizaci´on de un mejor estimador para la mediana. Palabras clave: mediana, cuantiles, estimador Harrell-Davis, simulaci´on estad´ısti- ca. Abstract The median, one of the most popular measures of central tendency widely-used in the statistical practice, is often described as the numerical value separating the higher half of the sample from the lower half. Despite its popularity and applica- tions, many people are not aware of the existence of several formulas to estimate this parameter. We present the results of a simulation study comparing the classic and the Harrell-Davis (Harrell & Davis 1982) estimators of the median for eight continuous statistical distributions. -
5. the Student T Distribution
Virtual Laboratories > 4. Special Distributions > 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 5. The Student t Distribution In this section we will study a distribution that has special importance in statistics. In particular, this distribution will arise in the study of a standardized version of the sample mean when the underlying distribution is normal. The Probability Density Function Suppose that Z has the standard normal distribution, V has the chi-squared distribution with n degrees of freedom, and that Z and V are independent. Let Z T= √V/n In the following exercise, you will show that T has probability density function given by −(n +1) /2 Γ((n + 1) / 2) t2 f(t)= 1 + , t∈ℝ ( n ) √n π Γ(n / 2) 1. Show that T has the given probability density function by using the following steps. n a. Show first that the conditional distribution of T given V=v is normal with mean 0 a nd variance v . b. Use (a) to find the joint probability density function of (T,V). c. Integrate the joint probability density function in (b) with respect to v to find the probability density function of T. The distribution of T is known as the Student t distribution with n degree of freedom. The distribution is well defined for any n > 0, but in practice, only positive integer values of n are of interest. This distribution was first studied by William Gosset, who published under the pseudonym Student. In addition to supplying the proof, Exercise 1 provides a good way of thinking of the t distribution: the t distribution arises when the variance of a mean 0 normal distribution is randomized in a certain way. -
Stable Components in the Parameter Plane of Transcendental Functions of Finite Type
Stable components in the parameter plane of transcendental functions of finite type N´uriaFagella∗ Dept. de Matem`atiquesi Inform`atica Univ. de Barcelona, Gran Via 585 Barcelona Graduate School of Mathematics (BGSMath) 08007 Barcelona, Spain [email protected] Linda Keen† CUNY Graduate Center 365 Fifth Avenue, New York NY, NY 10016 [email protected], [email protected] November 11, 2020 Abstract We study the parameter planes of certain one-dimensional, dynamically-defined slices of holomorphic families of entire and meromorphic transcendental maps of finite type. Our planes are defined by constraining the orbits of all but one of the singular values, and leaving free one asymptotic value. We study the structure of the regions of parameters, which we call shell components, for which the free asymptotic value tends to an attracting cycle of non-constant multiplier. The exponential and the tangent families are examples that have been studied in detail, and the hyperbolic components in those parameter planes are shell components. Our results apply to slices of both entire and meromorphic maps. We prove that shell components are simply connected, have a locally connected boundary and have no center, i.e., no parameter value for which the cycle is superattracting. Instead, there is a unique parameter in the boundary, the virtual center, which plays the same role. For entire slices, the virtual center is always at infinity, while for meromorphic ones it maybe finite or infinite. In the dynamical plane we prove, among other results, that the basins of attraction which contain only one asymptotic value and no critical points are simply connected. -
Bias, Mean-Square Error, Relative Efficiency
3 Evaluating the Goodness of an Estimator: Bias, Mean-Square Error, Relative Efficiency Consider a population parameter ✓ for which estimation is desired. For ex- ample, ✓ could be the population mean (traditionally called µ) or the popu- lation variance (traditionally called σ2). Or it might be some other parame- ter of interest such as the population median, population mode, population standard deviation, population minimum, population maximum, population range, population kurtosis, or population skewness. As previously mentioned, we will regard parameters as numerical charac- teristics of the population of interest; as such, a parameter will be a fixed number, albeit unknown. In Stat 252, we will assume that our population has a distribution whose density function depends on the parameter of interest. Most of the examples that we will consider in Stat 252 will involve continuous distributions. Definition 3.1. An estimator ✓ˆ is a statistic (that is, it is a random variable) which after the experiment has been conducted and the data collected will be used to estimate ✓. Since it is true that any statistic can be an estimator, you might ask why we introduce yet another word into our statistical vocabulary. Well, the answer is quite simple, really. When we use the word estimator to describe a particular statistic, we already have a statistical estimation problem in mind. For example, if ✓ is the population mean, then a natural estimator of ✓ is the sample mean. If ✓ is the population variance, then a natural estimator of ✓ is the sample variance. More specifically, suppose that Y1,...,Yn are a random sample from a population whose distribution depends on the parameter ✓.The following estimators occur frequently enough in practice that they have special notations. -
1 One Parameter Exponential Families
1 One parameter exponential families The world of exponential families bridges the gap between the Gaussian family and general dis- tributions. Many properties of Gaussians carry through to exponential families in a fairly precise sense. • In the Gaussian world, there exact small sample distributional results (i.e. t, F , χ2). • In the exponential family world, there are approximate distributional results (i.e. deviance tests). • In the general setting, we can only appeal to asymptotics. A one-parameter exponential family, F is a one-parameter family of distributions of the form Pη(dx) = exp (η · t(x) − Λ(η)) P0(dx) for some probability measure P0. The parameter η is called the natural or canonical parameter and the function Λ is called the cumulant generating function, and is simply the normalization needed to make dPη fη(x) = (x) = exp (η · t(x) − Λ(η)) dP0 a proper probability density. The random variable t(X) is the sufficient statistic of the exponential family. Note that P0 does not have to be a distribution on R, but these are of course the simplest examples. 1.0.1 A first example: Gaussian with linear sufficient statistic Consider the standard normal distribution Z e−z2=2 P0(A) = p dz A 2π and let t(x) = x. Then, the exponential family is eη·x−x2=2 Pη(dx) / p 2π and we see that Λ(η) = η2=2: eta= np.linspace(-2,2,101) CGF= eta**2/2. plt.plot(eta, CGF) A= plt.gca() A.set_xlabel(r'$\eta$', size=20) A.set_ylabel(r'$\Lambda(\eta)$', size=20) f= plt.gcf() 1 Thus, the exponential family in this setting is the collection F = fN(η; 1) : η 2 Rg : d 1.0.2 Normal with quadratic sufficient statistic on R d As a second example, take P0 = N(0;Id×d), i.e. -
Random Variables and Probability Distributions 1.1
RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS 1. DISCRETE RANDOM VARIABLES 1.1. Definition of a Discrete Random Variable. A random variable X is said to be discrete if it can assume only a finite or countable infinite number of distinct values. A discrete random variable can be defined on both a countable or uncountable sample space. 1.2. Probability for a discrete random variable. The probability that X takes on the value x, P(X=x), is defined as the sum of the probabilities of all sample points in Ω that are assigned the value x. We may denote P(X=x) by p(x) or pX (x). The expression pX (x) is a function that assigns probabilities to each possible value x; thus it is often called the probability function for the random variable X. 1.3. Probability distribution for a discrete random variable. The probability distribution for a discrete random variable X can be represented by a formula, a table, or a graph, which provides pX (x) = P(X=x) for all x. The probability distribution for a discrete random variable assigns nonzero probabilities to only a countable number of distinct x values. Any value x not explicitly assigned a positive probability is understood to be such that P(X=x) = 0. The function pX (x)= P(X=x) for each x within the range of X is called the probability distribution of X. It is often called the probability mass function for the discrete random variable X. 1.4. Properties of the probability distribution for a discrete random variable. -
On the Meaning and Use of Kurtosis
Psychological Methods Copyright 1997 by the American Psychological Association, Inc. 1997, Vol. 2, No. 3,292-307 1082-989X/97/$3.00 On the Meaning and Use of Kurtosis Lawrence T. DeCarlo Fordham University For symmetric unimodal distributions, positive kurtosis indicates heavy tails and peakedness relative to the normal distribution, whereas negative kurtosis indicates light tails and flatness. Many textbooks, however, describe or illustrate kurtosis incompletely or incorrectly. In this article, kurtosis is illustrated with well-known distributions, and aspects of its interpretation and misinterpretation are discussed. The role of kurtosis in testing univariate and multivariate normality; as a measure of departures from normality; in issues of robustness, outliers, and bimodality; in generalized tests and estimators, as well as limitations of and alternatives to the kurtosis measure [32, are discussed. It is typically noted in introductory statistics standard deviation. The normal distribution has a kur- courses that distributions can be characterized in tosis of 3, and 132 - 3 is often used so that the refer- terms of central tendency, variability, and shape. With ence normal distribution has a kurtosis of zero (132 - respect to shape, virtually every textbook defines and 3 is sometimes denoted as Y2)- A sample counterpart illustrates skewness. On the other hand, another as- to 132 can be obtained by replacing the population pect of shape, which is kurtosis, is either not discussed moments with the sample moments, which gives or, worse yet, is often described or illustrated incor- rectly. Kurtosis is also frequently not reported in re- ~(X i -- S)4/n search articles, in spite of the fact that virtually every b2 (•(X i - ~')2/n)2' statistical package provides a measure of kurtosis. -
Backpropagation TA: Yi Wen
Backpropagation TA: Yi Wen April 17, 2020 CS231n Discussion Section Slides credits: Barak Oshri, Vincent Chen, Nish Khandwala, Yi Wen Agenda ● Motivation ● Backprop Tips & Tricks ● Matrix calculus primer Agenda ● Motivation ● Backprop Tips & Tricks ● Matrix calculus primer Motivation Recall: Optimization objective is minimize loss Motivation Recall: Optimization objective is minimize loss Goal: how should we tweak the parameters to decrease the loss? Agenda ● Motivation ● Backprop Tips & Tricks ● Matrix calculus primer A Simple Example Loss Goal: Tweak the parameters to minimize loss => minimize a multivariable function in parameter space A Simple Example => minimize a multivariable function Plotted on WolframAlpha Approach #1: Random Search Intuition: the step we take in the domain of function Approach #2: Numerical Gradient Intuition: rate of change of a function with respect to a variable surrounding a small region Approach #2: Numerical Gradient Intuition: rate of change of a function with respect to a variable surrounding a small region Finite Differences: Approach #3: Analytical Gradient Recall: partial derivative by limit definition Approach #3: Analytical Gradient Recall: chain rule Approach #3: Analytical Gradient Recall: chain rule E.g. Approach #3: Analytical Gradient Recall: chain rule E.g. Approach #3: Analytical Gradient Recall: chain rule Intuition: upstream gradient values propagate backwards -- we can reuse them! Gradient “direction and rate of fastest increase” Numerical Gradient vs Analytical Gradient What about Autograd? -
The Probability Lifesaver: Order Statistics and the Median Theorem
The Probability Lifesaver: Order Statistics and the Median Theorem Steven J. Miller December 30, 2015 Contents 1 Order Statistics and the Median Theorem 3 1.1 Definition of the Median 5 1.2 Order Statistics 10 1.3 Examples of Order Statistics 15 1.4 TheSampleDistributionoftheMedian 17 1.5 TechnicalboundsforproofofMedianTheorem 20 1.6 TheMedianofNormalRandomVariables 22 2 • Greetings again! In this supplemental chapter we develop the theory of order statistics in order to prove The Median Theorem. This is a beautiful result in its own, but also extremely important as a substitute for the Central Limit Theorem, and allows us to say non- trivial things when the CLT is unavailable. Chapter 1 Order Statistics and the Median Theorem The Central Limit Theorem is one of the gems of probability. It’s easy to use and its hypotheses are satisfied in a wealth of problems. Many courses build towards a proof of this beautiful and powerful result, as it truly is ‘central’ to the entire subject. Not to detract from the majesty of this wonderful result, however, what happens in those instances where it’s unavailable? For example, one of the key assumptions that must be met is that our random variables need to have finite higher moments, or at the very least a finite variance. What if we were to consider sums of Cauchy random variables? Is there anything we can say? This is not just a question of theoretical interest, of mathematicians generalizing for the sake of generalization. The following example from economics highlights why this chapter is more than just of theoretical interest.