Sampling Student's T Distribution – Use of the Inverse Cumulative
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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]. -
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. -
4.2 Variance and Covariance
4.2 Variance and Covariance The most important measure of variability of a random variable X is obtained by letting g(X) = (X − µ)2 then E[g(X)] gives a measure of the variability of the distribution of X. 1 Definition 4.3 Let X be a random variable with probability distribution f(x) and mean µ. The variance of X is σ 2 = E[(X − µ)2 ] = ∑(x − µ)2 f (x) x if X is discrete, and ∞ σ 2 = E[(X − µ)2 ] = ∫ (x − µ)2 f (x)dx if X is continuous. −∞ ¾ The positive square root of the variance, σ, is called the standard deviation of X. ¾ The quantity x - µ is called the deviation of an observation, x, from its mean. 2 Note σ2≥0 When the standard deviation of a random variable is small, we expect most of the values of X to be grouped around mean. We often use standard deviations to compare two or more distributions that have the same unit measurements. 3 Example 4.8 Page97 Let the random variable X represent the number of automobiles that are used for official business purposes on any given workday. The probability distributions of X for two companies are given below: x 12 3 Company A: f(x) 0.3 0.4 0.3 x 01234 Company B: f(x) 0.2 0.1 0.3 0.3 0.1 Find the variances of X for the two companies. Solution: µ = 2 µ A = (1)(0.3) + (2)(0.4) + (3)(0.3) = 2 B 2 2 2 2 σ A = (1− 2) (0.3) + (2 − 2) (0.4) + (3− 2) (0.3) = 0.6 2 2 4 σ B > σ A 2 2 4 σ B = ∑(x − 2) f (x) =1.6 x=0 Theorem 4.2 The variance of a random variable X is σ 2 = E(X 2 ) − µ 2. -
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. -
Theoretical Statistics. Lecture 20
Theoretical Statistics. Lecture 20. Peter Bartlett 1. Recall: Functional delta method, differentiability in normed spaces, Hadamard derivatives. [vdV20] 2. Quantile estimates. [vdV21] 3. Contiguity. [vdV6] 1 Recall: Differentiability of functions in normed spaces Definition: φ : D E is Hadamard differentiable at θ D tangentially → ∈ to D D if 0 ⊆ φ′ : D E (linear, continuous), h D , ∃ θ 0 → ∀ ∈ 0 if t 0, ht h 0, then → k − k → φ(θ + tht) φ(θ) − φ′ (h) 0. t − θ → 2 Recall: Functional delta method Theorem: Suppose φ : D E, where D and E are normed linear spaces. → Suppose the statistic Tn : Ωn D satisfies √n(Tn θ) T for a random → − element T in D D. 0 ⊂ If φ is Hadamard differentiable at θ tangentially to D0 then ′ √n(φ(Tn) φ(θ)) φ (T ). − θ If we can extend φ′ : D E to a continuous map φ′ : D E, then 0 → → ′ √n(φ(Tn) φ(θ)) = φ (√n(Tn θ)) + oP (1). − θ − 3 Recall: Quantiles Definition: The quantile function of F is F −1 : (0, 1) R, → F −1(p) = inf x : F (x) p . { ≥ } Quantile transformation: for U uniform on (0, 1), • F −1(U) F. ∼ Probability integral transformation: for X F , F (X) is uniform on • ∼ [0,1] iff F is continuous on R. F −1 is an inverse (i.e., F −1(F (x)) = x and F (F −1(p)) = p for all x • and p) iff F is continuous and strictly increasing. 4 Empirical quantile function For a sample with distribution function F , define the empirical quantile −1 function as the quantile function Fn of the empirical distribution function Fn. -
A Tail Quantile Approximation Formula for the Student T and the Symmetric Generalized Hyperbolic Distribution
A Service of Leibniz-Informationszentrum econstor Wirtschaft Leibniz Information Centre Make Your Publications Visible. zbw for Economics Schlüter, Stephan; Fischer, Matthias J. Working Paper A tail quantile approximation formula for the student t and the symmetric generalized hyperbolic distribution IWQW Discussion Papers, No. 05/2009 Provided in Cooperation with: Friedrich-Alexander University Erlangen-Nuremberg, Institute for Economics Suggested Citation: Schlüter, Stephan; Fischer, Matthias J. (2009) : A tail quantile approximation formula for the student t and the symmetric generalized hyperbolic distribution, IWQW Discussion Papers, No. 05/2009, Friedrich-Alexander-Universität Erlangen-Nürnberg, Institut für Wirtschaftspolitik und Quantitative Wirtschaftsforschung (IWQW), Nürnberg This Version is available at: http://hdl.handle.net/10419/29554 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 IWQW Institut für Wirtschaftspolitik und Quantitative Wirtschaftsforschung Diskussionspapier Discussion Papers No. -
Stat 5102 Lecture Slides: Deck 1 Empirical Distributions, Exact Sampling Distributions, Asymptotic Sampling Distributions
Stat 5102 Lecture Slides: Deck 1 Empirical Distributions, Exact Sampling Distributions, Asymptotic Sampling Distributions Charles J. Geyer School of Statistics University of Minnesota 1 Empirical Distributions The empirical distribution associated with a vector of numbers x = (x1; : : : ; xn) is the probability distribution with expectation operator n 1 X Enfg(X)g = g(xi) n i=1 This is the same distribution that arises in finite population sam- pling. Suppose we have a population of size n whose members have values x1, :::, xn of a particular measurement. The value of that measurement for a randomly drawn individual from this population has a probability distribution that is this empirical distribution. 2 The Mean of the Empirical Distribution In the special case where g(x) = x, we get the mean of the empirical distribution n 1 X En(X) = xi n i=1 which is more commonly denotedx ¯n. Those with previous exposure to statistics will recognize this as the formula of the population mean, if x1, :::, xn is considered a finite population from which we sample, or as the formula of the sample mean, if x1, :::, xn is considered a sample from a specified population. 3 The Variance of the Empirical Distribution The variance of any distribution is the expected squared deviation from the mean of that same distribution. The variance of the empirical distribution is n 2o varn(X) = En [X − En(X)] n 2o = En [X − x¯n] n 1 X 2 = (xi − x¯n) n i=1 The only oddity is the use of the notationx ¯n rather than µ for the mean. -
On the Efficiency and Consistency of Likelihood Estimation in Multivariate
On the efficiency and consistency of likelihood estimation in multivariate conditionally heteroskedastic dynamic regression models∗ Gabriele Fiorentini Università di Firenze and RCEA, Viale Morgagni 59, I-50134 Firenze, Italy <fi[email protected]fi.it> Enrique Sentana CEMFI, Casado del Alisal 5, E-28014 Madrid, Spain <sentana@cemfi.es> Revised: October 2010 Abstract We rank the efficiency of several likelihood-based parametric and semiparametric estima- tors of conditional mean and variance parameters in multivariate dynamic models with po- tentially asymmetric and leptokurtic strong white noise innovations. We detailedly study the elliptical case, and show that Gaussian pseudo maximum likelihood estimators are inefficient except under normality. We provide consistency conditions for distributionally misspecified maximum likelihood estimators, and show that they coincide with the partial adaptivity con- ditions of semiparametric procedures. We propose Hausman tests that compare Gaussian pseudo maximum likelihood estimators with more efficient but less robust competitors. Fi- nally, we provide finite sample results through Monte Carlo simulations. Keywords: Adaptivity, Elliptical Distributions, Hausman tests, Semiparametric Esti- mators. JEL: C13, C14, C12, C51, C52 ∗We would like to thank Dante Amengual, Manuel Arellano, Nour Meddahi, Javier Mencía, Olivier Scaillet, Paolo Zaffaroni, participants at the European Meeting of the Econometric Society (Stockholm, 2003), the Sympo- sium on Economic Analysis (Seville, 2003), the CAF Conference on Multivariate Modelling in Finance and Risk Management (Sandbjerg, 2006), the Second Italian Congress on Econometrics and Empirical Economics (Rimini, 2007), as well as audiences at AUEB, Bocconi, Cass Business School, CEMFI, CREST, EUI, Florence, NYU, RCEA, Roma La Sapienza and Queen Mary for useful comments and suggestions. Of course, the usual caveat applies. -
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. -
Notes Mean, Median, Mode & Range
Notes Mean, Median, Mode & Range How Do You Use Mode, Median, Mean, and Range to Describe Data? There are many ways to describe the characteristics of a set of data. The mode, median, and mean are all called measures of central tendency. These measures of central tendency and range are described in the table below. The mode of a set of data Use the mode to show which describes which value occurs value in a set of data occurs most frequently. If two or more most often. For the set numbers occur the same number {1, 1, 2, 3, 5, 6, 10}, of times and occur more often the mode is 1 because it occurs Mode than all the other numbers in the most frequently. set, those numbers are all modes for the data set. If each number in the set occurs the same number of times, the set of data has no mode. The median of a set of data Use the median to show which describes what value is in the number in a set of data is in the middle if the set is ordered from middle when the numbers are greatest to least or from least to listed in order. greatest. If there are an even For the set {1, 1, 2, 3, 5, 6, 10}, number of values, the median is the median is 3 because it is in the average of the two middle the middle when the numbers are Median values. Half of the values are listed in order. greater than the median, and half of the values are less than the median. -
Depth, Outlyingness, Quantile, and Rank Functions in Multivariate & Other Data Settings Eserved@D = *@Let@Token
DEPTH, OUTLYINGNESS, QUANTILE, AND RANK FUNCTIONS – CONCEPTS, PERSPECTIVES, CHALLENGES Depth, Outlyingness, Quantile, and Rank Functions in Multivariate & Other Data Settings Robert Serfling1 Serfling & Thompson Statistical Consulting and Tutoring ASA Alabama-Mississippi Chapter Mini-Conference University of Mississippi, Oxford April 5, 2019 1 www.utdallas.edu/∼serfling DEPTH, OUTLYINGNESS, QUANTILE, AND RANK FUNCTIONS – CONCEPTS, PERSPECTIVES, CHALLENGES “Don’t walk in front of me, I may not follow. Don’t walk behind me, I may not lead. Just walk beside me and be my friend.” – Albert Camus DEPTH, OUTLYINGNESS, QUANTILE, AND RANK FUNCTIONS – CONCEPTS, PERSPECTIVES, CHALLENGES OUTLINE Depth, Outlyingness, Quantile, and Rank Functions on Rd Depth Functions on Arbitrary Data Space X Depth Functions on Arbitrary Parameter Space Θ Concluding Remarks DEPTH, OUTLYINGNESS, QUANTILE, AND RANK FUNCTIONS – CONCEPTS, PERSPECTIVES, CHALLENGES d DEPTH, OUTLYINGNESS, QUANTILE, AND RANK FUNCTIONS ON R PRELIMINARY PERSPECTIVES Depth functions are a nonparametric approach I The setting is nonparametric data analysis. No parametric or semiparametric model is assumed or invoked. I We exhibit the geometric structure of a data set in terms of a center, quantile levels, measures of outlyingness for each point, and identification of outliers or outlier regions. I Such data description is developed in terms of a depth function that measures centrality from a global viewpoint and yields center-outward ordering of data points. I This differs from the density function, -
The Smoothed Median and the Bootstrap
Biometrika (2001), 88, 2, pp. 519–534 © 2001 Biometrika Trust Printed in Great Britain The smoothed median and the bootstrap B B. M. BROWN School of Mathematics, University of South Australia, Adelaide, South Australia 5005, Australia [email protected] PETER HALL Centre for Mathematics & its Applications, Australian National University, Canberra, A.C.T . 0200, Australia [email protected] G. A. YOUNG Statistical L aboratory, University of Cambridge, Cambridge CB3 0WB, U.K. [email protected] S Even in one dimension the sample median exhibits very poor performance when used in conjunction with the bootstrap. For example, both the percentile-t bootstrap and the calibrated percentile method fail to give second-order accuracy when applied to the median. The situation is generally similar for other rank-based methods, particularly in more than one dimension. Some of these problems can be overcome by smoothing, but that usually requires explicit choice of the smoothing parameter. In the present paper we suggest a new, implicitly smoothed version of the k-variate sample median, based on a particularly smooth objective function. Our procedure preserves many features of the conventional median, such as robustness and high efficiency, in fact higher than for the conventional median, in the case of normal data. It is however substantially more amenable to application of the bootstrap. Focusing on the univariate case, we demonstrate these properties both theoretically and numerically. Some key words: Bootstrap; Calibrated percentile method; Median; Percentile-t; Rank methods; Smoothed median. 1. I Rank methods are based on simple combinatorial ideas of permutations and sign changes, which are attractive in applications far removed from the assumptions of normal linear model theory.