Multivariate Distributions
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Chapter 8 Fundamental Sampling Distributions And
CHAPTER 8 FUNDAMENTAL SAMPLING DISTRIBUTIONS AND DATA DESCRIPTIONS 8.1 Random Sampling pling procedure, it is desirable to choose a random sample in the sense that the observations are made The basic idea of the statistical inference is that we independently and at random. are allowed to draw inferences or conclusions about a Random Sample population based on the statistics computed from the sample data so that we could infer something about Let X1;X2;:::;Xn be n independent random variables, the parameters and obtain more information about the each having the same probability distribution f (x). population. Thus we must make sure that the samples Define X1;X2;:::;Xn to be a random sample of size must be good representatives of the population and n from the population f (x) and write its joint proba- pay attention on the sampling bias and variability to bility distribution as ensure the validity of statistical inference. f (x1;x2;:::;xn) = f (x1) f (x2) f (xn): ··· 8.2 Some Important Statistics It is important to measure the center and the variabil- ity of the population. For the purpose of the inference, we study the following measures regarding to the cen- ter and the variability. 8.2.1 Location Measures of a Sample The most commonly used statistics for measuring the center of a set of data, arranged in order of mag- nitude, are the sample mean, sample median, and sample mode. Let X1;X2;:::;Xn represent n random variables. Sample Mean To calculate the average, or mean, add all values, then Bias divide by the number of individuals. -
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. -
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. -
Permutation Tests
Permutation tests Ken Rice Thomas Lumley UW Biostatistics Seattle, June 2008 Overview • Permutation tests • A mean • Smallest p-value across multiple models • Cautionary notes Testing In testing a null hypothesis we need a test statistic that will have different values under the null hypothesis and the alternatives we care about (eg a relative risk of diabetes) We then need to compute the sampling distribution of the test statistic when the null hypothesis is true. For some test statistics and some null hypotheses this can be done analytically. The p- value for the is the probability that the test statistic would be at least as extreme as we observed, if the null hypothesis is true. A permutation test gives a simple way to compute the sampling distribution for any test statistic, under the strong null hypothesis that a set of genetic variants has absolutely no effect on the outcome. Permutations To estimate the sampling distribution of the test statistic we need many samples generated under the strong null hypothesis. If the null hypothesis is true, changing the exposure would have no effect on the outcome. By randomly shuffling the exposures we can make up as many data sets as we like. If the null hypothesis is true the shuffled data sets should look like the real data, otherwise they should look different from the real data. The ranking of the real test statistic among the shuffled test statistics gives a p-value Example: null is true Data Shuffling outcomes Shuffling outcomes (ordered) gender outcome gender outcome gender outcome Example: null is false Data Shuffling outcomes Shuffling outcomes (ordered) gender outcome gender outcome gender outcome Means Our first example is a difference in mean outcome in a dominant model for a single SNP ## make up some `true' data carrier<-rep(c(0,1), c(100,200)) null.y<-rnorm(300) alt.y<-rnorm(300, mean=carrier/2) In this case we know from theory the distribution of a difference in means and we could just do a t-test. -
Theoretical Statistics. Lecture 5. Peter Bartlett
Theoretical Statistics. Lecture 5. Peter Bartlett 1. U-statistics. 1 Outline of today’s lecture We’ll look at U-statistics, a family of estimators that includes many interesting examples. We’ll study their properties: unbiased, lower variance, concentration (via an application of the bounded differences inequality), asymptotic variance, asymptotic distribution. (See Chapter 12 of van der Vaart.) First, we’ll consider the standard unbiased estimate of variance—a special case of a U-statistic. 2 Variance estimates n 1 s2 = (X − X¯ )2 n n − 1 i n Xi=1 n n 1 = (X − X¯ )2 +(X − X¯ )2 2n(n − 1) i n j n Xi=1 Xj=1 n n 1 2 = (X − X¯ ) − (X − X¯ ) 2n(n − 1) i n j n Xi=1 Xj=1 n n 1 1 = (X − X )2 n(n − 1) 2 i j Xi=1 Xj=1 1 1 = (X − X )2 . n 2 i j 2 Xi<j 3 Variance estimates This is unbiased for i.i.d. data: 1 Es2 = E (X − X )2 n 2 1 2 1 = E ((X − EX ) − (X − EX ))2 2 1 1 2 2 1 2 2 = E (X1 − EX1) +(X2 − EX2) 2 2 = E (X1 − EX1) . 4 U-statistics Definition: A U-statistic of order r with kernel h is 1 U = n h(Xi1 ,...,Xir ), r iX⊆[n] where h is symmetric in its arguments. [If h is not symmetric in its arguments, we can also average over permutations.] “U” for “unbiased.” Introduced by Wassily Hoeffding in the 1940s. 5 U-statistics Theorem: [Halmos] θ (parameter, i.e., function defined on a family of distributions) admits an unbiased estimator (ie: for all sufficiently large n, some function of the i.i.d. -
Sampling Distribution of the Variance
Proceedings of the 2009 Winter Simulation Conference M. D. Rossetti, R. R. Hill, B. Johansson, A. Dunkin, and R. G. Ingalls, eds. SAMPLING DISTRIBUTION OF THE VARIANCE Pierre L. Douillet Univ Lille Nord de France, F-59000 Lille, France ENSAIT, GEMTEX, F-59100 Roubaix, France ABSTRACT Without confidence intervals, any simulation is worthless. These intervals are quite ever obtained from the so called "sampling variance". In this paper, some well-known results concerning the sampling distribution of the variance are recalled and completed by simulations and new results. The conclusion is that, except from normally distributed populations, this distribution is more difficult to catch than ordinary stated in application papers. 1 INTRODUCTION Modeling is translating reality into formulas, thereafter acting on the formulas and finally translating the results back to reality. Obviously, the model has to be tractable in order to be useful. But too often, the extra hypotheses that are assumed to ensure tractability are held as rock-solid properties of the real world. It must be recalled that "everyday life" is not only made with "every day events" : rare events are rarely occurring, but they do. For example, modeling a bell shaped histogram of experimental frequencies by a Gaussian pdf (probability density function) or a Fisher’s pdf with four parameters is usual. Thereafter transforming this pdf into a mgf (moment generating function) by mgf (z)=Et (expzt) is a powerful tool to obtain (and prove) the properties of the modeling pdf . But this doesn’t imply that a specific moment (e.g. 4) is effectively an accessible experimental reality. -
Arxiv:1804.01620V1 [Stat.ML]
ACTIVE COVARIANCE ESTIMATION BY RANDOM SUB-SAMPLING OF VARIABLES Eduardo Pavez and Antonio Ortega University of Southern California ABSTRACT missing data case, as well as for designing active learning algorithms as we will discuss in more detail in Section 5. We study covariance matrix estimation for the case of partially ob- In this work we analyze an unbiased covariance matrix estima- served random vectors, where different samples contain different tor under sub-Gaussian assumptions on x. Our main result is an subsets of vector coordinates. Each observation is the product of error bound on the Frobenius norm that reveals the relation between the variable of interest with a 0 1 Bernoulli random variable. We − number of observations, sub-sampling probabilities and entries of analyze an unbiased covariance estimator under this model, and de- the true covariance matrix. We apply this error bound to the design rive an error bound that reveals relations between the sub-sampling of sub-sampling probabilities in an active covariance estimation sce- probabilities and the entries of the covariance matrix. We apply our nario. An interesting conclusion from this work is that when the analysis in an active learning framework, where the expected number covariance matrix is approximately low rank, an active covariance of observed variables is small compared to the dimension of the vec- estimation approach can perform almost as well as an estimator with tor of interest, and propose a design of optimal sub-sampling proba- complete observations. The paper is organized as follows, in Section bilities and an active covariance matrix estimation algorithm. -
Examination of Residuals
EXAMINATION OF RESIDUALS F. J. ANSCOMBE PRINCETON UNIVERSITY AND THE UNIVERSITY OF CHICAGO 1. Introduction 1.1. Suppose that n given observations, yi, Y2, * , y., are claimed to be inde- pendent determinations, having equal weight, of means pA, A2, * *X, n, such that (1) Ai= E ai,Or, where A = (air) is a matrix of given coefficients and (Or) is a vector of unknown parameters. In this paper the suffix i (and later the suffixes j, k, 1) will always run over the values 1, 2, * , n, and the suffix r will run from 1 up to the num- ber of parameters (t1r). Let (#r) denote estimates of (Or) obtained by the method of least squares, let (Yi) denote the fitted values, (2) Y= Eai, and let (zt) denote the residuals, (3) Zi =Yi - Yi. If A stands for the linear space spanned by (ail), (a,2), *-- , that is, by the columns of A, and if X is the complement of A, consisting of all n-component vectors orthogonal to A, then (Yi) is the projection of (yt) on A and (zi) is the projection of (yi) on Z. Let Q = (qij) be the idempotent positive-semidefinite symmetric matrix taking (y1) into (zi), that is, (4) Zi= qtj,yj. If A has dimension n - v (where v > 0), X is of dimension v and Q has rank v. Given A, we can choose a parameter set (0,), where r = 1, 2, * , n -v, such that the columns of A are linearly independent, and then if V-1 = A'A and if I stands for the n X n identity matrix (6ti), we have (5) Q =I-AVA'. -
Lecture 14 Testing for Kurtosis
9/8/2016 CHE384, From Data to Decisions: Measurement, Kurtosis Uncertainty, Analysis, and Modeling • For any distribution, the kurtosis (sometimes Lecture 14 called the excess kurtosis) is defined as Testing for Kurtosis 3 (old notation = ) • For a unimodal, symmetric distribution, Chris A. Mack – a positive kurtosis means “heavy tails” and a more Adjunct Associate Professor peaked center compared to a normal distribution – a negative kurtosis means “light tails” and a more spread center compared to a normal distribution http://www.lithoguru.com/scientist/statistics/ © Chris Mack, 2016Data to Decisions 1 © Chris Mack, 2016Data to Decisions 2 Kurtosis Examples One Impact of Excess Kurtosis • For the Student’s t • For a normal distribution, the sample distribution, the variance will have an expected value of s2, excess kurtosis is and a variance of 6 2 4 1 for DF > 4 ( for DF ≤ 4 the kurtosis is infinite) • For a distribution with excess kurtosis • For a uniform 2 1 1 distribution, 1 2 © Chris Mack, 2016Data to Decisions 3 © Chris Mack, 2016Data to Decisions 4 Sample Kurtosis Sample Kurtosis • For a sample of size n, the sample kurtosis is • An unbiased estimator of the sample excess 1 kurtosis is ∑ ̅ 1 3 3 1 6 1 2 3 ∑ ̅ Standard Error: • For large n, the sampling distribution of 1 24 2 1 approaches Normal with mean 0 and variance 2 1 of 24/n 3 5 • For small samples, this estimator is biased D. N. Joanes and C. A. Gill, “Comparing Measures of Sample Skewness and Kurtosis”, The Statistician, 47(1),183–189 (1998). -
Covariances of Two Sample Rank Sum Statistics
JOURNAL OF RESEARCH of the National Bureou of Standards - B. Mathematical Sciences Volume 76B, Nos. 1 and 2, January-June 1972 Covariances of Two Sample Rank Sum Statistics Peter V. Tryon Institute for Basic Standards, National Bureau of Standards, Boulder, Colorado 80302 (November 26, 1971) This note presents an elementary derivation of the covariances of the e(e - 1)/2 two-sample rank sum statistics computed among aU pairs of samples from e populations. Key words: e Sample proble m; covariances, Mann-Whitney-Wilcoxon statistics; rank sum statistics; statistics. Mann-Whitney or Wilcoxon rank sum statistics, computed for some or all of the c(c - 1)/2 pairs of samples from c populations, have been used in testing the null hypothesis of homogeneity of dis tribution against a variety of alternatives [1, 3,4,5).1 This note presents an elementary derivation of the covariances of such statistics under the null hypothesis_ The usual approach to such an analysis is the rank sum viewpoint of the Wilcoxon form of the statistic. Using this approach, Steel [3] presents a lengthy derivation of the covariances. In this note it is shown that thinking in terms of the Mann-Whitney form of the statistic leads to an elementary derivation. For comparison and completeness the rank sum derivation of Kruskal and Wallis [2] is repeated in obtaining the means and variances. Let x{, r= 1,2, ... , ni, i= 1,2, ... , c, be the rth item in the sample of size ni from the ith of c populations. Let Mij be the Mann-Whitney statistic between the ith andjth samples defined by n· n· Mij= ~ t z[J (1) s=1 1' = 1 where l,xJ~>xr Zr~= { I) O,xj";;;x[ } Thus Mij is the number of times items in the jth sample exceed items in the ith sample. -
Analysis of Covariance (ANCOVA) with Two Groups
NCSS Statistical Software NCSS.com Chapter 226 Analysis of Covariance (ANCOVA) with Two Groups Introduction This procedure performs analysis of covariance (ANCOVA) for a grouping variable with 2 groups and one covariate variable. This procedure uses multiple regression techniques to estimate model parameters and compute least squares means. This procedure also provides standard error estimates for least squares means and their differences, and computes the T-test for the difference between group means adjusted for the covariate. The procedure also provides response vs covariate by group scatter plots and residuals for checking model assumptions. This procedure will output results for a simple two-sample equal-variance T-test if no covariate is entered and simple linear regression if no group variable is entered. This allows you to complete the ANCOVA analysis if either the group variable or covariate is determined to be non-significant. For additional options related to the T- test and simple linear regression analyses, we suggest you use the corresponding procedures in NCSS. The group variable in this procedure is restricted to two groups. If you want to perform ANCOVA with a group variable that has three or more groups, use the One-Way Analysis of Covariance (ANCOVA) procedure. This procedure cannot be used to analyze models that include more than one covariate variable or more than one group variable. If the model you want to analyze includes more than one covariate variable and/or more than one group variable, use the General Linear Models (GLM) for Fixed Factors procedure instead. Kinds of Research Questions A large amount of research consists of studying the influence of a set of independent variables on a response (dependent) variable. -
What Is Statistic?
What is Statistic? OPRE 6301 In today’s world. ...we are constantly being bombarded with statistics and statistical information. For example: Customer Surveys Medical News Demographics Political Polls Economic Predictions Marketing Information Sales Forecasts Stock Market Projections Consumer Price Index Sports Statistics How can we make sense out of all this data? How do we differentiate valid from flawed claims? 1 What is Statistics?! “Statistics is a way to get information from data.” Statistics Data Information Data: Facts, especially Information: Knowledge numerical facts, collected communicated concerning together for reference or some particular fact. information. Statistics is a tool for creating an understanding from a set of numbers. Humorous Definitions: The Science of drawing a precise line between an unwar- ranted assumption and a forgone conclusion. The Science of stating precisely what you don’t know. 2 An Example: Stats Anxiety. A business school student is anxious about their statistics course, since they’ve heard the course is difficult. The professor provides last term’s final exam marks to the student. What can be discerned from this list of numbers? Statistics Data Information List of last term’s marks. New information about the statistics class. 95 89 70 E.g. Class average, 65 Proportion of class receiving A’s 78 Most frequent mark, 57 Marks distribution, etc. : 3 Key Statistical Concepts. Population — a population is the group of all items of interest to a statistics practitioner. — frequently very large; sometimes infinite. E.g. All 5 million Florida voters (per Example 12.5). Sample — A sample is a set of data drawn from the population.