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Mathematical Statistics (Math 30) – Course Project for Spring 2011
Mathematical Statistics (Math 30) – Course Project for Spring 2011 Goals: To explore methods from class in a real life (historical) estimation setting To learn some basic statistical computational skills To explore a topic of interest via a short report on a selected paper The course project has been divided into 2 parts. Each part will account for 5% of your grade, because the project accounts for 10%. Part 1: German Tank Problem Due Date: April 8th Our Setting: The Purple (Amherst) and Gold (Williams) armies are at war. You are a highly ranked intelligence officer in the Purple army and you are given the job of estimating the number of tanks in the Gold army’s tank fleet. Luckily, the Gold army has decided to put sequential serial numbers on their tanks, so you can consider their tanks labeled as 1,2,3,…,N, and all you have to do is estimate N. Suppose you have access to a random sample of k serial numbers obtained by spies, found on abandoned/destroyed equipment, etc. Using your k observations, you need to estimate N. Note that we won’t deal with issues of sampling without replacement, so you can consider each observation as an independent observation from a discrete uniform distribution on 1 to N. The derivations below should lead you to consider several possible estimators of N, from which you will be picking one to propose as your estimate of N. Historical Setting: This was a real problem encountered by statisticians/intelligence in WWII. The Allies wanted to know how many tanks the Germans were producing per month (the procedure was used for things other than tanks too). -
Statistics Making an Impact
John Pullinger J. R. Statist. Soc. A (2013) 176, Part 4, pp. 819–839 Statistics making an impact John Pullinger House of Commons Library, London, UK [The address of the President, delivered to The Royal Statistical Society on Wednesday, June 26th, 2013] Summary. Statistics provides a special kind of understanding that enables well-informed deci- sions. As citizens and consumers we are faced with an array of choices. Statistics can help us to choose well. Our statistical brains need to be nurtured: we can all learn and practise some simple rules of statistical thinking. To understand how statistics can play a bigger part in our lives today we can draw inspiration from the founders of the Royal Statistical Society. Although in today’s world the information landscape is confused, there is an opportunity for statistics that is there to be seized.This calls for us to celebrate the discipline of statistics, to show confidence in our profession, to use statistics in the public interest and to champion statistical education. The Royal Statistical Society has a vital role to play. Keywords: Chartered Statistician; Citizenship; Economic growth; Evidence; ‘getstats’; Justice; Open data; Public good; The state; Wise choices 1. Introduction Dictionaries trace the source of the word statistics from the Latin ‘status’, the state, to the Italian ‘statista’, one skilled in statecraft, and on to the German ‘Statistik’, the science dealing with data about the condition of a state or community. The Oxford English Dictionary brings ‘statistics’ into English in 1787. Florence Nightingale held that ‘the thoughts and purpose of the Deity are only to be discovered by the statistical study of natural phenomena:::the application of the results of such study [is] the religious duty of man’ (Pearson, 1924). -
Local Variation As a Statistical Hypothesis Test
Int J Comput Vis (2016) 117:131–141 DOI 10.1007/s11263-015-0855-4 Local Variation as a Statistical Hypothesis Test Michael Baltaxe1 · Peter Meer2 · Michael Lindenbaum3 Received: 7 June 2014 / Accepted: 1 September 2015 / Published online: 16 September 2015 © Springer Science+Business Media New York 2015 Abstract The goal of image oversegmentation is to divide Keywords Image segmentation · Image oversegmenta- an image into several pieces, each of which should ideally tion · Superpixels · Grouping be part of an object. One of the simplest and yet most effec- tive oversegmentation algorithms is known as local variation (LV) Felzenszwalb and Huttenlocher in Efficient graph- 1 Introduction based image segmentation. IJCV 59(2):167–181 (2004). In this work, we study this algorithm and show that algorithms Image segmentation is the procedure of partitioning an input similar to LV can be devised by applying different statisti- image into several meaningful pieces or segments, each of cal models and decisions, thus providing further theoretical which should be semantically complete (i.e., an item or struc- justification and a well-founded explanation for the unex- ture by itself). Oversegmentation is a less demanding type of pected high performance of the LV approach. Some of these segmentation. The aim is to group several pixels in an image algorithms are based on statistics of natural images and on into a single unit called a superpixel (Ren and Malik 2003) a hypothesis testing decision; we denote these algorithms so that it is fully contained within an object; it represents a probabilistic local variation (pLV). The best pLV algorithm, fragment of a conceptually meaningful structure. -
School of Social Sciences Economics Division University of Southampton Southampton SO17 1BJ, UK
School of Social Sciences Economics Division University of Southampton Southampton SO17 1BJ, UK Discussion Papers in Economics and Econometrics Professor A L Bowley’s Theory of the Representative Method John Aldrich No. 0801 This paper is available on our website http://www.socsci.soton.ac.uk/economics/Research/Discussion_Papers ISSN 0966-4246 Key names: Arthur L. Bowley, F. Y. Edgeworth, , R. A. Fisher, Adolph Jensen, J. M. Keynes, Jerzy Neyman, Karl Pearson, G. U. Yule. Keywords: History of Statistics, Sampling theory, Bayesian inference. Professor A. L. Bowley’s Theory of the Representative Method * John Aldrich Economics Division School of Social Sciences University of Southampton Southampton SO17 1BJ UK e-mail: [email protected] Abstract Arthur. L. Bowley (1869-1957) first advocated the use of surveys–the “representative method”–in 1906 and started to conduct surveys of economic and social conditions in 1912. Bowley’s 1926 memorandum for the International Statistical Institute on the “Measurement of the precision attained in sampling” was the first large-scale theoretical treatment of sample surveys as he conducted them. This paper examines Bowley’s arguments in the context of the statistical inference theory of the time. The great influence on Bowley’s conception of statistical inference was F. Y. Edgeworth but by 1926 R. A. Fisher was on the scene and was attacking Bayesian methods and promoting a replacement of his own. Bowley defended his Bayesian method against Fisher and against Jerzy Neyman when the latter put forward his concept of a confidence interval and applied it to the representative method. * Based on a talk given at the Sample Surveys and Bayesian Statistics Conference, Southampton, August 2008. -
History of Health Statistics
certain gambling problems [e.g. Pacioli (1494), Car- Health Statistics, History dano (1539), and Forestani (1603)] which were first of solved definitively by Pierre de Fermat (1601–1665) and Blaise Pascal (1623–1662). These efforts gave rise to the mathematical basis of probability theory, The field of statistics in the twentieth century statistical distribution functions (see Sampling Dis- (see Statistics, Overview) encompasses four major tributions), and statistical inference. areas; (i) the theory of probability and mathematical The analysis of uncertainty and errors of mea- statistics; (ii) the analysis of uncertainty and errors surement had its foundations in the field of astron- of measurement (see Measurement Error in omy which, from antiquity until the eighteenth cen- Epidemiologic Studies); (iii) design of experiments tury, was the dominant area for use of numerical (see Experimental Design) and sample surveys; information based on the most precise measure- and (iv) the collection, summarization, display, ments that the technologies of the times permit- and interpretation of observational data (see ted. The fallibility of their observations was evident Observational Study). These four areas are clearly to early astronomers, who took the “best” obser- interrelated and have evolved interactively over the vation when several were taken, the “best” being centuries. The first two areas are well covered assessed by such criteria as the quality of obser- in many histories of mathematical statistics while vational conditions, the fame of the observer, etc. the third area, being essentially a twentieth- But, gradually an appreciation for averaging observa- century development, has not yet been adequately tions developed and various techniques for fitting the summarized. -
Contents 3 Parameter and Interval Estimation
Probability and Statistics (part 3) : Parameter and Interval Estimation (by Evan Dummit, 2020, v. 1.25) Contents 3 Parameter and Interval Estimation 1 3.1 Parameter Estimation . 1 3.1.1 Maximum Likelihood Estimates . 2 3.1.2 Biased and Unbiased Estimators . 5 3.1.3 Eciency of Estimators . 8 3.2 Interval Estimation . 12 3.2.1 Condence Intervals . 12 3.2.2 Normal Condence Intervals . 13 3.2.3 Binomial Condence Intervals . 16 3 Parameter and Interval Estimation In the previous chapter, we discussed random variables and developed the notion of a probability distribution, and then established some fundamental results such as the central limit theorem that give strong and useful information about the statistical properties of a sample drawn from a xed, known distribution. Our goal in this chapter is, in some sense, to invert this analysis: starting instead with data obtained by sampling a distribution or probability model with certain unknown parameters, we would like to extract information about the most reasonable values for these parameters given the observed data. We begin by discussing pointwise parameter estimates and estimators, analyzing various properties that we would like these estimators to have such as unbiasedness and eciency, and nally establish the optimality of several estimators that arise from the basic distributions we have encountered such as the normal and binomial distributions. We then broaden our focus to interval estimation: from nding the best estimate of a single value to nding such an estimate along with a measurement of its expected precision. We treat in scrupulous detail several important cases for constructing such condence intervals, and close with some applications of these ideas to polling data. -
Unpicking PLAID a Cryptographic Analysis of an ISO-Standards-Track Authentication Protocol
A preliminary version of this paper appears in the proceedings of the 1st International Conference on Research in Security Standardisation (SSR 2014), DOI: 10.1007/978-3-319-14054-4_1. This is the full version. Unpicking PLAID A Cryptographic Analysis of an ISO-standards-track Authentication Protocol Jean Paul Degabriele1 Victoria Fehr2 Marc Fischlin2 Tommaso Gagliardoni2 Felix Günther2 Giorgia Azzurra Marson2 Arno Mittelbach2 Kenneth G. Paterson1 1 Information Security Group, Royal Holloway, University of London, U.K. 2 Cryptoplexity, Technische Universität Darmstadt, Germany {j.p.degabriele,kenny.paterson}@rhul.ac.uk, {victoria.fehr,tommaso.gagliardoni,giorgia.marson,arno.mittelbach}@cased.de, [email protected], [email protected] October 27, 2015 Abstract. The Protocol for Lightweight Authentication of Identity (PLAID) aims at secure and private authentication between a smart card and a terminal. Originally developed by a unit of the Australian Department of Human Services for physical and logical access control, PLAID has now been standardized as an Australian standard AS-5185-2010 and is currently in the fast-track standardization process for ISO/IEC 25185-1. We present a cryptographic evaluation of PLAID. As well as reporting a number of undesirable cryptographic features of the protocol, we show that the privacy properties of PLAID are significantly weaker than claimed: using a variety of techniques we can fingerprint and then later identify cards. These techniques involve a novel application of standard statistical -
Uniform Distribution
Uniform distribution The uniform distribution is the probability distribution that gives the same probability to all possible values. There are two types of Uniform distribution uniform distributions: Statistics for Business Josemari Sarasola - Gizapedia x1 x2 x3 xN a b Discrete uniform distribution Continuous uniform distribution We apply uniform distributions when there is absolute uncertainty about what will occur. We also use them when we take a random sample from a population, because in such cases all the elements have the same probability of being drawn. Finally, they are also usedto create random numbers, as random numbers are those that have the same probability of being drawn. Statistics for Business Uniform distribution 1 / 17 Statistics for Business Uniform distribution 2 / 17 Discrete uniform distribution Discrete uniform distribution Distribution function i Probability function F (x) = P [X x ] = ; x = x , x , , x ≤ i N i 1 2 ··· N 1 P [X = x] = ; x = x1, x2, , xN N ··· 1 (N 1)/N − 1 1 1 1 1 1 1 1 1 1 N N N N N N N N N N i/N x1 x2 x3 xN 3/N 2/N 1/N x1 x2 x3 xi xN 1xN − Statistics for Business Uniform distribution 3 / 17 Statistics for Business Uniform distribution 4 / 17 Discrete uniform distribution Discrete uniform distribution Distribution of the maximum We draw a value from a uniform distribution n times. How is Notation, mean and variance distributed the maximum among those n values? Among n values, the maximum will be less than xi when all of them are less than xi: x1 + xN µ = 2 n i i i i 2 (xN x1 + 2)(xN x1) P [Xmax xi] = = X U(x1, x2, . -
Milton Friedman Papers 1931-2006
http://oac.cdlib.org/findaid/ark:/13030/tf7t1nb2hx Online items available Register of the Milton Friedman Papers 1931-2006 Processed by Linda Bernard, Dana M. Harris, and Elizabeth Konzak Hoover Institution Archives Stanford University Stanford, California 94305-6010 Phone: (650) 723-3563 Fax: (650) 725-3445 Email: [email protected] URL: http://www.hoover.org/library-and-archives © 1997, 2008 Hoover Institution Archives. All rights reserved. Register of the Milton Friedman 77011 1 Papers 1931-2006 Register of the Milton Friedman Papers 1931-1991 Hoover Institution Archives Stanford University Stanford, California Contact Information Hoover Institution Archives Stanford University Stanford, California 94305-6010 Phone: (650) 723-3563 Fax: (650) 725-3445 Email: [email protected] URL: http://www.hoover.org/library-and-archives/ Processed by: Linda Bernard, Dana M. Harris, and Elizabeth Konzak Date Completed: 1992, revised 2008 Encoded by: Brooke Dykman Dockter and Elizabeth Konzak © 1997, 2008 Hoover Institution Archives. All rights reserved. Descriptive Summary Title: Milton Friedman papers Date (inclusive): 1931-2006 Collection number: 77011 Creator: Friedman, Milton, 1912-2006 Creator: Friedman, Rose D. Physical Description: 228 manuscript boxes, 2 oversize boxes, 4 card file boxes, 1 slide box, 1 envelope(94 linear feet) Repository: Hoover Institution Archives Stanford, California 94305-6010 Abstract: Speeches and writings, correspondence, notes, statistics, printed matter, sound recordings, videotapes, and photographs, relating to economic theory, economic conditions in the United States, and governmental economic policy. Digitized copies of many of the sound and video recordings in this collection, as well as some of Friedman's writings, are available at http://hoohila.stanford.edu/friedman/ . -
Professor A. L. Bowley's Theory of the Representative Method *
Key names: Arthur L. Bowley, F. Y. Edgeworth, , R. A. Fisher, Adolph Jensen, J. M. Keynes, Jerzy Neyman, Karl Pearson, G. U. Yule. Keywords: History of Statistics, Sampling theory, Bayesian inference. Professor A. L. Bowley’s Theory of the Representative Method * John Aldrich Economics Division School of Social Sciences University of Southampton Southampton SO17 1BJ UK e-mail: [email protected] Abstract Arthur. L. Bowley (1869-1957) first advocated the use of surveys–the “representative method”–in 1906 and started to conduct surveys of economic and social conditions in 1912. Bowley’s 1926 memorandum for the International Statistical Institute on the “Measurement of the precision attained in sampling” was the first large-scale theoretical treatment of sample surveys as he conducted them. This paper examines Bowley’s arguments in the context of the statistical inference theory of the time. The great influence on Bowley’s conception of statistical inference was F. Y. Edgeworth but by 1926 R. A. Fisher was on the scene and was attacking Bayesian methods and promoting a replacement of his own. Bowley defended his Bayesian method against Fisher and against Jerzy Neyman when the latter put forward his concept of a confidence interval and applied it to the representative method. * Based on a talk given at the Sample Surveys and Bayesian Statistics Conference, Southampton, August 2008. I am grateful to Andrew Dale for his comments on an earlier draft. December 2008 1 Introduction “I think that if practical statistics has acquired something valuable in the represen- tative method, this is primarily due to Professor A. -
ELEC-E7130 Examples on Parameter Estimation
ELEC-E7130 Examples on parameter estimation Samuli Aalto Department of Communications and Networking Aalto University September 25, 2018 1 Discrete uniform distribution U d(1;N), one unknown parameter N Let U d(1;N) denote the discrete uniform distribution, for which N 2 f1; 2;:::g and the point probabilities are given by 1 p(i) = PfX = ig = ; i 2 f1;:::;Ng: N It follows that N N X 1 X 1 N(N + 1) N + 1 E(X) = PfX = igi = i = = ; N N 2 2 i=1 i=1 N N X 1 X 1 N(N + 1)(2N + 1) E(X2) = PfX = igi2 = i2 = N N 6 i=1 i=1 (N + 1)(2N + 1) = ; 6 (N + 1)(2N + 1) N + 12 V(X) = E(X2) − (E(X))2 = − 6 2 (N + 1)(N − 1) N 2 − 1 = = : 12 12 1 1.1 Estimation of N: Method of Moments (MoM) d Consider an IID sample (x1; : : : ; xn) of size n from distribution U (1;N). The first (theoretical) moment equals N + 1 µ = E(X) = ; 1 2 and the corresponding sample moment is the sample meanx ¯n: n 1 X µ^ =x ¯ = x : 1 n n i i=1 Let N^ denote the estimator of the unknown parameter N. MoM estimation: By solving the requirement (for the first moment) that N^ + 1 µ^ = ; 1 2 we get the estimator MoM N^ = 2^µ1 − 1 = 2¯xn − 1: (1) 1.2 Estimation of N: Maximum Likelihood (ML) d Consider again an IID sample (x1; : : : ; xn) of size n from distribution U (1;N). -
1. Revivew of Normal Distributions (1) True/False Practice: (A) F(X)
MATH 10B DISCUSSION SECTION PROBLEMS 4/11 { SOLUTIONS JAMES ROWAN 1. Revivew of normal distributions (1) True/False practice: 2 (a) f(x) = p 1 e−(x−4) =18 is a PDF for a random variable with mean 4 and standard deviation 3. 2π3 True . Recall that a normal random variable with mean µ and standard deviation σ has PDF 2 2 f(x) = p 1 e−(x−µ) =(2σ ). 2πσ (2) (Stewart/Day 12.5.73) The normal distribution models dispersion along a one-dimensional habitat like a coastline. Suppose the mean dispersal distance is 0 meters and the variance is 2 square meters. (a) What is the probability that an individual disperses more than 2 meters? We have a normal random variable with mean 0 and variance 2, and want to find the probability P (jXj ≥ 2). By transforming X into a standard normal variable (i.e. one withp mean 0 and variance 1), we can use a z-score table to look up probabilities.p We have thatpZ = X= 2 is a standard normal random variable, so P (jXj ≥ 2) = P (jZj ≥ 2) = 1 − 2P (0 ≤ Z ≤ 2). Looking at our z-score table, p we find that P (0 ≤ Z ≤ 2) ≈ 0:42, so that P (jXj ≥ 2) ≈ 0:16 . (b) What is the probability that an individual disperses less than 1 meter? We have a normal random variable with mean 0 and variance 2, and want to find the probability P (jXj ≤ 1). By transforming X into a standard normal variable (i.e. one withp mean 0 and variance 1), we can use a z-score table to look up probabilities.