chapter1 - SMSL/2011 THE UNIVERSITY OF HONG KONG DEPARTMENT...

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Unformatted text preview: SMSL/2011 THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1302 PROBABILITY AND STATISTICS II § 1 An Overview § 1.1 Statistics: the discipline 1.1.1 Broadly speaking, statistics as a discipline seeks to make sense of observations (or data , or samples ) to obtain a reliable “portrait” of reality. Key steps: 1. problem setup — identify the questions we wish to answer; 2. observation — acquire numeric (or even graphical) data by physical means; 3. modelling — provide a connection between each possible answer to all possible observa- tions; 4. inference — make decision on the basis of observed information and provide justification. 1.1.2 Pure scientists — seek to “discover” reality or “invent” technology. Statisticians — seek to “portray” reality by statistical reasoning but make no claim to have actually discovered it. In a sense, statisticians are more akin to people like novelists, painters, movie directors, com- posers etc., despite their use of an entirely different means to portray reality. 1.1.3 Statistical reasoning relies on scientific argumentation, drawing knowledge from different fields of mathematics and, in particular, probability theory . In this sense, statisticians are akin to physicists, astronomers, engineers etc. 1.1.4 Statistics is both a subject of arts and sciences . § 1.2 Statistical model 1.2.1 What are data ? • numeric information observed in real life 1 • a source for understanding the unknown nature 1.2.2 Data can be observed, but the unknown nature cannot. How can we tell the “nature” from “data”? Statistical reasoning provides a channel for communication between them. Figure § 1.2.1 sum- marizes the general process. Nature / Population Observed Data Parameters Random variables Distributional assumptions Statistical Model Modelling Interpretation Sampling Inference Realization Figure § 1.2.1: Flow of statistical reasoning 1.2.3 Sample — set of observations { x 1 ,...,x n } , which are taken as realisations of a set of random variables X 1 ,...,X n , respectively. They constitute our data . [“Realisation” means the assignment of an actual numerical value to the random “variable”, often as a result of an experiment or observation study.] 1.2.4 Statistical model — collection of probability distributions, each of which is a candidate for the true, but unknown, distribution of ( X 1 ,...,X n ). 2 1.2.5 Parametric statistical model or parametric family — statistical model indexed by some parameter θ in an index set Θ. Under this model, the random variables X 1 ,...,X n are assumed to be generated from a mem- ber of the parametric family corresponding to an unknown value of θ , i.e....
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chapter1 - SMSL/2011 THE UNIVERSITY OF HONG KONG DEPARTMENT...

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