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Chapter 1 - SMSL/2010 THE UNIVERSITY OF HONG KONG...

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SMSL/2010 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 “portray” reality on the basis of observa- tions (or data , or samples ). Key steps: 1. modelling — conceptualize reality with regard to problem being investigated; 2. observation — acquire numeric (or even visual) data by physical means; 3. data reduction — extract information/evidence from data which is relevant to investiga- tion; 4. inference — make decision on the basis of relevant information; 5. confidence assessment — justify decision by quantifying confidence in it. 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
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§ 1.2 Statistical model 1.2.1 What are data ? numeric information observed in real life 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.] 2
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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 ) 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. X 1 , . . . , X n have joint probability function p ( x 1 , . . . , x n | θ ), for some unknown θ Θ.
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