NCSU ST 522 - Principles of Data Reduction (37 pages)

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Principles of Data Reduction



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Principles of Data Reduction

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Lecture Notes


Pages:
37
School:
North Carolina State University
Course:
St 522 - Statistical Theory II
Statistical Theory II Documents

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Chapter 6 Principles of Data Reduction 1 Statistical Inference Data X X1 Xn from a probability distribution f x with unknown Our task is Examples to estimate to estimate to estimate to estimate based on data the success probability p in a Bernoulli trial the supporting rate p of a president candidate the average SAT score of the freshmen at a national level Three types of methods to estimate point estimation Chapter 7 hypothesis testing Chapter 8 interval estimation Chapter 9 Two Steps for Statistical Inference Step 1 Data reduction summarizing information about in data with one or a few statistics T T X Data X1 Xn contains much information some are relevant for and some are not Dropping irrelevant information is desirable but dropping relevant information is undesirable the dimension of T is generally smaller than the sample size n Step 2 Estimator construction using T to construct point estimators test statistics upper lower confidence limit 8 2 Statistics and Partition Def A statistic T X is a function of the sample X1 Xn Examples sample mean X sample variance S 2 the largest order statistic X n the smallest order statistic X 1 Partition of Sample Space by T X Consider the discrete case For any possible value t of T there is a corresponding set At x T x t The set collection At all t makes a partition on the sample space of X Note X P X x P T X t x At The event X x is the subset of T X T x i e X x T X T x Example Toss a coin n 3 times and let X1 X3 be respectively the outcome of each toss Let T the total number of heads obtained i e P T 3i 1 Xi Write down the partition of the sample space given by T Remark Often T has a simpler data structure and distribution than the original sample X X1 Xn so it would be nice if we can use T X to summarize and then replace the entire data 9 Important Issues in Data Reduction We should think about the following questions carefully before the simplification process Is there any loss of information due to summarization How to



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