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# Chap6 - Statistics 211 Prof Emanuel Parzen Chapter 6...

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Statistics 211 Prof. Emanuel Parzen Chapter 6 STATISTICAL INFERENCE, HYPOTHESIS TESTS, CONFIDENCE INTERVALS Statistical inference is the science of learning from data. Its strategy (long range plan) is to determine probability models which fit the observed data. When our data is assumed to be a random sample of a random variable Y , our goal is to learn the true distribution function ( ( [ ] F F ; Pr , y y Y Y y y = = - ∞ < < ∞ Our notation ( F ; y Y is read: “distribution function (evaluated at a real number y ) of the random variable Y .” A random variable Y has quantile function ( ( ; , 0 1 Q P Q P Y P = < < intuitively satisfies ( ( F Q P P = or ( y Q P = is a solution of ( [ ] F Pr . P y Y y = = We often assume the distribution of Y is ( , Normal μ σ , equivalently ( , 0,1 Y Z Z Normal μ σ = + with distribution ( [ ] Pr x Z x Φ = . Then ( 29 F ; Pr , Y y y y Y Z μ μ μ σ σ σ - - - = = = Φ ( ( ( 1 ; ; , Q P Y P Q P Z μ σ μ σ - = + Φ = + In general, our tactic to learn about unknown ( F ; y Y is to assume a family of distributions ( F ; y θ depending on a parameter θ . The notation ( F ; y θ is read “ distribution function (evaluated at real number y) corresponding to parameter value θ ”. Statistical inference, the problems of learning from data, can be stated to be the problem of learning the parameters θ such that ( F ; y θ fits the data; more precisely fits the sample distribution function, for y -∞ < < ∞ , ( [ ] F ; Proportion . y sample sample y = The process of statistical inference consists of a series of actions (or phases): Action 0: Q. Questions scientific are posed; design data to be observed. ----------------------------------------------------------------------------------------------- Action .5: S. Sample summary and graphs to look in order to look at the data; particularly informative is the practical sample quantile function ( ; Q P sample , defined as a continuous function connecting with lines the points ((j-.5)/n, Y(j;n)) where Y(j;n) is j-th order statistic of the sample. ----------------------------------------------------------------------------------------------

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Action 1: I. Identification of parametric model for true distribution function, denoted ( F ; y θ or ( ; Q P θ . Frequently assumed probability models are: ( , Normal μ σ for continuous data, with parameters \mu and \sigma, and ( Bernoulli p for 0-1 binary data, with parameter p on interval 0<p<1. -------------------------------------------------------------------------------------------- Action 2: E. Estimation of parameter θ by estimator ɵ θ , which we call a point estimator to distinguish it from an interval estimator provided by a confidence interval. We use estimator to define pivot ɵ ( , Tin θ θ , a function of the parameter and the estimator, which is a random variable whose distribution given θ does not depend on θ and which is increasing function of θ . For location parameters θ similar to mean μ and proportion probability p we define pivot ɵ ( ɵ ( ɵ ( , Tin SE θ θ θ θ θ = - where ɵ ( SE θ is standard error (estimator of standard deviation) of ɵ θ given that θ is the true parameter value.
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Chap6 - Statistics 211 Prof Emanuel Parzen Chapter 6...

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