Intro_Inference.pdf

Sometimes θ is a vector that we are only interested

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Sometimes, θ is a vector that we are only interested in one (or some) component of θ . In this case, the remaining parameters are referred to as nuisance parameter . Jimin Ding, Math WUSTL Math 494 Spring 2018 9 / 44

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Type of Statistical Models I Nonparametric model: goal is F or f I Parametric model: goal is θ Since the second question has a smaller space of candidates, we first start with question 2 (simpler). Sometimes, θ is a vector that we are only interested in one (or some) component of θ . In this case, the remaining parameters are referred to as nuisance parameter . Semiparametric model is a combination of parametric and nonparametric model and the goal is f ( x ; θ, g ) which is only partially specified by the parameter θ , and g is a unspecified function (infinite-dimension). Jimin Ding, Math WUSTL Math 494 Spring 2018 9 / 44
What is Statistical Inference? Statistical inference is the process of using data to infer the distribution that generated data. Jimin Ding, Math WUSTL Math 494 Spring 2018 10 / 44

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What is Statistical Inference? Statistical inference is the process of using data to infer the distribution that generated data. Three components of statistical inference: I Point Estimation I Confidence Interval I Hypothesis Testing Jimin Ding, Math WUSTL Math 494 Spring 2018 10 / 44
Point Estimation Jimin Ding, Math WUSTL Math 494 Spring 2018 11 / 44

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Point Estimation I Parameter of Interest : A fixed and unknown population parameter, θ , or a function of model parameter, g ( θ ) . Eg: population mean μ , population standard deviation σ ,... I Point Estimator : A statistic from a sample that is used to estimate the parameter of interest, ˆ θ (which is a r.v.). Eg: sample mean ¯ X = i X i /n for μ , sample standard deviation S = q i ( X i - ¯ X ) 2 ( n - 1) for σ . I Estimate : The numerical value of an estimator in an observed sample. Eg: observed sample mean ¯ x = i x i /n for μ , observed sample standard deviation s = q i ( x i - ¯ x ) 2 ( n - 1) for σ . I Point estimation : the process of providing a point estimator. Jimin Ding, Math WUSTL Math 494 Spring 2018 12 / 44
Example 1: Poisson Model (Ex 4.1.3) Suppose the number of customers X that enter a store during 9-10am. follows a Poisson distribution with parameter θ . Suppose a random sample of the number of customers that enter the store during 9-10am. for 10 days results in the values: 9 , 7 , 9 , 15 , 10 , 13 , 11 , 7 , 2 , 12 . What is a good guess of θ ? go back to CI Jimin Ding, Math WUSTL Math 494 Spring 2018 13 / 44

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Maximum Likelihood Estimation (MLE) I The observed values in a random sample, x 1 , x 2 , · · · , x n , can be called as realizations of the sample . I Likelihood function: joint pdf of the realizations (observed data) L ( θ ) = f 1 ( x 1 ; θ ) f 2 ( x 2 ; θ ) · · · f n ( x n ; θ ) n i =1 f ( x i ; θ ) ( if iid ) I The larger L ( θ ) is, the more possible that we observe these realizations x 1 , x 2 , · · · , x n .
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