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Point Estimation Notes of STAT 6205 by Dr. Fan

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Overview Section 6.1 Point estimation Maximum likelihood estimation Methods of moments Sufficient statistics o Definition o Exponential family o Mean square error (how to choose an estimator) 6205-Ch6 2

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Big Picture Goal: To study the unknown distribution of a population Method: Get a representative/random sample and use the information obtained in the sample to make statistical inference on the unknown features of the distribution Statistical Inference has two parts: o Estimation (of parameters) o Hypothesis testing Estimation: o Point estimation: use a single value to estimate a parameter o Interval estimation: find an interval covering the unknown parameter 6205-Ch6 3

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Point Estimator Biased/unbiased: an estimator is called unbiased if its mean is equal to the parameter of estimate; otherwise, it is biased Example: X_bar is unbiased for estimating mu 6205-Ch6 4

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Maximum Likelihood Estimation Given a random sample X1, X2, …, Xn from a distribution f(x; ) where is a (unknown) value in the parameter space W. Likelihood function vs. joint pdf Maximum Likelihood Estimator (m.l.e.) of , denoted as is the value which maximizes the likelihood function, given the sample X1, X2, …, Xn. 6205-Ch6 5

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Examples/Exercises Problem 1: To estimate p, the true probability of heads up for a given coin. Problem 2: Let X1, X2, …, Xn be a random sample from a Exp(mu) distribution. Find the m.l.e. of mu. Problem 3: Let X1, X2, …, Xn be a random sample from a Weibull(a=3,b) distribution. Find the m.l.e. of b. Problem 4: Let X1, X2, …, Xn be a random sample from a N( , ^2) distribution. Find the m.l.e. of and . Problem 5: Let X1, X2, …, Xn be a random sample from a Weibull(a,b) distribution. Find the m.l.e. of a and b. 6205-Ch6 6

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Method of Moments Idea: Set population moments = sample moments and solve for parameters Formula: When the parameter q is r-dimensional, solve the following equations for q : 6205-Ch6 7

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Examples/Exercises Given a random sample from a population Problem 1: Find the m.m.e. of m for a Exp( m ) population. Exercise 1: Find the m.m.e. of m and s for a N( m,s ^2) population. 6205-Ch6 8

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Sufficient Statistics Idea: The “sufficient” statistic contains all information about the unknown parameter; no other statistic can provide additional information as to the unknown parameter. If for any event A, P[A|Y=y] does not depend on the unknown parameter, then the statistic Y is called “sufficient” (for the unknown parameter). Any one-to-one mapping of a sufficient statistic Y is also sufficient. Sufficient statistics do not need to be estimators of the parameter. 6205-Ch6 9

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Sufficient Statistics 6205-Ch6 10

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Examples/Exercises Let X1, X2, …, Xn be a random sample from f(x) Problem: Let f be Poisson(a). Prove that 1.X-bar is sufficient for the parameter a 2.The m.l.e. of a is a function of the sufficient statistic Exercise: Let f be Bin(n, p). Prove that X-bar is sufficient for p (n is known). Hint: find a sufficient statistic Y for p and then show that X-bar is a function of Y 6205-Ch6 11

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Exponential Family 6205-Ch6 12

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Examples/Exercises Example 1: Find a sufficient statistic for p for Bin(n, p) Example 2: Find a sufficient statistic for a for Poisson(a) Exercise: Find a sufficient statistic for m for Exp( m ) 6205-Ch6 13

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Joint Sufficient Statistics Example: Prove that X-bar and S^2 are joint sufficient statistics for m and s of N( m, s ^2) 6205-Ch6 14

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Application of Sufficience 6205-Ch6 15

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Example Consider a Weibull distribution with parameter(a=2, b) 1)Find a sufficient statistic for b 2)Find an unbiased estimator which is a function of the sufficient statistic found in 1) 6205-Ch6 16

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Good Estimator? Criterion: mean square error 6205-Ch6 17

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Example Which of the following two estimator of variance is better? 6205-Ch6 18

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