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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 Definition Exponential family Mean square error (how to choose an estimator) 6205-Ch6

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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: Estimation (of parameters) Hypothesis testing Estimation: Point estimation: use a single value to estimate a parameter Interval estimation: find an interval covering the unknown parameter 6205-Ch6

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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

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**Maximum Likelihood Estimation**

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

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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(m,s^2) distribution. Find the m.l.e. of m and s. 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

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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

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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

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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

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**Sufficient Statistics**

6205-Ch6

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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 X-bar is sufficient for the parameter a 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

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Exponential Family 6205-Ch6

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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

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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

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**Application of Sufficience**

6205-Ch6

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**Example Consider a Weibull distribution with parameter(a=2, b)**

Find a sufficient statistic for b Find an unbiased estimator which is a function of the sufficient statistic found in 1) 6205-Ch6

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Good Estimator? Criterion: mean square error 6205-Ch6

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**Example Which of the following two estimator of variance is better?**

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