Point Estimation

1. Point Estimation Notes of STAT 6205 by Dr. Fan

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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

10. Sufficient Statistics 6205-Ch6

11. 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

12. Exponential Family 6205-Ch6

13. 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

14. 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

15. Application of Sufficience 6205-Ch6

16. 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

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

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