lec6print

# lec6print - General concepts Properties of estimators...

This preview shows pages 1–11. Sign up to view the full content.

General concepts Properties of estimators Maximum Likelihood estimation Bayesian inference Point estimation Artin Armagan Sta. 113 Chapter 6 of Devore February 12, 2009 Artin Armagan Point estimation

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
General concepts Properties of estimators Maximum Likelihood estimation Bayesian inference Table of contents 1 General concepts 2 Properties of estimators 3 Maximum Likelihood estimation 4 Bayesian inference Artin Armagan Point estimation
General concepts Properties of estimators Maximum Likelihood estimation Bayesian inference A point estimate Definition A point estimate of a parameter θ is a single number that is a reasonable value for θ . The point estimate is given by a suitable statistic and computing this statistic from data. This statistic is the point estimator of θ . Artin Armagan Point estimation

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
General concepts Properties of estimators Maximum Likelihood estimation Bayesian inference Example A car manufacturer makes cars that explode or not explode with probability p . What is a point estimator of p ? If we observe cars 30 cars and 22 of them explode then ˆ p = 22 30 . Artin Armagan Point estimation
General concepts Properties of estimators Maximum Likelihood estimation Bayesian inference Estimators of the mean Given x 1 , ..., x n the following are estimators of the population mean μ and assume n is odd. 1 sample mean ¯ x = 1 n i x i 2 sample median (order data) ˜ x = x ( n +1) / 2 3 average of extremes ˇ x = min( x i )+max( x i ) 2 4 trimmed mean ¯ x tr (10) = 1 n i x i where n are the observations not in the largest and smallest 10% Artin Armagan Point estimation

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
General concepts Properties of estimators Maximum Likelihood estimation Bayesian inference Unbiased estimators Definition A point estimator ˆ θ is said to be an unbiased estimator of θ if E ( ˆ θ ) = θ for every possible value of θ . If ˆ θ is not unbiased then the bias is E ( ˆ θ ) θ. Artin Armagan Point estimation
General concepts Properties of estimators Maximum Likelihood estimation Bayesian inference Gamma example X 1 , ..., X 200 drawn from a Gamma distribution with α = 6 and β = 2 p ( x ) = 1 2 6 Γ(2) x 5 e x / 2 , x 0 , the mean is μ = α × β = 12. Artin Armagan Point estimation

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
General concepts Properties of estimators Maximum Likelihood estimation Bayesian inference Binomial Proposition If X is a binomial rv with parameters n and p then the sample proportion ˆ p = X n is an unbiased estimate of p. Artin Armagan Point estimation
General concepts Properties of estimators Maximum Likelihood estimation Bayesian inference Principle of unbiased estimation (PUE) When choosing among several estimators of θ select one that is unbiased. Artin Armagan Point estimation

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
General concepts Properties of estimators Maximum Likelihood estimation Bayesian inference Estimates of variance Proposition Let X 1 , ..., X n be a random sample from a distribution with mean μ and variance σ 2 . The estimator ˆ σ 2 = s 2 = i ( X i ¯ X ) 2 n 1 , is an unbiased estimator of σ 2 .
This is the end of the preview. Sign up to access the rest of the document.
• Fall '08
• Staff
• Maximum likelihood, Estimation theory, maximum likelihood estimation, estimators Maximum Likelihood, Artin Armagan, estimation Bayesian inference

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern