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 Point estimation Sayan Mukherjee Sta. 113 Chapter 6 of Devore October 18, 2007 Sayan Mukherjee 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 Table of contents 1 General concepts 2 Properties of estimators 3 Maximum Likelihood estimation Sayan Mukherjee Point estimation
General concepts Properties of estimators Maximum Likelihood estimation A point estimate Defnition 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 . Sayan Mukherjee 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 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 . Sayan Mukherjee Point estimation
General concepts Properties of estimators Maximum Likelihood estimation 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 0 i x i where n 0 are the observations not in the largest and smallest 10% Sayan Mukherjee 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 Gaussian -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 estimators of mean mean median extreme trmmed mean Sayan Mukherjee Point estimation
General concepts Properties of estimators Maximum Likelihood estimation Cauchy 0 500 1000 1500 0 0.002 0.004 0.006 0.008 0.01 mean 1.5 2 2.5 3 0 0.5 1 1.5 2 median 0 2 4 6 x 10 4 0 1 x 10 -4 extreme 2 3 4 5 6 0 0.5 1 1.5 trimmed Sayan Mukherjee 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 Uniform 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0 10 20 30 40 50 60 estimators of mean mean median extreme trmmed mean Sayan Mukherjee Point estimation
General concepts Properties of estimators Maximum Likelihood estimation Unbiased estimators Defnition 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. Sayan Mukherjee 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 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.
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 64

lec6print - General concepts Properties of estimators...

This preview shows document pages 1 - 11. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online