1. Stat 321 – Lecture 26 Estimators (cont.) The judge asked the statistician if she promised to tell the truth, the whole truth, and nothing but the truth? The statistician replied, “Yes, 95% of the time.”

2. Last Time – Finding Point Estimators Method of Moments  General idea: Equate “moments” of probability distribution with moments of the observed data set  Simple case: Find E(X) in terms of unknown parameter, solve for  in terms of E(X), substitute sample mean for E(X) Maximum Likelihood  General idea: Choose the parameter value that maximizes the likelihood function (f(x;  ) as a function of  ) given the observed data

3. Example 1 (a) If x =.75, then what does (  +1).75  look like? (b) What if we have more than one observation?

4. Joint pdf Since independent, take the product of the marginal pdf’s f(x 1, …, x n ;  ) = (  +1) n (  x i )  How maximize? ln f(x 1, …, x n ;  ) = nln(  +1) +  ln(  x i ) d/d  = n/  +ln(  x i ) = 0 => estimator = -1 –n/ln(  x i ) For this particular sample

5. Bootstrapping Especially useful in determining the sampling variability in an estimator Turns out, if take repeated samples from original sample, with replacement, get a reasonable estimate of the standard deviation of estimator!

6. Bootstrapping Population (  = 402.6,  = 267.9) Sample mean should have mean 402.6 and standard deviation 84.7

7. Bootstrapping And what about something crazy like a trimmed mean?

8. Bootstrapping And what about our MOME and MLE estimators for our Beta(3,1) pdf? Compare empirical sampling distribution of these estimators from Beta(3,1) to bootstrap samples from a sample of 10 observations.

9. Confidence intervals?

10. Larger sample size?

11. For Thursday Quiz