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# 23 Estimation

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### 23 Estimation

1. 1. Stat310 Estimation Hadley Wickham Saturday, 11 April 2009
2. 2. 1. What’s next 2. Recap 3. Example: speed of sound 4. T distribution 5. Comparing estimators 6. Interval estimators Saturday, 11 April 2009
3. 3. What’s next Today: ﬁnish of estimation Thursday & Tuesday: testing Last class: where next? Other stats courses and why you should bother Help session tomorrow. Saturday, 11 April 2009
4. 4. Recap What is the distribution of the average of n iid normal random variables with the same mean and variance? How can you form a 95% ci for a random variable with that distribution? Saturday, 11 April 2009
5. 5. Example We want to ﬁgure out what the speed of sound is. We do this by performing an experiment with our velocitometer. A velocitometer can measure the speed of anything, but has normally distributed error with standard deviation 10 meters per second. How can we decrease this error? How can we frame this problem statistically? Saturday, 11 April 2009
6. 6. Your turn We perform the experiment 10 times and get the following 10 speeds: 340 333 334 332 333 336 350 348 331 344 (mean: 338, sd: 7.01) What is our estimate of the speed of sound? What is the error (sd) of this estimate? Give an interval that we’re 95% certain the true speed of sound lies in. Saturday, 11 April 2009
7. 7. Hat notation Usually write the estimate of a parameter with a little hat over it. Subscript identiﬁes type of estimator used. ˆ µM M ˆ2 σ 2 ˆ σˆ µ ˆ µM L Saturday, 11 April 2009
8. 8. 355 350 345 upper 340 335 330 325 50 100 150 200 expt Saturday, 11 April 2009
9. 9. 355 350 345 upper 340 335 330 325 50 100 150 200 expt Saturday, 11 April 2009
10. 10. 355 350 345 lower 340 335 330 325 50 100 150 200 expt Saturday, 11 April 2009
11. 11. Example We want to ﬁgure out what the speed of sound is. We do this by performing an experiment with our velocitometer. A velocitometer can measure the speed of anything, but has normally distributed error with standard deviation 10 meters per second. Why is this example not realistic? Saturday, 11 April 2009
12. 12. Some reasons No such thing as a velocitometer! Scientiﬁc experiments usually much more complicated Don’t normally know the errors are normally distributed. Don’t normally know the standard deviation of the errors. Saturday, 11 April 2009
13. 13. Resolution Possible to overcome all of these problems, but we’re going to focus on just one. What happens if we don’t know the standard deviation, but have to estimate it? Saturday, 11 April 2009
14. 14. Your turn What is an estimate for the standard deviation of a normal distribution? When we have to estimate the sd, what do you think happens to the distribution of our estimate of the mean? (Would it get more or less accurate? What will happen to the conﬁdence interval?) What about as n gets bigger? Saturday, 11 April 2009
15. 15. t-distribution Xi ∼ Normal(µ, σ )2 ¯n − µ ¯n − µ X X √ ∼Z √ ∼ tn−1 σ/ n s/ n Parameter called degrees of freedom Saturday, 11 April 2009
16. 16. 0.3 df 1 dens 0.2 2 15 Inf 0.1 −3 −2 −1 0 1 2 3 x Saturday, 11 April 2009
17. 17. Properties of the t-dist Heavier tails compared to the normal distribution. lim tn = Z n→∞ Practically, if n > 30, the t distribution is practically equivalent to the normal. Saturday, 11 April 2009
18. 18. t-tables Basically the same as the standard normal. But one table for each value of degrees of freedom. Easiest to use calculator or computer: http://www.stat.tamu.edu/~west/applets/ tdemo.html (For homework, use this applet, for ﬁnal, I’ll give you a small table, if necessary) Saturday, 11 April 2009
19. 19. Example Back to the example. 340 333 334 332 333 336 350 348 331 344 (mean: 338, sd: 7.01) If sd is known: (332, 344) If not known: (333, 342) (2.23) Saturday, 11 April 2009
20. 20. Constructing interval ¯n − µ X √ ∼Z σ/ n ¯n − µ X √ ∼ tn−1 s/ n Saturday, 11 April 2009
21. 21. Steps Form conﬁdence interval for standardised distribution. Write as probability statement. Back transform. Write as interval. Saturday, 11 April 2009
22. 22. More complicated case (n − 1)S 2 ∼ χ (n − 1) 2 2 σ Find 95% conﬁdence interval for standard deviation in previous case (sd = 7.01, n = 10) Saturday, 11 April 2009
23. 23. Standard deviation Find conﬁdence interval for χ2(9). Generally want the shortest conﬁdence interval, but hard to ﬁnd when not symmetric. Any of the following are correct. The best has the smallest interval. Saturday, 11 April 2009
24. 24. 0.10 0.08 0.06 0.04 0.02 0.00 0 5 10 15 20 25 30 Saturday, 11 April 2009
25. 25. 0.10 (0.05, 1) (3.33,Inf) Length: Inf 0.08 0.06 0.04 0.02 0.00 0 5 10 15 20 25 30 Saturday, 11 April 2009
26. 26. 0.10 (0.03, 0.99) (2.85,21.67) Length: 18.8 0.08 0.06 0.04 0.02 0.00 0 5 10 15 20 25 30 Saturday, 11 April 2009
27. 27. 0.10 (0.025, 0.975) (2.7,19.0) Length: 16.3 0.08 0.06 0.04 0.02 0.00 0 5 10 15 20 25 30 Saturday, 11 April 2009
28. 28. 0.10 (0.01, 0.96) (2.09,17.61) Length: 15.5 0.08 0.06 0.04 0.02 0.00 0 5 10 15 20 25 30 Saturday, 11 April 2009
29. 29. 0.10 (0, 0.95) (0.0,16.9) Length: 16.9 0.08 0.06 0.04 0.02 0.00 0 5 10 15 20 25 30 Saturday, 11 April 2009
30. 30. Steps Form conﬁdence interval for standardised distribution. Write as probability statement. Back transform. Write as interval. Saturday, 11 April 2009
31. 31. Comparing estimators We know two ways of estimating the standard deviation of the normal distribution: the usual standard deviation of the data, or the estimate from maximum likelihood or method of moments How can we compare the two? Saturday, 11 April 2009
32. 32. What are other estimators of the standard deviation? Saturday, 11 April 2009
33. 33. Your turn How can we compare different point estimators? Say I have u1 and u2 which are functions of the X's, trying to estimate some θ. Based on what properties could I choose between u1 and u2? (probability, mean and variance) Saturday, 11 April 2009
34. 34. ˆn ) = θ E(θ Unbiased ˆ1 ) < V ar(θ2 ) ˆ V ar(θ Minimum variance Common problem is to ﬁnd UMVE (unbiased minimum variance estimator) across all possible estimators Saturday, 11 April 2009
35. 35. Low bias, low variance Low bias, high variance High bias, low variance High bias, high variance Saturday, 11 April 2009