Pivotal Quantities. Mathematics 47: Lecture 16. Dan Sloughter. Furman University. March 30, 2006

Transcription

1 Pivotal Quantities Mathematics 47: Lecture 16 Dan Sloughter Furman University March 30, 2006 Dan Sloughter (Furman University) Pivotal Quantities March 30, / 10

2 Pivotal quantities Definition Suppose X 1, X 2,..., X n is a random sample from a distribution with parameter θ. If Y = g(x 1, X 2,..., X n, θ) is a random variable whose distribution does not depend on θ, then we call Y a pivotal quantity for θ. Dan Sloughter (Furman University) Pivotal Quantities March 30, / 10

3 Pivotal quantities Definition Suppose X 1, X 2,..., X n is a random sample from a distribution with parameter θ. If Y = g(x 1, X 2,..., X n, θ) is a random variable whose distribution does not depend on θ, then we call Y a pivotal quantity for θ. Example In finding confidence intervals for µ given a random sample X 1, X 2,..., X n from N(µ, σ 2 ), we used the fact that X µ S n is a pivotal quantity for µ. Dan Sloughter (Furman University) Pivotal Quantities March 30, / 10

4 Example Dan Sloughter (Furman University) Pivotal Quantities March 30, / 10

5 Example Suppose X 1, X 2,..., X n is a random sample from N(µ, σ 2 ). Dan Sloughter (Furman University) Pivotal Quantities March 30, / 10

6 Example Suppose X 1, X 2,..., X n is a random sample from N(µ, σ 2 ). Then (n 1)S 2 is χ 2 (n 1), and so is a pivotal quantity for σ 2. σ 2 Dan Sloughter (Furman University) Pivotal Quantities March 30, / 10

7 Example Suppose X 1, X 2,..., X n is a random sample from N(µ, σ 2 ). Then (n 1)S 2 is χ 2 (n 1), and so is a pivotal quantity for σ 2. Let χ 2 n,α be the α quantile of χ 2 (n). σ 2 Dan Sloughter (Furman University) Pivotal Quantities March 30, / 10

8 Example Suppose X 1, X 2,..., X n is a random sample from N(µ, σ 2 ). Then (n 1)S 2 is χ 2 (n 1), and so is a pivotal quantity for σ 2. σ 2 Let χ 2 n,α be the α quantile of χ 2 (n). Then P ( χ 2 n 1, α 2 ) (n 1)S 2 σ 2 χ 2 n 1,1 α = 1 α. 2 Dan Sloughter (Furman University) Pivotal Quantities March 30, / 10

9 Example (cont d) Dan Sloughter (Furman University) Pivotal Quantities March 30, / 10

10 Example (cont d) It follows that P ( (n 1)S 2 χ 2 n 1,1 α 2 σ 2 ) (n 1)S 2 χ 2 = 1 α. n 1, α 2 Dan Sloughter (Furman University) Pivotal Quantities March 30, / 10

11 Example (cont d) It follows that P ( (n 1)S 2 χ 2 n 1,1 α 2 σ 2 ) (n 1)S 2 χ 2 = 1 α. n 1, α 2 Hence ( (n 1)S 2 χ 2, n 1,1 α 2 ) (n 1)S 2 χ 2 n 1, α 2 is a 100(1 α)% confidence interal for σ 2. Dan Sloughter (Furman University) Pivotal Quantities March 30, / 10

12 Example Dan Sloughter (Furman University) Pivotal Quantities March 30, / 10

13 Example The estimated ages of 19 mineral samples from the Black Forest of Germany were found to be (in millions of years): Dan Sloughter (Furman University) Pivotal Quantities March 30, / 10

14 Example The estimated ages of 19 mineral samples from the Black Forest of Germany were found to be (in millions of years): Then s 2 = Dan Sloughter (Furman University) Pivotal Quantities March 30, / 10

15 Example The estimated ages of 19 mineral samples from the Black Forest of Germany were found to be (in millions of years): Then s 2 = We find (either with R, or with Table Va) that χ 2 18,.025 = 8.23, and χ 2 18,.975 = Dan Sloughter (Furman University) Pivotal Quantities March 30, / 10

16 Example Dan Sloughter (Furman University) Pivotal Quantities March 30, / 10

17 Example So, assuming the sample is from N(µ, σ 2 ), we see that ( (18)(733.4), (18)(733.4) ) = (419.1, ) is a 95% confidence interval for σ 2. Dan Sloughter (Furman University) Pivotal Quantities March 30, / 10

18 Example So, assuming the sample is from N(µ, σ 2 ), we see that ( (18)(733.4), (18)(733.4) ) = (419.1, ) is a 95% confidence interval for σ 2. Note: Taking square roots, it follows that (20.47, 40.05) is a 95% confidence interval for σ. Dan Sloughter (Furman University) Pivotal Quantities March 30, / 10

19 Example Dan Sloughter (Furman University) Pivotal Quantities March 30, / 10

20 Example Let X 1, X 2,..., X n be a random sample from a uniform distribution on (0, θ). Dan Sloughter (Furman University) Pivotal Quantities March 30, / 10

21 Example Let X 1, X 2,..., X n be a random sample from a uniform distribution on (0, θ). Recall: the density of T = X (n) is { n θ n tn 1, if 0 < t < θ, f T (t) = 0, otherwise. Dan Sloughter (Furman University) Pivotal Quantities March 30, / 10

22 Example Let X 1, X 2,..., X n be a random sample from a uniform distribution on (0, θ). Recall: the density of T = X (n) is { n θ n tn 1, if 0 < t < θ, f T (t) = 0, otherwise. If we let Y = T θ, then Y has density { ny n 1, if 0 < y < 1, f Y (y) = θf T (θy) = 0, otherwise. Dan Sloughter (Furman University) Pivotal Quantities March 30, / 10

23 Example Let X 1, X 2,..., X n be a random sample from a uniform distribution on (0, θ). Recall: the density of T = X (n) is { n θ n tn 1, if 0 < t < θ, f T (t) = 0, otherwise. If we let Y = T θ, then Y has density { ny n 1, if 0 < y < 1, f Y (y) = θf T (θy) = 0, otherwise. Hence Y is a pivotal quantity for θ. Dan Sloughter (Furman University) Pivotal Quantities March 30, / 10

24 Example (cont d) Dan Sloughter (Furman University) Pivotal Quantities March 30, / 10

25 Example (cont d) Now the distribution function for Y is 0, if y 0, F Y (y) = y n, if 0 < y 1, 1, if y 1. Dan Sloughter (Furman University) Pivotal Quantities March 30, / 10

26 Example (cont d) Now the distribution function for Y is 0, if y 0, F Y (y) = y n, if 0 < y 1, 1, if y 1. So, in particular, ( ) ( ) P Y (0.95) 1 n = F Y (0.95) 1 n = Dan Sloughter (Furman University) Pivotal Quantities March 30, / 10

27 Example (cont d) Dan Sloughter (Furman University) Pivotal Quantities March 30, / 10

28 Example (cont d) Thus P ( X(n) (0.95) 1 n θ ) = Dan Sloughter (Furman University) Pivotal Quantities March 30, / 10

29 Example (cont d) Thus P ( X(n) (0.95) 1 n θ ) = And so ( ) X(n), (0.95) 1 n is a 95% confidence interval for θ. Dan Sloughter (Furman University) Pivotal Quantities March 30, / 10

30 One-sided confidence intervals Note: the interval in the previous example is an example of a one-sided confidence interval, as opposed to the two-sided confidence intervals of our previous examples. Dan Sloughter (Furman University) Pivotal Quantities March 30, / 10

31 One-sided confidence intervals Note: the interval in the previous example is an example of a one-sided confidence interval, as opposed to the two-sided confidence intervals of our previous examples. In particular, we call a lower confidence bound for θ. X (n) (0.95) 1 n Dan Sloughter (Furman University) Pivotal Quantities March 30, / 10

32 One-sided confidence intervals Note: the interval in the previous example is an example of a one-sided confidence interval, as opposed to the two-sided confidence intervals of our previous examples. In particular, we call Example a lower confidence bound for θ. X (n) (0.95) 1 n Dan Sloughter (Furman University) Pivotal Quantities March 30, / 10

33 One-sided confidence intervals Note: the interval in the previous example is an example of a one-sided confidence interval, as opposed to the two-sided confidence intervals of our previous examples. In particular, we call Example a lower confidence bound for θ. X (n) (0.95) 1 n Suppose in a sample of size 20 from a uniform distribution on the interval (0, θ), we find x (20) = Dan Sloughter (Furman University) Pivotal Quantities March 30, / 10

34 One-sided confidence intervals Note: the interval in the previous example is an example of a one-sided confidence interval, as opposed to the two-sided confidence intervals of our previous examples. In particular, we call Example a lower confidence bound for θ. X (n) (0.95) 1 n Suppose in a sample of size 20 from a uniform distribution on the interval (0, θ), we find x (20) = Then (0.95) , so ( , ) is a 95% confidence interval for θ. Dan Sloughter (Furman University) Pivotal Quantities March 30, / 10