Point and Interval Estimation II Bios 662

Transcription

1 Point and Interval Estimation II Bios 662 Michael G. Hudgens, Ph.D. mhudgens :17 BIOS Point and Interval Estimation II

2 Nonparametric CI for the Median Suppose X 1,..., X n iid according to continuous distribution F Let ζ 1/2 be the population median We will show Pr[X (r) < ζ 1/2 < X (n r+1) ] = 1 2 n n r i=r ( ) n i Therefore, for fixed n, we choose largest r such that n r 1 ( ) n 2 n 1 α i i=r BIOS Point and Interval Estimation II

3 Let Y be a Bernoulli r.v. Bernoulli RV Y can take on two values, 0 or 1 Pr[Y = 1] = π; Pr[Y = 0] = 1 π E(Y ) = π; V ar(y ) = π(1 π) BIOS Point and Interval Estimation II

4 Binomial RV Process that produces independent Bernoulli RVs with the same probability of success π Let Y count the number of successes in n trials Y Binomial(n,π) Pr[Y = y] = ( ) n π y (1 π) n y y E(Y ) = nπ; V ar(y) = nπ(1 π) BIOS Point and Interval Estimation II

5 Derivation of CI for Median CDF Pr[X i x] = F (x) Therefore Pr[X (r) x] = Pr[at least r of the X i x] = n i=r ( ni ) F (x) i {1 F (x)} n i BIOS Point and Interval Estimation II

6 Derivation of CI for Median By law of total probability Pr[X (r) ζ p ] = Pr[X (r) ζ p, X (s) ζ p ] + Pr[X (r) ζ p, X (s) < ζ p ] If s > r, then X (s) < ζ p X (r) < ζ p Therefore Pr[X (r) ζ p ] = Pr[X (r) ζ p X (s) ] + Pr[X (s) < ζ p ] BIOS Point and Interval Estimation II

7 Derivation of CI for Median Pr[X (r) ζ p X (s) ] = Pr[X (r) ζ p ] Pr[X (s) < ζ p ] = n i=r ( ni ) F (ζp ) i {1 F (ζ p )} n i n i=s ( ni ) F (ζp ) i {1 F (ζ p )} n i = s 1 i=r If p = 1/2; F (ζ p ) = 1/2, such that Pr[X (r) ζ 0.5 X (s) ] = 1 2 n ( ni ) F (ζp ) i {1 F (ζ p )} n i s 1 i=r ( ) n i BIOS Point and Interval Estimation II

8 Derivation of CI for Median We could choose any r and s such that Pr(X (r) ζ 0.5 X (s) ) = 1 2 n s 1 i=r ( n i But the best choice for s is n r + 1 (why?) Thus we choose r such that n r 1 ( ) n 2 n i i=r 1 α ) 1 α BIOS Point and Interval Estimation II

9 Derivation of CI for Median Values of r for 95% CI for Median n r Cf page van Belle et al. BIOS Point and Interval Estimation II

10 95% CI for Betacarotene Example For n = 23, choose r = 7 such that n r + 1 = 17 Therefore (y (7) = 106, y (17) = 186) gives a 95% CI for the median betacarotene value This CI makes no assumptions about the distribution of the Y s Note: i=7 > sum(dbinom(7:16,23,1/2)) [1] ( ) 23 i = α BIOS Point and Interval Estimation II

11 SAS Code and Output proc univariate data=beta cipctldf; var base1; run; 95% Confidence Limits Order Statistics Quantile Distribution Free LCL Rank UCL Rank Coverage 99% % % % Q % Median % Q % % % % Min BIOS Point and Interval Estimation II

12 Large sample CI for median If n is sufficiently large, say n > 25, we can get an approximate 100(1 α)% CI for the median by counting n z 1 α/2 2 ordered observations to the left and right of the median and rounding out to the next integer Cf Lehmann (1998, p.84) BIOS Point and Interval Estimation II

13 Large Sample CI for Median: Example Suppose n = 100 and α = 0.05 Then n z 1 α/2 2 Rounding yields: = 5(1.96) = ± 9.8 (y (40), y (61) ) Can show r = 40 using exact method > sum(dbinom(40:60,100,1/2)) [1] > sum(dbinom(41:59,100,1/2)) [1] BIOS Point and Interval Estimation II

14 If general, Large sample CI for any quantile Pr[ζ p < Z (r) ] = r 1 where q = 1 p i=0 = r 1 i=0 ( ni ) F (ζp ) i {1 F (ζ p )} n i ( ni ) p i q n i From CLT, if Y Bin(n, p), then Y np + 1/2 npq N(0, 1) Thus Pr[ζ p Z (r) ] = Pr[Y r 1] = Pr[Z (r 1) np+1/2 npq ] = Φ( r np 1/2 npq ) BIOS Point and Interval Estimation II

15 Large sample CI for any quantile Goal is symmetric (1 α)% CI, so want r np 1/2 α/2 = Pr[ζ p < Z (r) ] = Φ( ) npq That is z 1 α/2 = r np 1/2 npq Implying r = np z 1 α/2 npq For p = 1/2, yields r = n z 1 α/2 n 2 BIOS Point and Interval Estimation II

16 Large sample CI for any quantile Similar reasoning yields s = np z 1 α/2 npq Thus (1 α)% CI for ζ p is given by (X ( r ), X ( s ) ) Note n large enough ensures r, s {1,..., n} BIOS Point and Interval Estimation II

17 References for Order Statistics A. E. Sarhan and B. G. Greenberg, Contributions to Order Statistics, H. A. David, Order Statistics D. B. Owen, Handbook of Statistical Tables E. L. Lehmann Nonparametrics: Statistical Methods Based on Ranks, BIOS Point and Interval Estimation II

18 CI for Variance Recall (result 4.4 p.95 text) Therefore (n 1)s 2 σ 2 1 α = Pr[χ 2 α/2,n 1 Implying 1 α = Pr (n 1)s2 χ 2 1 α/2,n 1 χ 2 n 1 (n 1)s2 σ 2 χ 2 1 α/2,n 1 ] σ 2 (n 1)s2 χ 2 α/2,n 1 BIOS Point and Interval Estimation II

19 CI for Variance Since the χ 2 distribution is not symmetric, need to look up both χ 2 α/2,n 1 and χ2 1 α/2,n 1 This CI is dependent on the Y s being from a normal distribution BIOS Point and Interval Estimation II

20 CI for Variance for Betacarotene Example n = 23; s 2 = χ 2.025,22 = 10.98; χ2.975,22 = Therefore, 95% CI for σ 2 (22( )/36.78, 22( )/10.98) = ( , ) 95% CI for σ = (47.05, 86.12) BIOS Point and Interval Estimation II

21 SAS Code and Output proc univariate data=beta cibasic; var base1; run; Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean Std Deviation Variance BIOS Point and Interval Estimation II

22 CI for Variance - Nonnormal data Large sample theory n(s 2 n σ 2 ) d N(0, (α 4 1)σ 4 ) where α 4 = E(X µ) 4 /σ 4 is the kurtosis (cf. Dudewicz and Mishra Modern Mathematical Statistics, p. 325) Crude approximation : replace usual CI with (n 1)s 2 χ 2 1 α/2,n 1 (1 + g 2/n), (n 1)s 2 χ 2 α/2,n 1 (1 + g 2/n) where g 2 = b 2 3 and b 2 is an estimate of α 4 (cf Solomon and Stephens, Encyc of Stat Sci) BIOS Point and Interval Estimation II

23 CI for Variance - Nonnormal data Nonparametric approach such as bootstrap (cf Efron and Tibshirani An Introduction to the Bootstrap, Ch 14) Software? BIOS Point and Interval Estimation II