Bootstrap Method > # Purpose: understand how bootstrap method works > obs=c(11.96, 5.03, 67.40, 16.07, 31.50, 7.73, 11.10, 22.38) > n=length(obs) >
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1 Btstrap Methd > # Purpse: understand hw btstrap methd wrks > bs=c(11.96, 5.03, 67.40, 16.07, 31.50, 7.73, 11.10, 22.38) > n=length(bs) > mean(bs) [1] > # estimate f lambda > lambda = 1/mean(bs); lambda [1] > # The expnential distributin with rate=lambda has density f(x) = lambda e^(- lambda x) > # draw a randm sample f size n frm expnential(lambda) distributin, where lambda is estimated frm data > x=rexp(n, lambda); x [1] [1] > 1/mean(x) # Btstrap estimate f lambda [1] > # d it again > x=rexp(n, lambda); x [1] [1] > 1/mean(x) # Btstrap estimate f lambda [1] > # here we d a nnparametrical btstrape by replacing x=rexp(n, lambda) with x=sample(bs, n, replace=t) > x=sample(bs, n, replace=t); x [1] [1] > 1/mean(x) # Btstrap estimate f lambda [1] > # d it again > x=sample(bs, n, replace=t); x [1] [1] > 1/mean(x) # Btstrap estimate f lambda [1] > # repeat the prcedure B times, save the B sample means in xbar and sample sd in ss[] > B=200; xbar = rep(0, B); ss = rep(0, B) > fr(i in 1:B) { x=sample(bs, n, replace=t); xbar[i] = mean(x) } > # the B btstrap estimate f lambda are > lambda.bt = 1/xbar > summary(lambda.bt) Min. 1st Qu. Median Mean 3rd Qu. Max > # btstrap estimate f standard errr > sd(lambda.bt) [1]
2 > # distributin f the btstrap estimates > stem(lambda.bt) The decimal pint is 2 digit(s) t the left f the > hist(lambda.bt) > lambda.bt.sub = lambda.bt-mean(lambda.bt) > hist(lambda.bt.sub) Histgram f lambda.bt Histgram f lambda.bt.sub lambda.bt > # 5 and 95 percentiles f lambda.bt-mean(lambda.bt) > quantile(lambda.bt.sub, c(.05,.95) ) 5% 95% > # 90% btstrap CI > c(lambda-quantile(lambda.bt.sub,.95), lambdaquantile(lambda.bt.sub,.05) ) 95% 5% lambda.bt.sub
3 Btstrap Estimate f the Standard Errr (CD Only) There are situatins in which the standard errr f the pint estimatr is unknwn. Usually, these are cases where the frm f ˆ is cmplicated, and the standard expectatin and variance peratrs are difficult t apply. A cmputer-intensive technique called the btstrap that was develped in recent years can be used fr this prblem. Suppse that we are sampling frm a ppulatin that can be mdeled by the prbability distributin f 1x; 2. The randm sample results in data values x 1, x 2, p, x n and we btain ˆ as the pint estimate f. We wuld nw use a cmputer t btain btstrap samples frm the distributin f 1x; ˆ 2, and fr each f these samples we calculate the btstrap estimate ˆ f. This results in Btstrap Sample Observatins Btstrap Estimate 1 2 B x 1, x 2, p, x n x 1, x 2, p, x n x 1, x 2, p, x n ˆ 1 ˆ 2 ˆ B Usually B 100 r 200 f these btstrap samples are taken. Let 11 B2 g B i 1 ˆ i be the sample mean f the btstrap estimates. The btstrap estimate f the standard errr f ˆ is just the sample standard deviatin f the ˆi, r B a 1 ˆ i 2 2 i 1 s ˆ R B 1 (S7-1) In the btstrap literature, B 1 in Equatin S7-1 is ften replaced by B. Hwever, fr the large values usually emplyed fr B, there is little difference in the estimate prduced fr. s ˆ EXAMPLE S7-1 The time t failure f an electrnic mdule used in an autmbile engine cntrller is tested at an elevated temperature in rder t accelerate the failure mechanism. The time t failure is expnentially distributed with unknwn parameter. Eight units are selected at randm and tested, with the resulting failure times (in hurs): x , x , x , x , x , x , x , and x Nw the mean f an expnential distributin is 1, s E(X ) 1, and the expected value f the sample average is E1X2 1. Therefre, a reasnable way t estimate is with ˆ 1 X. Fr ur sample, x 21.65, s ur estimate f is ˆ T find the btstrap standard errr we wuld nw btain B 200 (say) samples f n 8 bservatins each frm an expnential distributin with parameter The fllwing table shws sme f these results: Btstrap Sample Observatins Btstrap Estimate , 28.85, 14.14, 59.12, 3.11, 32.19, 5.26, , 2.10, 40.17, 32.43, 6.94, 30.66, 18.99, 5.61 ˆ ˆ , 39.26, 19.59, 43.53, 9.55, 7.07, 6.03, 8.94 ˆ
4 7-3 The sample average f the ˆ i (the btstrap estimates) is , and the standard deviatin f these btstrap estimates is Therefre, the btstrap standard errr f ˆ is In this case, estimating the parameter in an expnential distributin, the variance f the estimatr we used, ˆ, is knwn. When n is large, V 1 ˆ 2 2 n. Therefre the estimated standard errr f ˆ is 2 ˆ 2 n Ntice that this result agrees reasnably clsely with the btstrap standard errr. Smetimes we want t use the btstrap in situatins in which the frm f the prbability distributin is unknwn. In these cases, we take the n bservatins in the sample as the ppulatin and select B randm samples each f size n, with replacement, frm this ppulatin. Then Equatin S7-1 can be applied as described abve. The bk by Efrn and Tibshirani (1993) is an excellent intrductin t the btstrap Bayesian Estimatin f Parameters (CD Only) This bk uses methds f statistical inference based n the infrmatin in the sample data. In effect, these methds interpret prbabilities as relative frequencies. Smetimes we call prbabilities that are interpreted in this manner bjective prbabilities. There is anther apprach t statistical inference, called the Bayesian apprach, that cmbines sample infrmatin with ther infrmatin that may be available prir t cllecting the sample. In this sectin we briefly illustrate hw this apprach may be used in parameter estimatin. Suppse that the randm variable X has a prbability distributin that is a functin f ne parameter. We will write this prbability distributin as f 1x 0 2. This ntatin implies that the exact frm f the distributin f X is cnditinal n the value assigned t. The classical apprach t estimatin wuld cnsist f taking a randm sample f size n frm this distributin and then substituting the sample values x i int the estimatr fr. This estimatr culd have been develped using the maximum likelihd apprach, fr example. Suppse that we have sme additinal infrmatin abut and that we can summarize that infrmatin in the frm f a prbability distributin fr, say, f( ). This prbability distributin is ften called the prir distributin fr, and suppse that the mean f the prir is 0 and the variance is 2 0. This is a very nvel cncept insfar as the rest f this bk is cncerned because we are nw viewing the parameter as a randm variable. The prbabilities assciated with the prir distributin are ften called subjective prbabilities, in that they usually reflect the analyst s degree f belief regarding the true value f. The Bayesian apprach t estimatin uses the prir distributin fr, f( ), and the jint prbability distributin f the sample, say f 1x 1, x 2, p, x n 0 2, t find a psterir distributin fr, say, f 1 0 x 1, x 2, p, x n 2. This psterir distributin cntains infrmatin bth frm the sample and the prir distributin fr. In a sense, it expresses ur degree f belief regarding the true value f after bserving the sample data. It is easy cnceptually t find the psterir distributin. The jint prbability distributin f the sample X 1, X 2, p, X n and the parameter (remember that is a randm variable) is f 1x 1, x 2, p, x n, 2 f 1x 1, x 2, p, x n 0 2 f 1 2 and the marginal distributin f X 1, X 2, p, X n is a f 1x 1, x 2, p, x n, 2, discrete f 1x 1, x 2, p, x n 2 µ f 1x 1, x 2, p, x n, 2 d, cntinuus
5 8-2.6 Btstrap Cnfidence Intervals (CD Only) In Sectin we shwed hw a technique called the btstrap culd be used t estimate the standard errr ˆ, where ˆ is an estimate f a parameter. We can als use the btstrap t find a cnfidence interval n the parameter. T illustrate, cnsider the case where is the mean f a nrmal distributin with knwn. Nw the estimatr f is X. Als ntice that z 2 1n is the 100(1 /2) percentile f the distributin f X, and z 2 1n is the 100( 2) percentile f this distributin. Therefre, we can write the prbability statement assciated with the 100(1 )% cnfidence interval as r P percentile X percentile2 1 P1X percentile X percentile2 1 This last prbability statement implies that the lwer and upper 100(1 )% cnfidence limits fr are L X percentile f X X z 2 1n U X percentile f X X z 2 1n We may generalize this t an arbitrary parameter. The 100(1 )% cnfidence limits fr are L ˆ percentile f ˆ U ˆ percentile f ˆ Unfrtunately, the percentiles f ˆ may nt be as easy t find as in the case f the nrmal distributin mean. Hwever, they culd be estimated frm btstrap samples. Suppse we find B btstrap samples and calculate ˆ,, p, and and then calculate ˆ 2, p, ˆ B 1 ˆ 2 ˆ B ˆ 1,. The required percentiles can be btained directly frm the differences. Fr example, if B 200 and a 95% cnfidence interval n is desired, the fifth smallest and fifth largest f the differences ˆ i are the estimates f the necessary percentiles. We will illustrate this prcedure using the situatin first described in Example 7-3, invlving the parameter f an expnential distributin. Fllwing that example, a randm sample f n 8 engine cntrller mdules were tested t failure, and the estimate f btained was ˆ , where ˆ 1 X is a maximum likelihd estimatr. We used 200 btstrap samples t btain an estimate f the standard errr fr ˆ. Figure S8-1(a) is a histgram f the 200 btstrap estimates ˆ i, i 1, 2, p, 200. Ntice that the histgram is nt symmetrical and is skewed t the right, indicating that the sampling distributin f ˆ als has this same shape. We subtracted the sample average f these btstrap estimates frm each ˆ i. The histgram f the differences ˆ i, i 1, 2, p, 200, is shwn in Figure S8-1(b). Suppse we wish t find a 90% cnfidence interval fr. Nw the fifth percentile f the btstrap samples ˆ i is and the ninetyfifth percentile is Therefre the lwer and upper 90% btstrap cnfidence limits are L ˆ 95 percentile f ˆ i U ˆ 5 percentile f ˆ i
6 _ λ i ^ λ _ i λ (a) Histgram f the btstrap estimate (b) Histgram f the differences ^ λ _ i λ Figure S8-1 Histgrams f the btstrap estimates f and the differences ˆ i ˆ used in finding the btstrap cnfidence interval. Therefre, ur 90% btstrap cnfidence interval fr is There is an exact cnfidence interval fr the parameter in an expnential distributin. Fr the engine cntrller failure data fllwing Example 7-3, the exact 90% cnfidence interval fr is Ntice that the tw cnfidence intervals are very similar. The length f the exact cnfidence interval is , while the length f the btstrap cnfidence interval is , which is nly slightly lnger. The percentile methd fr btstrap cnfidence intervals wrks well when the estimatr is unbiased and the standard errr f ˆ is apprximately cnstant (as a functin f ). An imprvement, knwn as the bias-crrected and accelerated methd, adjusts the percentiles in mre general cases. It culd be applied in this example (because ˆ is a biased estimatr), but at the cst f additinal cmplexity Develpment f the t-distributin (CD Only) We will give a frmal develpment f the t-distributin using the techniques presented in Sectin 5-8. It will be helpful t review that material befre reading this sectin. First cnsider the randm variable This quantity can be written as T X S 1n T X 1 n 2S 2 2 (S8-1) The cnfidence interval is 2 2,2n 12g x where 2 and 2 i ,2n 12g x i 2 2,2n 1 2,2n are the lwer and upper 2 percentage pints f the chi-square distributin (which was intrduced briefly in Chapter 4 and discussed further in Sectin 8-4), and the x i are the n sample bservatins.
, which yields. where z1. and z2
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