4.1 Non-parametric computational estimation
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1 Chapter 4 Resamplig Methods 4.1 No-parametric computatioal estimatio Let x 1,...,x be a realizatio of the i.i.d. r.vs X 1,...,X with a c.d.f. F. We are iterested i the precisio of estimatio of a populatio parameter θ F. Oe possibility is to estimate θ F by θ ˆF, where ˆF is the empirical distributio futio. We will deote a estimator of a parameter θ by ˆθ. Examples where fx) = dfx) dx. The θ F = EX) = θ ˆF = 1 x i, xfx)dx, which is the sample mea. Here we assig equal probability, 1, to each realizatio of X. The θ F = varx) = θ ˆF = 1 x EX)) 2 fx)dx. x i x) 2, which is the variace of the sample. 33
2 34 CHAPTER 4. RESAMPLING METHODS 3. θ F = Fc) = PX c) The θ ˆF = 1 #{i : x i c} Oe might ask: How good is ˆθ = θ ˆF as a estimator of θ F? Three commo measures of goodess are: It is easy to see that Bias θ ˆθ) = E F ˆθ) θ 4.1) se θ ˆθ) = varˆθ) 4.2) [ ] MSE θ ˆθ) = E F ˆθ θ) 2 4.3) MSE θ ˆθ) = varˆθ) + Bias θ ˆθ)) 2 4.4) Namely: ] E [ˆθ θ) 2 = Eˆθ 2 2ˆθθ + θ 2 ) = [ ) ] 2 E ˆθ 2 2ˆθEˆθ) Bias θ ˆθ)) + Eˆθ) Bias θ ˆθ) = [ 2 E ˆθ 2 2ˆθEˆθ) 2ˆθBias θ ˆθ) + Eˆθ)) 2Eˆθ)Biasθ ˆθ) + [ ) ] 2 2 E ˆθ 2 2ˆθEˆθ) + Eˆθ) + Bias θ ˆθ)) = varˆθ) + ) 2 Bias θ ˆθ) Also ote that MSE θ ˆθ) = varˆθ) + [ ) ] 2 se θ ˆθ) Biasθ ˆθ) 2 se θ ˆθ) 2 Bias θ ˆθ)) = seθ ˆθ) 1 + ) ] 2 Bias θ ˆθ) = ) 2 Biasθ ˆθ). se θ ˆθ) =
3 4.2. BOOTSTRAP ESTIMATES OF BIAS, STANDARD ERROR AND MSE35 Problem How to calculate Bias θ ˆθ), se θ ˆθ) ad MSE θ ˆθ)? If we kew the distributio F the we could calculate expected value ad variace of the estimator ˆθ directly from defiitios. It may be difficult if fx) = df is complicated. The a practical alterative is simulatio: dx Geerate a large umber of radom samples from the populatio with the c.d.f. F ad calculate a value of ˆθ for each sample. The mea ad variace of the set of geerated values of ˆθ will give a good approximatio to E F ˆθ) ad var F ˆθ). What if F is ukow? The the simulatio from F is impossible. I such situatios a further approximatio is to replace F by ˆF. Let θ be the value of θ calculated at the radom sample from ˆF. The idea is that ad Bias θ ˆθ) Biasˆθθ ) var F ˆθ) var ˆFθ ). The heuristic reasoig is that ˆF is close to F ad so the relatioship of θ ˆF to θ F should be close to the relatioship of θ F to θ ˆF, as show i the diagram below. True ukow Empirical Resampled F data ˆF resamplig F θ F θ ˆF θ F 4.2 Bootstrap estimates of bias, stadard error ad MSE Assume we do ot kow F. The bootstrap estimates of Bias θ ˆθ), se θ ˆθ) ad MSE θ ˆθ) are obtaied by substitutig ˆF for F i 4.1), 4.2) ad 4.3), respectively. ˆF is the distributio which assigs probability 1 to each observatio x i. So, a radom sample from ˆF is just a radom sample from the set {x 1,...,x } with replacemet. The procedure to calculate the estimates is the followig:
4 36 CHAPTER 4. RESAMPLING METHODS costruct N samples of size from {x 1,...,x } with replacemet; deote the bootstrap samples by {x 1,...,x } i, i = 1,...,N; deote by θ i the value of the estimator calculated for the i-th bootstrap sample; calculate sample mea ad variace of bootstrap estimates θ i, i = 1,...,, that is θ = 1 N N θ i, 1 N 1 N θ i θ ) 2 ; Bias θ ˆθ), var F ˆθ), se F ˆθ) ad MSE θ ˆθ) are approximated, respectively, by Biasˆθθ ), var ˆFθ ), seˆθθ ) ad MSEˆθθ ) which are further approximated by Biasˆθθ ) = θ ˆθ, var ˆFθ ) = 1 N N 1 θ i θ ) 2, ŝe ˆFθ 1 ) = N N 1 θ i θ ) 2, MSEˆθθ ) = 1 N 1 N θ i θ ) 2 + θ ˆθ) 2. The followig diagram represets the bootstrap resamplig method: Empirical Bootstrap Bootstrap Bootstrap distributio samples of size replicatios of ˆθ estimates ˆF {x 1,...,x } 1 θ1 {x 1,...,x } 2 θ2 {x 1,...,x }... {x 1,...,x } N θn bias: θ ˆθ variace: 1 N 1 N θ i θ ) 2 Example: sample mea ad sample media Let x 1,...,x be a realizatio of the i.i.d. r.vs X 1,...,X with a c.d.f. F. Cosider θ F = E F X i ) ad let θ ˆF = X. We kow that the sample mea X is a ubiased estimator of EX i ). What is the bootstrap bias of the mea?
5 4.2. BOOTSTRAP ESTIMATES OF BIAS, STANDARD ERROR AND MSE37 Biasˆθθ ) = E ˆFθ ) ˆθ = E ˆF X) X = 0 Is the estimate of the bootstrap bias of the sample mea, Biasˆθ X), equal to zero as well? Let { } be a sample from a populatio with a c.d.f. F. Here the sample mea is 2.87 ad the sample media is bootstrap samples replicates of sample mea media { } { } { } { } { } { } { } { } { } { } { } { } { } { } { } The bootstrap estimate of the sample mea is the average of the replicates θ i, that is θ = ) = Hece, the estimate of bootstrap bias of the sample mea is Biasˆθθ ) = θ ˆθ = = So, the aswer to the questio if the estimate of the bootstrap bias of the sample mea, Biasˆθ X), is equal to zero, is GENERALLY NOT.
6 38 CHAPTER 4. RESAMPLING METHODS Similar calculatios for the above data ad the bootstrap samples show that the estimate of the bootstrap bias for the sample media is Biasˆθθ ) = θ ˆθ = = However, we do ot kow what is the true bias of the sample media, so we do ot kow how good this estimate is. The followig questio arises: How big should the bootstrap sample be to get a high probability that the estimate of the bootstrap bias of a estimator is close to the true value of the bias kow or ukow)? The Cetral Limit Theorem says that the distributio of a average is approximately ormal if the sample size is large ad the variace is fiite. Applyig it to the bootstrap replicates θ i we get P θ E ˆFθ ) < 2ŝe ˆFθ ) N ) = ) This meas P Biasˆθθ ) Biasˆθθ ) < 2ŝe ˆFθ ) N ) = Hece, we eed N such that 2ŝe ˆF θ ) N is very small, for example Example: The patch data Efro, Tibshirai, 1993, page 127) Eight subjects wore medical patches desiged to icrease the blood levels of a certai atural hormoe. Each subject had his blood levels of the hormoe measured after wearig three differet patches: a placebo patch, which had o medicie i it, a old patch which was from a lot maufactured at a old plat, ad a ew patch, which was from a lot maufactured at a ewly opeed plat. The purpose of the experimet was to show that the ew plat was producig patches equivalet to those from the old plat. The observatios are i the table below.
7 4.2. BOOTSTRAP ESTIMATES OF BIAS, STANDARD ERROR AND MSE39 subject placebo old patch ew patch oldpatch - placebo ewpatch - oldpatch p i old i ew i z i = old i p i y i = ew i old i mea: The Food ad Drug Admiistratio FDA) criterio for the bioequivalece is that the expected value of the ew patches match that of the old patches i the sese that Eew) Eold) Eold) Eplacebo) 0.2 This meas that the ew patch should match the old oe withi 20% of the amout of hormoe the old drug adds to placebo blood levels. Deote the parameter of iterest by θ, i.e., θ = Eew) Eold) Eold) Eplacebo) ad let us assume that the pairs x i = z i,y i ) are realizatio of i.i.d. bivariate r.vs X i = Z i,y i ) with ukow c.d.f. F. The the parameter θ is θ = E FY ) E F Z). The atural estimator of θ is ˆθ = Ȳ Z ad its value for the give observatios is whose absolute value is less the 0.2. ˆθ = = GeStat program doig the bootstrap calculatios will be show durig lectures.
8 40 CHAPTER 4. RESAMPLING METHODS 4.3 The Jackkife The Jackkife is oe of the oldest resamplig methods. Here we get replicatios of a estimator ˆθ by costructig ew samples simply omittig oe observatio at a time. So, we get samples of size 1. Here is the procedure: Empirical Jackkife Jackkife Jackkife distributio samples of size 1 replicatios of ˆθ estimates of ˆF {x 2, x 3,...,x } θ1) {x 1, x 3...,x } θ2) {x 1,...,x }... {x 1,...,x 1 } θ) bias: 1) θ ˆθ) where θ = 1 θ i) variace: θ i) θ ) 2 Here, θ i) is calculated i the same way as ˆθ except that the i th observatio is omitted. The Jackkife estimator of bias ad variace of ˆθ are defied to be: Biasˆθθ Jack) = 1) θ ˆθ) 4.6) varˆθθ Jack) = 1 1 θi) θ ) 2 4.7) For simple types of θ the Jackkife estimator ca be calculated explicitly. Example: Jackkife estimator of the mea ad of its variace Mea Let θ be the expected value of a r.v. X with a c.d.f. F ad let the estimator of θ be the average of a radom sample of size, i.e., ˆθ = X. The Jackkife replicatios of ˆθ are calculated as: θi) = 1 X j. 1 j=1,j i Is the Jackkife estimate of bias of the mea equal to zero? θ = 1 θi) = 1 1 X j = j=1,j i j=1,j i X j =
9 4.3. THE JACKKNIFE 41 The Jackkife estimate of bias is ) X i = X = ˆθ. Biasˆθθ Jack) = 1) θ ˆθ) = 1) X X) = 0. Variace of the mea Here we calculate the Jackkife variace of the mea. varˆθθ Jack) = 1 θi) θ ) 2 = ) X X i ) X = 1 1 1) X i X) 2 = 1 S2. j=1,j i X j X) 2 = X X i ) 2 1) 2 = So, the Jackkife estimator of the variace of the mea is the familiar oe. Example: Opiio survey A opiio survey asked a radom sample of 200 people a yes/o questio of whom 75 aswered yes. The estimate of the populatio proportio p of those who would aswer yes is estimated as ˆp = 75 = 3. A social sciece researcher is iterested i a parameter θ = p1 p) = pq. A atural estimate of the parameter is Is the estimator ˆθ = ˆpˆq biased? ˆθ = ˆpˆq = = The calculatios will be show durig lectures.
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