Chapter 9 Bootstrap Confidence Intervals William Q. Meeker and Luis A. Escobar Iowa State University and Louisiana State University 9-1 Copyright 1998-2008 W. Q. Meeker and L. A. Escobar. Based on the authors’ text Statistical Methods for Reliability Data, John Wiley & Sons Inc. 1998. December 14, 2015 8h 9min Bootstrap Sampling and Bootstrap Confidence Intervals 9-3 b − µ)/sc • Instead of assuming Zµb = (µ eµb ∼ ˙ NOR(0, 1), use Monte Carlo simulation to approximate the distribution of Zµb. b ∗ − µ)/ b sc • Simulate B = 4000 values of Zµb∗ = (µ eµb∗ . • Some bootstrap approximations: ◮ Zµb ∼ ˙ Zµb∗ ◮ Zlog(σb) ∼ ˙ Zlog(σb∗) ◮ Zlogit[Fb(t)] ∼ ˙ Zlogit[Fb∗(t)] Simulated Censored Samples From DATA*1 (Draw B samples, each of size n) DATA ^* θ 1 ^) F(t; θ when computing confidence intervals for µ, σ, and F . n units ^) F(t; θ Actual Sample Data From Population or Process (Used to Estimate Model Parameters) A Simple Parametric Bootstrap Sampling Method Population or Process F(t; θ) DATA*2 ^* θ 2 . . . DATA*B ^* θ B 9-5 Bootstrap Confidence Intervals Chapter 9 Objectives • Explain basic ideas behind the use of computer simulation to obtain bootstrap confidence intervals. • Explain different methods for generating bootstrap samples. ^* θ 1 DATA*1 9-4 9-2 • Obtain and interpret simulation-based pointwise parametric bootstrap confidence intervals. n units ^ θ ^* θ B DATA*B . .. ^* θ 2 DATA*2 DATA Actual Sample Data From Resample with Replacement from DATA Population or Process (Draw B Samples, each of Size n) (Used to Estimate Model Parameters) A Simple Bootstrap Re-Sampling Method F(t; θ) Population or Process 0.8 0.6 0.4 µ • • 11.0 11.5 • Scatterplot of 1,000 (Out of B =10,000) Bootstrap b ∗ and σ b ∗ for Shock Absorber Estimates µ σ 0.2 10.5 • • • • • • • • • • •• • • • • • •• • •• ••• • • • • • • •• • • • • • • • • • •••• ••••••••••••••••••• • • •••• • •• • • • • • • • • • ••• ••• • • • ••••••••••••••••••••••••••••••••• •••••• ••• • ••• • •• • • ••• • ••••••••• ••••• •••••••••••••• ••••••• •• •••• • • • • • • ••• ••• • •••••••••••••••••••••••••••••••• ••••••••• •••• •• ••• •• • •• • •• •••••••••••••••••••••••••••••• ••••• ••••••••••• • •• • •• • •• •••• •••••••••••••••••••••••••••••••••••••••••••• ••• ••• •••••••••••• • • • •••••• •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• •• • • • •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• • •• •• • • •• • • • • •• ••••••••••••••••••• • •••• •••••••••••••••••••••••••••••• ••• • • • •• ••••••••••••••••••••••••••••••••••• ••• • • • • •••• • •• •• • • • ••••• ••••••••••••• •••• • •• •• • •• 10.0 9-6 10000 15000 Kilometers 20000 25000 9-7 b σ b ) from the Original Sample Weibull Plot of F (t; µ, b ∗, σ b ∗) (dark line) and 50 (Out of B =10,000) F (t; µ Computed from Bootstrap Samples for the Shock Absorber .7 .5 .3 .2 .1 .05 .03 .02 .01 .005 .003 .001 10.0 1 .5 0 muhat* 10.4 10.6 Bootstrap Estimates 10.2 -2 Z-muhat* -1 0 1 Bootstrap-t Untransformed -3 -2 Z-muhat* -1 0 1 Bootstrap-t Untransformed -3 9-9 b ∗ and Zµ Bootstrap Distributions of Weibull µ b∗ Based on B=10,000 Bootstrap Samples for the Shock Absorber .0005 Proportion Failing 0.4 sigmahat* 0.3 0.5 Bootstrap Estimates 0.2 -1 0 Z-log(sigmahat*) -2 1 2 Bootstrap-t log-transform -3 Bootstrap cdf 0 .5 1 -3 -2 Z-sigmahat* -1 0 0 Z-log(sigmahat*) -2 1 2 9 - 11 2 Bootstrap-t log-transform -4 Bootstrap-t Untransformed b ∗ , Zσ Bootstrap Distributions of σ b ∗ , and Zlog(σ b ∗) Based on B=10,000 Bootstrap Samples Bootstrap cdf Bootstrap Confidence Interval for µ j sceµb∗ b j∗ − µ b µ • With complete data or Type II censoring, j Zµb∗ = (1−α/2) sceµb] =1−α (α/2) b − zµ µ b∗ has a distribution that does not depend on any unknown parameters. Such a quantity is called a pivotal quantity. j sceµb, < Zµb∗ ≤ zµb∗ • By the definition of quantiles, then (α/2) Pr zµb∗ (1−α/2) b − zµ µ̃] = [µ b∗ • Simple algebra shows that e [µ, 9-8 provides an exact 95% confidence interval for µ. With other kinds of censoring, the interval is, in general, only approximate. Bootstrap Confidence Interval for σ b ∗) − log(σ b) log(σ scelog(σb∗) • With complete data or Type II censoring, Zlog(σb∗) = j b /w] e σ < Zlog(σb∗) ≤ zlog(σb∗) e b /w , σ̃] = [σ (1−α/2) =1−α has a distribution that does not depend on any unknown parameters. Such a quantity is called a pivotal quantity. (α/2) • By the definition of quantiles, then Pr zlog(σb∗) e [σ , • Simple algebra shows that (1−α/2) e = exp zlog(σ scelog(σb) and w b∗) (α/2) scelog(σb) e provides an exact 95% confidence interval for σ, where w = exp zlog(σb∗) 9 - 10 With other kinds of censoring, the interval is, in general, only approximate. Bootstrap Confidence Interval for F (te) logit(Fb ∗) − logit(Fb ) scelogit(Fb∗) • With complete data or Type II censoring [using F = F (te)], Zlogit(Fb∗) = j < Zlogit(Fb∗) ≤ zlogit(Fb∗) (α/2) (1−α/2) ! =1−α has a distribution that does not depend on any unknown parameters. Such a quantity is called a pivotal quantity. (α/2) • By the definition of quantiles, then Pr zlogit(Fb∗) • Simple algebra shows that ∗ Fb Fb , [F , F̃ ] = Fb + (1 − Fb ) × w Fb + (1 − Fb ) × w e e e where provides an exact 95% confidence interval for F , where w = e i i h h ∗ (1−α/2) 9 - 12 b b se e = exp zlogit(Fb ) se exp zlogit(Fb ) b) With other b) and w logit(F logit(F kinds of censoring, the interval is, in general, only approximate. 0.06 0.10 Bootstrap Estimates 0.02 F(10000)hat* 0 1 2 3 0 .5 1 -2 -15 -10 -5 Z-F(10000)hat* -1 0 1 2 Z-logit(F(10000)hat*) 0 9 - 13 Bootstrap-t logit-transformed -20 Bootstrap-t Untransformed Bootstrap Distributions of Fb (te)∗, ZFb(t )∗ , and e Zlogit[Fb(t )∗] for te=10,000 km Based on B=10,000 e Bootstrap Samples 0.0 -1 Bootstrap-t logit-transformed -2 Z-logit(F(10000)hat*) 14000 18000 Bootstrap Estimates 10000 t0.1hat* 2 Z-log(t0.1hat*) 0 4 Bootstrap-t log-transform -2 -2 -1 -1 0 1 Z-t0.1hat* 0 1 2 2 Z-log(t0.1hat*) 3 3 4 9 - 15 Bootstrap-t log-transform -2 Bootstrap-t Untransformed 1 .5 0 4 te=10,000 km Based on B=10,000 Bootstrap Samples Bootstrap Distributions of tbp∗, Zbt∗ , and Zlog[bt∗] for p p Bootstrap cdf Bootstrap cdf Bootstrap Confidence Interval for tp p scelog(bt∗) log(tbp∗) − log(tbp)] • With complete data or Type II censoring, p Zlog(bt∗) = e tbp/w] p (1−α/2) =1−α has a distribution that does not depend on any unknown parameters. Such a quantity is called a pivotal quantity. p j t˜p] = [tbp/w, e < Zlog(bt∗) ≤ zlog(bt∗) • By the definition of quantiles, then p (α/2) Pr zlog(bt∗) e [tp, • Simple algebra shows that p (1−α/2) p p (α/2) e = exp z scelog(bt ) and w log(b t∗ ) p scelog(bt ) e provides an exact 95% confidence interval for tp, where w = exp zlog(bt∗) 9 - 14 With other kinds of censoring, the interval is, in general, only approximate.