Chapter 9 Bootstrap Confidence Intervals William Q. Meeker and Luis A. Escobar Iowa State University and Louisiana State University 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. January 13, 2014 3h 41min 9-1
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. Obtain and interpret simulation-based pointwise parametric bootstrap confidence intervals. 9-2
Bootstrap Sampling and Bootstrap Confidence Intervals Instead of assuming Z µ = ( µ µ)/ŝe µ NOR(0,1), use Monte Carlo simulation to approximate the distribution of Z µ. Simulate B = 4000 values of Z µ = ( µ µ)/ŝe µ. Some bootstrap approximations: Z µ Z µ Z log( σ) Z log( σ ) Z logit[ F(t)] Z logit[ F (t)] when computing confidence intervals for µ, σ, and F. 9-3
A Simple Bootstrap Re-Sampling Method Population or Process Actual Sample Data From Population or Process (Used to Estimate Model Parameters) Resample with Replacement from DATA (Draw B Samples, each of Size n) n units F(t; θ) DATA DATA * 1 θ^ θ ^* 1 DATA* 2 θ^* 2.. DATA * B θ^* B 9-4
A Simple Parametric Bootstrap Sampling Method Population or Process Actual Sample Data From Population or Process (Used to Estimate Model Parameters) Simulated Censored Samples From (Draw B samples, each of size n) F(t; θ^ ) n units F(t; θ) DATA ^ F(t; θ) DATA* 1 θ ^* 1 DATA* 2 θ ^* 2. DATA * B θ ^* B 9-5
Scatterplot of 1,000 (Out of B =10,000) Bootstrap Estimates µ and σ for Shock Absorber 10.0 10.5 11.0 11.5 0.2 0.4 0.6 0.8 µ σ 9-6
Weibull Plot of F(t; µ, σ) from the Original Sample (dark line) and 50 (Out of B =10,000) F(t; µ, σ ) Computed from Bootstrap Samples for the Shock Absorber.7.5.3.2.1 Proportion Failing.05.03.02.01.005.003.001.0005 10000 15000 20000 25000 Kilometers 9-7
Bootstrap Confidence Interval for µ With complete data or Type II censoring, Z µ j = µ j µ ŝe µ j has a distribution that does not depend on any unknown parameters. Such a quantity is called a pivotal quantity. By the definition of quantiles, then ) Pr (z µ < Z µ (α/2) j z µ (1 α/2) = 1 α Simple algebra shows that [µ, µ] = [ µ z µ (1 α/2) ŝe µ, µ z µ (α/2) ŝe µ ] provides an exact 95% confidence interval for µ. With other kinds of censoring, the interval is, in general, only approximate. 9-8
Bootstrap Distributions of Weibull µ and Z µ Based on B=10,000 Bootstrap Samples for the Shock Absorber Bootstrap Estimates Bootstrap-t Untransformed 10.0 10.2 10.4 10.6 muhat* -3-2 -1 0 1 Z-muhat* Bootstrap-t Untransformed 1 Bootstrap cdf.5 0-3 -2-1 0 1 Z-muhat* 9-9
Bootstrap Confidence Interval for σ With complete data or Type II censoring, Z log( σ ) = log( σ ) log( σ) ŝe log( σ ) has a distribution that does not depend on any unknown parameters. Such a quantity is called a pivotal quantity. By the definition of quantiles, then ) Pr (z log( σ )(α/2) < Z log( σ j ) z log( σ )(1 α/2) = 1 α Simple algebra shows that [σ, σ] = [ σ/w, σ/ w] provides an exact 95% confidence interval for σ, where = ] w ] exp [z log( σ )(1 α/2) ŝe log( σ) and w = exp [z log( σ )(α/2) ŝe log( σ) With other kinds of censoring, the interval is, in general, only approximate. 9-10
Bootstrap Distributions of σ, Z σ, and Z log( σ ) Based on B=10,000 Bootstrap Samples Bootstrap Estimates Bootstrap-t Untransformed 0.2 0.3 0.4 0.5 sigmahat* Bootstrap-t log-transform -4-2 0 2 Z-sigmahat* Bootstrap-t log-transform 1 Bootstrap cdf.5 0-3 -2-1 0 1 2 Z-log(sigmahat*) -3-2 -1 0 1 2 Z-log(sigmahat*) 9-11
Bootstrap Confidence Interval for F(t e ) With complete data or Type II censoring [using F = F(t e )], Z logit( F ) = logit( F ) logit( F) ŝe logit( F ) has a distribution that does not depend on any unknown parameters. Such a quantity is called a pivotal quantity. By the definition of quantiles, then ) (z logit( F )(α/2) < Z logit( F j ) z logit( F )(1 α/2) Pr = 1 α Simple algebra shows that [F, F] = F F +(1 F) w, F F +(1 F) w where provides an exact 95% confidence interval for F, where = [ ] w exp z logit( F ŝe ) (1 α/2) logit( F) and w = exp [ z logit( F ) (α/2) ŝe logit( F)] With other kinds of censoring, the interval is, in general, only approximate. 9-12
Bootstrap Distributions of F(t e ), Z F(te ), and Z logit[ F(te ) ] for t e=10,000 km Based on B=10,000 Bootstrap Samples Bootstrap Estimates Bootstrap-t Untransformed 0.0 0.02 0.06 0.10 F(10000)hat* Bootstrap-t logit-transformed -20-15 -10-5 0 Z-F(10000)hat* Bootstrap-t logit-transformed 1 Bootstrap cdf.5 0-2 -1 0 1 2 3 Z-logit(F(10000)hat*) -2-1 0 1 2 Z-logit(F(10000)hat*) 9-13
Bootstrap Confidence Interval for t p With complete data or Type II censoring, Z log( t p) = log( t p ) log( t p )] ŝe log( t p) has a distribution that does not depend on any unknown parameters. Such a quantity is called a pivotal quantity. By the definition of quantiles, then ) Pr (z log( t p)(α/2) < Z log( t p)j z log( t p)(1 α/2) = 1 α Simple algebra shows that exp [t p, t p ] = [ t p /w, t p / w] provides an exact 95% confidence interval for t p, where = [ ] w ] z log( t ŝe p) (1 α/2) log( t and w = exp p ) [ z log( t p) (α/2) ŝe log( t p ) With other kinds of censoring, the interval is, in general, only approximate. 9-14
Bootstrap Distributions of t p, Z t p, and Z log[ t p] for t e =10,000 km Based on B=10,000 Bootstrap Samples Bootstrap Estimates Bootstrap-t Untransformed 10000 14000 18000 t0.1hat* Bootstrap-t log-transform -2-1 0 1 2 3 4 Z-t0.1hat* Bootstrap-t log-transform 1 Bootstrap cdf.5 0-2 0 2 4 Z-log(t0.1hat*) -2-1 0 1 2 3 4 Z-log(t0.1hat*) 9-15