Chapter 9. Bootstrap Confidence Intervals. William Q. Meeker and Luis A. Escobar Iowa State University and Louisiana State University

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1 Chapter 9 Bootstrap Confidence Intervals William Q. Meeker and Luis A. Escobar Iowa State University and Louisiana State University Copyright W. Q. Meeker and L. A. Escobar. Based on the authors text Statistical Methods for Reliability Data, John Wiley & Sons Inc January 13, h 41min 9-1

2 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

3 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

4 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

5 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

6 Scatterplot of 1,000 (Out of B =10,000) Bootstrap Estimates µ and σ for Shock Absorber µ σ 9-6

7 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 Proportion Failing Kilometers 9-7

8 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

9 Bootstrap Distributions of Weibull µ and Z µ Based on B=10,000 Bootstrap Samples for the Shock Absorber Bootstrap Estimates Bootstrap-t Untransformed muhat* Z-muhat* Bootstrap-t Untransformed 1 Bootstrap cdf Z-muhat* 9-9

10 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

11 Bootstrap Distributions of σ, Z σ, and Z log( σ ) Based on B=10,000 Bootstrap Samples Bootstrap Estimates Bootstrap-t Untransformed sigmahat* Bootstrap-t log-transform Z-sigmahat* Bootstrap-t log-transform 1 Bootstrap cdf Z-log(sigmahat*) Z-log(sigmahat*) 9-11

12 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

13 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 F(10000)hat* Bootstrap-t logit-transformed Z-F(10000)hat* Bootstrap-t logit-transformed 1 Bootstrap cdf Z-logit(F(10000)hat*) Z-logit(F(10000)hat*) 9-13

14 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

15 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 t0.1hat* Bootstrap-t log-transform Z-t0.1hat* Bootstrap-t log-transform 1 Bootstrap cdf Z-log(t0.1hat*) Z-log(t0.1hat*) 9-15