THE BOOTSTRAP RESAMPLING METHODS AND THE MONTE-

CARLO PROCEDURE

There are two different methods for generating bootstrap sample data. One is the parametric bootstrap, where once the parameters of the underlying distribution are estimated, they are plugged into the assumed distribution and pseudo random numbers then drawn from this estimated distribution to produce the bootstrap sample. The non- parametric bootstrap does not assume a set underlying distribution, but resample from the sample data to produce new samples. The resampling procedure, of course, should be adopted to fit the underlying structure of the problem. For example, in a regression setting, resampling must be done on the residuals of a fitted model rather than from the original data.

In the following, we combine the bootstrap steps with the steps needed to carry out a Monte-Carlo comparison of the proposed methods of building confidence bounds and intervals. The steps for the parametric bootstrap and the non-parametric bootstrap are given separately. Note that the confidence bounds and intervals based on the asymptotic distribution can be computed at each Monte-Carlo simulation sample and does no require bootstrap resampling.

2.3.1 The Proposed Parametric Bootstrap Method and the Monte-Carlo Procedure for Studying its Performance.The Monte-Carlo procedure employed to study the performance of the parametric bootstrap method is described below. The steps

for the parametric bootstrap method for obtaining confidence bounds for α λ, , and β and lower bounds for the mean life are embedded in this procedure and are given in italics. Note that distributional parameters are varied in the study as follows:

(α =1.5,nad2,λ=1,β =1.5,and2) _{with𝜇𝜇= 𝜆𝜆Γ �1 +} 1

𝛼𝛼�. The censoring time was set at 𝜏𝜏=1, and1.5. Note that without loss of generality, scale parameter λcan be set at 1.

(1) For fixed values of n and 𝜋𝜋, generate a random sample 𝑥𝑥_𝑖𝑖,𝑖𝑖= 1,2, … ,𝑛𝑛𝜋𝜋 from the Weibull (𝛼𝛼,𝜆𝜆) distribution. This would be considered data from the normal use sample. Similarly, generate the data set𝑦𝑦_𝑗𝑗,𝑗𝑗 = 1,2, … ,𝑛𝑛𝜋𝜋, representing the sample under the high stress condition, from the Weibull (𝛼𝛼,𝛽𝛽𝜆𝜆) distribution. (2) Use the ML method to estimate the parameters with the same censoring time τ

used for both samples. In this study, the nonlinear equations of the maximum likelihood estimates were solved iteratively using the Newton Raphson method.

(3) Employ the resulting estimates of the parameters and acceleration factor to construct asymptotic confidence limits with confidence level at 𝛾𝛾 = 0.95. Also, plug-in the MLEs into the Fisher Information matrix to obtain the asymptotic variance and covariance matrix of the estimators and then use them in the delta method to compute the lower bound for mean life.

(4) Replace the unknown parameters,𝛼𝛼,𝜆𝜆, in the Weibull distribution for the normal use case with their MLEs, 𝛼𝛼�,𝜆𝜆̂, and utilize the estimated distribution to generate a bootstrap sample 𝑥𝑥_𝑖𝑖∗ ,𝑖𝑖 = 1,2, … ,𝑛𝑛𝜋𝜋 of size 𝑛𝑛𝜋𝜋.Censor the data based on the censoring time τ.

(5) Similarly replace the unknown parameters,𝛼𝛼,𝜆𝜆,𝛽𝛽 in the Weibull distribution for the high stress case with their MLEs, 𝛼𝛼�,𝜆𝜆̂,𝛽𝛽̂ and utilize the estimated distribution to generate a bootstrap sample 𝑦𝑦_𝑗𝑗∗ ,𝑗𝑗= 1,2, … ,𝑛𝑛𝜋𝜋 of size 𝑛𝑛𝜋𝜋. Censor the data based on the censoring time τ.

(6) Re-estimate the Weibull parameters of the normal use distribution were using the combined bootstrap samples. Denote the bootstrap sample-based MLEs of

𝛼𝛼,𝜆𝜆,𝛽𝛽𝑎𝑎𝑛𝑛𝑎𝑎𝜇𝜇 obtained at bootstrap step k by 𝛼𝛼�∗(𝑘𝑘),𝜆𝜆̂∗(𝑘𝑘),𝛽𝛽̂∗(𝑘𝑘)𝑎𝑎𝑛𝑛𝑎𝑎𝜇𝜇̂∗(𝑘𝑘) respectively.

(7) Repeat Steps (4) to (6) 1,000 times. Construct the empirical distributions of the bootstrap estimates 𝛼𝛼�∗(𝑘𝑘),𝜆𝜆̂∗(𝑘𝑘),𝛽𝛽̂∗(𝑘𝑘)𝑎𝑎𝑛𝑛𝑎𝑎𝜇𝜇̂∗(𝑘𝑘), k=1, 2, …, 1,000

(8) Use the empirical distributionsobtained from bootstrap estimates to construct, confidence interval for 𝛼𝛼,𝜆𝜆,𝛽𝛽using quantiles at �1−𝛾𝛾

2 �100% and 1− � 1−𝛾𝛾

2 �100% of the

respective empirical distribution as the lower and upper bounds respectively. Use the (1− 𝛾𝛾)100% quantile of the empirical distribution of 𝜇𝜇̂∗(𝑘𝑘) , k=1, 2, …, 1,000, as the lower bound for𝜇𝜇.

(9) Repeat steps (1) through (8) 1,000 times and compute the average number of times each parameter fell within the bound(s). This would yield an estimate of the

expected coverage for each interval. For each parameter except𝜇𝜇, the widths of the two sided interval computed in Steps (3) and (8) are averaged to obtained an estimate of the expected, width.

2.3.2 The Proposed Nonparametric Bootstrap Method and the Monte-

Carlo Procedure for Studying its Performance.The Monte-Carlo procedure employed to study the performance of the nonparametric bootstrap method is described below. The steps for the parametric bootstrap method for obtaining confidence bounds for

𝜶𝜶,𝝀𝝀,𝒂𝒂𝒂𝒂𝒂𝒂𝜷𝜷 and lower binds for the mean life are imbedded in this procedure and given in italics.

(1) For fixed values of n and 𝜋𝜋, generate a random sample 𝑥𝑥_𝑖𝑖,𝑖𝑖 = 1,2, … ,𝑛𝑛𝜋𝜋 from the Weibull (𝛼𝛼,𝜆𝜆) distribution. This would be considered data from the normal use sample. Similarly, generate the data set𝑦𝑦_𝑗𝑗,𝑗𝑗= 1,2, … ,𝑛𝑛𝜋𝜋, representing the sample under the high stress condition, from the Weibull (𝛼𝛼,𝛽𝛽𝜆𝜆) distribution.

(2) Obtain a bootstrap resample from each of the two samples generated in Step (1) above, with each bootstrap sample of size πn (or πn) obtained by sampling with replacement from the respective sample obtained in (1).

(3) New “bootstrap estimates” 𝛼𝛼�∗,𝜆𝜆̂∗,𝑎𝑎𝑛𝑛𝑎𝑎𝛽𝛽̂∗ are computed from the combined bootstrap sample using the ML method as described in Step (2) given in Section 2.3.1. Also estimate the mean life µ under normal conditions, accounting for the censoring.

(4) Repeat the process given in Steps (2) and (3) 1,000 times and obtain the empirical distributions of 𝛼𝛼�∗,𝜆𝜆̂∗, 𝑎𝑎𝑛𝑛𝑎𝑎�𝛽𝛽∗..

(5) Using the empirical distributions of the 𝛼𝛼�∗,𝜆𝜆̂∗, 𝑎𝑎𝑛𝑛𝑎𝑎 𝛽𝛽̂∗ obtained from bootstrap estimates, construct confidence interval for 𝛼𝛼,𝜆𝜆,𝑎𝑎𝑛𝑛𝑎𝑎𝛽𝛽using respective quantiles at �1−𝛾𝛾

2 �100% and 1− � 1−𝛾𝛾

2 �100%.

(6) Using the empirical distributions of the mean 𝜇𝜇̂∗ obtained from bootstrap estimates, construct the lower confidence bound for 𝜇𝜇 is using quantile at

(1− 𝛾𝛾)100% .

(7) Coverage probabilities were computed based on 1,000 simulation runs by repeating Steps (1) – (6).

Population Or Process

Actual Sample Data From Population Or Process Used to estimate model

parameters

Simulated censored samples from F�t,�

Draw 1000 samples, each of size n

Figure 2.1.Illustrates the parametric bootstrap resampling method

𝐹𝐹(𝑡𝑡,𝜃𝜃�) 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷1∗ 𝜃𝜃�1∗ ……… 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷2∗ 𝜃𝜃�2∗ ……… 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐵𝐵∗ 𝜃𝜃�𝐵𝐵∗ ……… 𝐹𝐹(𝑡𝑡,𝜃𝜃�) DATA …………

inference.

Population Or Process

Actual Sample Data From Population Or

Process

Used to estimate model parameters

Simulated censored samples from F�t,�

Draw 1000 samples, each of size n

Figure 2.2. Illustrates the nonparametric bootstrap resampling for parametric inference. 𝐹𝐹𝐹𝐹(𝑡𝑡,𝜃𝜃�) 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷1∗ 𝜃𝜃��1∗ ……… 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷2∗ 𝜃𝜃��2∗ ……… 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷∗𝐵𝐵 𝜃𝜃��𝐵𝐵∗ ……… 𝜃𝜃� DATA …………