1. Fractional-Random-Weight Bootstrap
Applications of the
Fractional-Random-Weight Bootstrap
Chris Gotwalt (JMP)
Li Xu (Virginia Tech)
Yili Hong (Virginia Tech)
William Q. Meeker (Iowa State University)
Copyright © 2013, SAS Institute Inc. All rights reserved.

2. Fractional Weight Bootstrap
Overview
Bootstrap background
Weighted estimation and random-weight bootstrap
Integer weights
Fractional weights
Fractional-random-weight methods
Uniform Dirichlet weights
iid weights
Motivating example: Bearing cage field failure data
Other examples
Rocket motor failures
Power transformer failure prediction
Generalized gamma fitting to limited data
Other potential applications
Concluding remarks

3. Fractional Weight Bootstrap
Bootstrap Background
Popular statistical tool:
Leads to improved inferences (e.g., tests and confidence intervals)
Requires mild assumptions
Easy to implement
Applies even to procedures where there is less classical theory to offer guidance

4. Fractional Weight Bootstrap
Common Bootstrap—Basic Ideas
The most common bootstrapping approach is done via a Monte Carlo simulation:
A new dataset is created by
Resampling the rows of the original data with replacement or
Parametrically simulating
A statistical procedure (e.g., model fitting, a hypothesis test, or confidence interval) is applied to the simulated data set and the some results (estimates, p-values, etc.) are stored
This repeated a number of times (e.g., 2,500 times) and then the saved results are processed (…somehow, depends on the application) to make inferences (e.g., construct confidence intervals)
Copyright © 2013, SAS Institute Inc. All rights reserved.

5. Fractional Weight Bootstrap
Weighted Data and Weighted Estimation
It is often convenient to put “weights” (or frequencies or counts) on observations
Binary data (replace with number of 0s and 1s)
Censored life test data (e.g., failure times plus the number of units that survived a 1,000 hour test)
Data compression where similar observations are put into bins (e.g., a histogram)
Observations with non-constant variance (weights inversely proportional to variance)
Many estimation methods allow specification of weights or frequencies
Sample moments (means and variances)
Weighted least squares
Likelihood
Sample mean and standard deviation with weights
Weighted likelihood

6. n=15 Integer Weight Bootstrap Samples
Fractional Weight Bootstrap
n=15 Integer Weight Bootstrap Samples

7. n=15 Integer and Fractional Random-Weight Bootstrap Samples
Fractional Weight Bootstrap
n=15 Integer and Fractional Random-Weight Bootstrap Samples

8. Choosing Bootstrap Random Weights
Fractional Weight Bootstrap
Choosing Bootstrap Random Weights
Integer weights (resampling)
Sampling observations with replacement
Equivalent to drawing a sample weights from a uniform multinomial distribution
The integer weights have a mean and variance equal to 1
Fractional (non-integer) weights
Sample from a uniform Dirichlet distribution (weights will sum to n). Suggested by Rubin (1981) as the “Bayesian bootstrap”
Sample from some other distribution that has a mean and a variance of 1 (expected value of the sum of the weights will be equal to n)
There is large-sample theory to support both methods

9. Fractional-Random Weight Bootstrap Background
Fractional Weight Bootstrap
The fractional-random-weight bootstrap is also known as the
Random-weight bootstrap
Weighted likelihood bootstrap
Weighted bootstrap
Perturbation bootstrap
Bayesian bootstrap
Operationally, the fractional-random-weight bootstrap is used in the same way as the resampling bootstrap
The fractional-random-weight bootstrap has important advantages in many applications
Like resampling, the method is nonparametric
All original observations remain in the bootstrap samples
Estimation difficulties do not arise

10. Fractional Weight Bootstrap
BOOTSTRAPPING IN JMP PRO
In JMP PRO, it is easy to bootstrap almost any analysis:
Fractional Weights are not the default in JMP PRO, but easy to select
Copyright © 2013, SAS Institute Inc. All rights reserved.

11. Bearing Cage Field Failure Data Event Plot
Fractional Weight Bootstrap
Bearing Cage Field Failure Data Event Plot
6 Failures
1697 Right-censored observations
Will the bootstrap work?

12. Weibull Analysis Using Life Distribution
Fractional Weight Bootstrap
Weibull Analysis Using Life Distribution

13. Will the Bootstrap Work with Heavy Censoring?
Fractional Weight Bootstrap
Will the Bootstrap Work with Heavy Censoring?
If the expected number failing is too small there could be bootstrap samples with only 0 or 1 failure, causing JMP’s ML algorithm to fail
For the Bearing Cage example, the probability of obtaining a bootstrap sample with 0 or 1 failure using the integer weight method is 0.017
Using the fractional weight method, the probability is 0

14. Fractional Weight Bootstrap
Bootstrap Confidence Limits
Bootstrap Confidence Limits
Bootstrap Confidence Limits
Fractional Weight Bootstrap
Bootstrap Results for Estimating the Bearing Cage Weibull Distribution Shape Parameter
Integer Weights (resampling)
Fractional Weights

15. Rocket Motor Field Data Event Plot
Fractional Weight Bootstrap
Rocket Motor Field Data Event Plot
3 Left-censored observations
1934 Right-censored observations
The usual resampling (integer weight) bootstrap will not work

16. Rocket Motor Weibull Analysis
Fractional Weight Bootstrap
Rocket Motor Weibull Analysis

17. Rocket Motor Weibull Analysis Resampling and Bootstrap Results
Bootstrap Confidence Limits
Bootstrap Confidence Limits
Fractional Weight Bootstrap
Rocket Motor Weibull Analysis Resampling and Bootstrap Results
Fractional Weights
Integer Weights (resampling)
Weibull β
Weibull β

18. Power Transformer Data from Hong, Meeker, and McCalley (2009)
Fractional Weight Bootstrap
Power Transformer Data from Hong, Meeker, and McCalley (2009)
710 Power transformers with 62 failed units
Some units more than 60 years old
Units still in service at the “data freeze” date in March 2008 are right censored
Data records begin in 1980
Units installed before 1980 are observations from a truncated distribution
Several explanatory variables
Purpose: predict future failures

19. Event Plot of a Subset of the Power Transformer Field Failure Data
Fractional Weight Bootstrap
Event Plot of a Subset of the Power Transformer Field Failure Data

20. Power Transformer Model and Likelihood
Fractional Weight Bootstrap
Power Transformer Model and Likelihood
Fit Weibull and lognormal distributions
Stratification based on whether manufactured before or after 1987
The likelihood functions is
where tij is the failure tij is the censoring time is the lower truncation time and are censoring and truncation indicators for transformer i in stratum j
How to bootstrap/simulate to get prediction intervals?

21. Fractional Weight Bootstrap
Power Transformer Fleet Predictions Based on the Fractional-Random-Weight Bootstrap

22. Fractional Weight Bootstrap
Power Transformer Individual Predictions based on the Fractional-Weight Bootstrap

23. Fractional Weight Bootstrap
Concluding Remarks
With vastly improved computing capabilities and developed theory, bootstrapping provides an important useful tool for obtaining
Trustworthy confidence intervals
Trustworthy prediction intervals
Better regression models
Should also be useful for Generalized Pivotal Quantity (aka Generalized Fiducial) statistical methods.
The Fractional-random-weight bootstrap tremendously expands the potential areas of application of the bootstrap

24. Fractional Weight Bootstrap
Some References
Rubin, D. B. (1981). The Bayesian bootstrap. Annals of Statistics 9, 130–134.
Newton, M. A. and A. E. Raftery (1994). Approximate Bayesian inference with the weighted likelihood bootstrap. Journal of the Royal Statistical Society, Series B 56, 3–48.
Jin, Z., Z. Ying, and L. Wei (2001). A simple resampling method by perturbing the minimand. Biometrika 88, 381–390.
Hong, Y., W. Q. Meeker, and J. D. McCalley (2009). Prediction of remaining life of power transformers based on left truncated and right censored lifetime data. Annals of Applied Statistics 3, 857–879.
Meeker, W. Q., Hahn, G.J., and L. A. Escobar (2017) Statistical Intervals: A Guide for Practitioners and Researchers, Wiley.