Predicting a Future Lifetime through Box-Cox Transformation

In predicting a future lifetime based on a sample of past lifetimes, the Box-Cox transformation method provides a simple and unified procedure that is shown in this article to meet or often outperform the corresponding frequentist solution in terms of coverage probability and average length of prediction intervals. Kullback-Leibler information and second-order asymptotic expansion are used to justify the Box-Cox procedure. Extensive Monte Carlo simulations are also performed to evaluate the small sample behavior of the procedure. Certain popular lifetime distributions, such as Weibull, inverse Gaussian and Birnbaum-Saunders are served as illustrative examples. One important advantage of the Box-Cox procedure lies in its easy extension to linear model predictions where the exact frequentist solutions are often not available.

[1]  D. Hinkley,et al.  The Analysis of Transformed Data , 1984 .

[2]  Nancy R. Mann,et al.  On Constructing Prediction Intervals for Samples from a Weibull or Extreme Value Distribution , 1980 .

[3]  David Ruppert,et al.  On prediction and the power transformation family , 1981 .

[4]  D. Cox,et al.  An Analysis of Transformations , 1964 .

[5]  James R. Rieck,et al.  A log-linear model for the Birnbaum-Saunders distribution , 1991 .

[6]  Lee J. Bain,et al.  On Prediction Limits for Samples From a Weibull or Extreme-Value Distribution , 1982 .

[7]  N. L. Johnson,et al.  Continuous Univariate Distributions. , 1995 .

[8]  Sean M. Collins,et al.  Prediction Techniques for Box-Cox Regression Models , 1991 .

[9]  N. Duan.  Sensitivity analysis for Box-Cox power transformation model: Contrast parameters , 1993 .

[10]  W. J. Padgett.  An approximate prediction linterval for the mean of future observations from the inverse gaussian distribution , 1982 .

[11]  A. Desmond,et al.  Shortest prediction intervals for the birnbaum-saunders distribution , 1995 .

[12]  R. Serfling.  Approximation Theorems of Mathematical Statistics , 1980 .

[13]  Jagdish K. Patel,et al.  Prediction intervals - a review , 1989 .

[14]  Russell C. H. Cheng,et al.  Embedded Models in Three-parameter Distributions and their Estimation , 1990 .

[15]  W. J. Padgett,et al.  Prediction Intervals for Future Observations from the Inverse Gaussian Distribution , 1986, IEEE Transactions on Reliability.

[16]  J. Leroy Folks,et al.  The Inverse Gaussian Distribution , 1989 .

[17]  A. Basu,et al.  The Inverse Gaussian Distribution , 1993 .

[18]  I. Guttman,et al.  Prediction Limits for the Inverse Gaussian Distribution , 1982 .

[19]  Luis A. Escobar,et al.  Statistical Intervals: A Guide for Practitioners , 1991 .

[20]  Zhenlin Yang.  On Robustness of Usual Confidence Region under Transformation Misspecification , 1998 .

[21]  D. Cox,et al.  Parameter Orthogonality and Approximate Conditional Inference , 1987 .

[22]  Robert W. Mee,et al.  Prediction limits for the Weibull distribution utilizing simulation , 1994 .

[23]  Peter M. Hooper,et al.  Confidence intervals following Box‐Cox transformation , 1997 .

[24]  J. L. Folks,et al.  The Inverse Gaussian Distribution as a Lifetime Model , 1977 .

[25]  Lee J. Bain,et al.  Prediction Limits and Two-Sample Problems with Complete or Censored Weibull Data , 1979 .

[26]  Raj S. Chhikara,et al.  The Inverse Gaussian Distribution , 1990 .

[27]  Gerald J. Hahn,et al.  A Survey of Prediction Intervals and Their Applications , 1973 .

[28]  Richard A. Johnson,et al.  The Large-Sample Behavior of Transformations to Normality , 1980 .

[29]  Sam C. Saunders,et al.  Estimation for a family of life distributions with applications to fatigue , 1969, Journal of Applied Probability.

[30]  J. Leroy Folks,et al.  The Inverse Gaussian Distribution: Theory: Methodology, and Applications , 1988 .

[31]  Zhenlin Yang.  Some asymptotic results on box-cox transformation methodology , 1996 .