1 / 14

Renewal Estimation Without Renewal Counts

Renewal Estimation Without Renewal Counts. Larry George Problem Solving Tools INFORMS San Jose Nov. 18, 2002. Estimate the inter-renewal time distribution . From grouped counts Without knowledge of how many renewals have occurred

shawna
Download Presentation

Renewal Estimation Without Renewal Counts

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Renewal Estimation Without Renewal Counts Larry GeorgeProblem Solving Tools INFORMS San Jose Nov. 18, 2002

  2. Estimate the inter-renewal time distribution • From grouped counts • Without knowledge of how many renewals have occurred • Kaplan-Meier (life table) npmle is for censored failure data, either: • Failures stay dead or • Ages between renewals and survivors’ ages are known

  3. Review Inter-Renewal cdf est. • Inter-renewal age data • Pena, Strawderman, and Hollander, JASA • Denby and Vardi shortcut to K-M method • Frees and Schneider, Lin, and O’Cinneide estimate renewal function as Fn(k)(t) • Baxter and Li: np CI on renewal function • Count data • Dattero: interval counts limited to 0 or 1, based on forward recurrence density • Tortorella: npmle from grouped data • http://amap.cirad.fr/amapmod/refermanual15/stat/estimate2.html

  4. Lucent Renewable Part Reliability

  5. Identifiability • “The superposition of independent renewal processes is not renewal unless both are Poisson.” O’Cinneide • “Let N denote the the superposition of two stationary independent renewal processes based on analytic distributions. Then, from the knowledge of the distribution of N alone, the two distribution functions may be determined, except when N is a Poisson process.”

  6. Identifiability alternatives A A A • O’Cinneide’s is the first pair • Mine is the second, “modified” RP B B time A B B time A B

  7. Data Descriptions • Installed base and failures in each calendar interval. Were failures were first, second, or …? • Wards lists year, make, and model counts. NHTSA lists complaint data by year, make, model, and age at complaint, sometimes. Failure = OEM tire? • M88 A1 data is installed base and rebuilds in 1990s • Ford published monthly drivetrain warranty returns/1000 * installed base => absolute monthly return

  8. Firestone Tires • 1978 recall, 40 dead • Aug. 2000 recall, 170 dead • Nov. 2000 TREAD Act • Transportation Recall Enhancement, Accountability, and Documentation • Feb. 2001 Sent reliability analysis to Ford • March 2001 NHTSA TREAD Act Insurance Study • NHTSA asks for personal id, VIN, crash data • Companies say that violates privacy, risks liability • April 2001 Ford recalls Firestone tires • June 2002 proposal to estimate from counts • Aug. 2002 rejection

  9. M88 A1 engine annual renewal rate; $116k each plus labor

  10. 1988 Ford V-8 460 drivetrain

  11. Npmle for renewal process • L = f*k(j)(tj) (1-Fres*k(j)(tj)) • First product is failures and second product is survivors, tj are ages • *k(j) denotes convolution, k() unknown! • Fres*k(j)(tj), last cdf in k(j) is residual cdf • Solve {lnL/  f*k(j)(tj) = 0} for f(t) • Discretize and group data • Solve high order polynomials simultaneously. May be only one real, nonnegative pdf solution for small problems. WIP

  12. Semiparametric mle • For M/G/ estimate G(t) • Output of Mt/G/ is Poisson ltG(t) • IFR PP rate estimator gives G(t) = lout(t)/l • Recycling M/G/  gives: • Spmle: G1 = l1out/l, Gj = lJout/(l+p lJ-1out) • Spmle Gj is cdf if Gj < 1 and Gj/Gj+1<1

  13. Nplse • Min(Obs. returns – E[returns])2 where • E[returns] = S N(k-j)*D(j) (actuarial fcst) • N(k-j) is ships j periods ago • D(j) is demand rate for j period old unit • D(j) = M(j) – M(j-1), the discrete renewal density • M(j) = S F*k(j), k = 1,…

  14. References • George, L. L. and A. Agrawal, “Estimation of a Hidden Service Distribution of an M/G/ Service System,” Naval Research Logistics Quarterly, pp. 549-555, September, 1973, Vol. 20, No. 3 • Turnbull, Bruce W., “Nonparametric estimation of a survivorship function with doubly censored data,” J. Amer. Statist. Assn., Vol. 69, No. 345, pp 169-174, March 1974 • Tortorella, Michael, “Life estimation from pooled discrete renewal counts,” Lifetime Data: Models in Reliability and Survival Analysis, N. P. Jewell et al. (eds.), pp331-338, Kluwer, Netherlands, 1996 • Dattero, Ronald, “Non-parametric estimation for renewal processes from event count data,” Applied Stochastic Models and Data Analysis, Vol. 5, pp 1-12, 1989 • Vardi, Y. “Nonparametric estimation in renewal processes,” The Annals of Statistics, Vol. 10, No. 3, pp772-785, 1982 • Schneider, Helmut, Bin-Shan Lin, and Colm O’Cinneide, “Comparison of nonparametric estimators for the renewal function,” Applied Statist., Vol. 39, No. 1, pp 55-61, 1990 • Frees, Edward, “Nonparametric renewal function estimation,” The Annals of Statistics, Vol. 14, No. 4, pp1366-1378, 1986 • Pena, E. A., Strawderman. R. L., and Hollander, M., “Nonparametric estimationwith recurrent event data,” Journal of the AmericanStatistical Association, Vol. 96, pp 1299-1315, 2001. • O’Cinneide, C., “Identifiability of superpositions of renewal processes,” Stochastic Models, Vol. 7 Dec. 1991, pp 603-614 • Nelson, Wayne, Recurrent-Events Data Analysis for Product Repairs, Disease Recurrences, and Other Applications, ASA-SIAM, 2003 • Kovbassa, Sergei, “Statistical estimation of the independence of intervals in dynamic models of trains of actions,” http://www.actuaries.org/members/en/ASTIN/ colloquia/Porto_Cervo/Kovbassa.pdf

More Related