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Renewal Estimation Without Renewal Counts

Renewal Estimation Without Renewal Counts

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Renewal Estimation Without Renewal Counts

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  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