Nonparametric maximum likelihood computation of a \(U\)-shaped hazard function. (English) Zbl 1384.62120
Summary: A new algorithm is presented and studied in this paper for fast computation of the nonparametric maximum likelihood estimate of a \(U\)-shaped hazard function. It successfully overcomes a difficulty when computing a \(U\)-shaped hazard function, which is only properly defined by knowing its anti-mode, and the anti-mode itself has to be found during the computation. Specifically, the new algorithm maintains the constant hazard segment, regardless of its length being zero or positive. The length varies naturally, according to what mass values are allocated to their associated knots after each updating. Being an appropriate extension of the constrained Newton method, the new algorithm also inherits its advantage of fast convergence, as demonstrated by some real-world data examples. The algorithm works not only for exact observations, but also for purely interval-censored data, and for data mixed with exact and interval-censored observations.
MSC:
| 62G05 | Nonparametric estimation |
| 62N02 | Estimation in survival analysis and censored data |
| 62N05 | Reliability and life testing |
| 62G07 | Density estimation |
Keywords:
algorithms; lifetime and survival analysis; nonparametric methods; shape-restricted estimation; numerical optimizationReferences:
| [1] | Ayer, M., Brunk, H.D., Ewing, G.M., Reid, W.T., Silverman, E.: An empirical distribution function for sampling with incomplete information. Ann. Math. Stat. 26, 641-647 (1955) · Zbl 0066.38502 · doi:10.1214/aoms/1177728423 |
| [2] | Banerjee, M.: Estimating monotone, unimodal and U-shaped failure rates using asymptotic pivots. Stat. Sin. 18, 467-492 (2008) · Zbl 1135.62079 |
| [3] | Bray, T.A., Crawford, G.B., Proschan, F.: Maximum Likelihood Estimation of a U-shaped Failure Rate Function. Defense Technical Information Center, Mathematical Note 534, Boeing Research Laboratories, Seattle (1967) · Zbl 1072.62023 |
| [4] | Dümbgen, L., Freitag-Wolf, S., Jongbloed, G.: Estimating a unimodal distribution from interval-censored data. J. Am. Stat. Assoc. 101, 1094-1106 (2006) · Zbl 1120.62313 · doi:10.1198/016214506000000032 |
| [5] | Grenander, U.: On the theory of mortality measurement. II. Skand. Aktuarietidskr. 39, 125-153 (1956) · Zbl 0077.33715 |
| [6] | Groeneboom, P., Jongbloed, G.: Nonparametric Estimation under Shape Constraints: Estimators, Algorithms and Asymptotics. Cambridge University Press, Cambridge (2014) · Zbl 1338.62008 · doi:10.1017/CBO9781139020893 |
| [7] | Groeneboom, P., Jongbloed, G., Wellner, J.A.: The support reduction algorithm for computing non-parametric function estimates in mixture models. Scand. J. Stat. 35, 385-399 (2008) · Zbl 1199.65017 · doi:10.1111/j.1467-9469.2007.00588.x |
| [8] | Hall, P., Huang, L.S., Gifford, J.A., Gijbels, I.: Nonparametric estimation of hazard rate under the constraint of monotonicity. J. Comput. Graph. Stat. 10, 592-614 (2001) · doi:10.1198/106186001317115135 |
| [9] | Huang, J., Wellner, J.A.: Estimation of a monotone density or monotone hazard under random censoring. Scand. J. Stat. 22, 3-33 (1995) · Zbl 0827.62032 |
| [10] | Jankowski, H., Wang, I., McCague, H., Wellner, J.A.: R Package ConvexHaz: Nonparametric MLE/LSE of Convex Hazard (Version 0.2). http://cran.r-project.org/web/packages/convexHaz/index.html (2009) · Zbl 1200.62025 |
| [11] | Jankowski, H.K., Wellner, J.A.: Computation of nonparametric convex hazard estimators via profile methods. J. Nonparametric Stat. 21, 505-518 (2009a) · Zbl 1161.62014 · doi:10.1080/10485250902745359 |
| [12] | Jankowski, H.K., Wellner, J.A.: Nonparametric estimation of a convex bathtub-shaped hazard function. Bernoulli 15, 1010-1035 (2009b) · Zbl 1200.62025 · doi:10.3150/09-BEJ202 |
| [13] | Kaplan, E.L., Meier, P.: Nonparametric estimation from incomplete observations. J. Am. Stat. Assoc. 53, 457-481 (1958) · Zbl 0089.14801 · doi:10.1080/01621459.1958.10501452 |
| [14] | Klein, J .P., Moeschberger, M.L.: Survival Analysis: Techniques for Censored and Truncated Data, 2nd edn. Springer, Berlin (2003) · Zbl 1011.62106 |
| [15] | Lawson, C .L., Hanson, R .J.: Solving Least Squares Problems. Prentice-Hall, Inc, Englewood Cliffs (1974) · Zbl 0860.65028 |
| [16] | Lee, E .T., Wang, J .W.: Statistical Methods for Survival Data Analysis, 3rd edn. Wiley, London (2003) · Zbl 1026.62103 · doi:10.1002/0471458546 |
| [17] | Meyer, M.C., Habtzghi, D.: Nonparametric estimation of density and hazard rate functions with shape restrictions. J. Nonparametric Stat. 23, 455-470 (2011) · Zbl 1359.62416 · doi:10.1080/10485252.2010.531133 |
| [18] | Mykytyn, S.W., Santner, T.J.: Maximum likelihood estimation of the survival function based on censored data under hazard rate assumptions. Commun. Stat. Theory Methods 10, 1369-1387 (1981) · Zbl 0496.62037 · doi:10.1080/03610928108828120 |
| [19] | Peto, R.: Experimental survival curves for interval-censored data. J. R. Stat. Soc. Ser. C 22, 86-91 (1973) |
| [20] | Proschan, F.: Theoretical explanation of observed decreasing failure rate. Technometrics 5, 375-383 (1963) · doi:10.1080/00401706.1963.10490105 |
| [21] | R Core Team: R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna (2015) · Zbl 1135.62079 |
| [22] | Reboul, L.: Estimation of a function under shape restrictions: applications to reliability. Ann. Stat. 33, 1330-1356 (2005) · Zbl 1072.62023 · doi:10.1214/009053605000000138 |
| [23] | Schick, A., Yu, Q.: Consistency of the GMLE with mixed case interval-censored data. Scand. J. Stat. 27, 45-55 (2000) · Zbl 0938.62109 · doi:10.1111/1467-9469.00177 |
| [24] | Tsai, W.-Y.: Estimation of the survival function with increasing failure rate based on left truncated and right censored data. Biometrika 75, 319-324 (1988) · Zbl 0639.62085 · doi:10.1093/biomet/75.2.319 |
| [25] | Turnbull, B.W.: Nonparametric estimation of a survivorship function with doubly censored data. J. Am. Stat. Assoc. 69, 169-173 (1974) · Zbl 0281.62044 · doi:10.1080/01621459.1974.10480146 |
| [26] | Wang, Y.: On fast computation of the non-parametric maximum likelihood estimate of a mixing distribution. J. R. Stat. Soc. Ser. B 69, 185-198 (2007) · Zbl 1120.62022 · doi:10.1111/j.1467-9868.2007.00583.x |
| [27] | Wang, Y.: Dimension-reduced nonparametric maximum likelihood computation for interval-censored data. Comput. Stat. Data Anal. 52, 2388-2402 (2008) · Zbl 1452.62257 · doi:10.1016/j.csda.2007.10.018 |
| [28] | Wang, Y.: npsurv: Non-parametric Survival Analysis (R Package Version 0.3-4). http://cran.r-project.org/package=npsurv (2015) · Zbl 1120.62022 |
| [29] | Wellner, JA; Koul, H. (ed.); Deshpande, JV (ed.), Interval censoring case 2: alternative hypotheses, No. 27, 271-291 (1995), Pune · Zbl 0876.62044 · doi:10.1214/lnms/1215452225 |
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