Nonparametric estimation in renewal theory II: Solutions of renewal-type equations

Annals of Statistics

Volumn 28, Issue 1, 2000, Pages 88-115

  Politis, Konstadinos ^a   Pitts, Susan M ^a  

^a NONE

Author keywords

Asymptotic efficiency; Bootstrap; Functional central limit theorems; Renewal function; Renewal type equations

Indexed keywords

EID: 0034413232     PISSN: 00905364     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (34)

1. ALSMEYER, G. (1991). Erneuerungstheorie: Analyse stochastischer Regenerationsschemata. Teubner, Stutgart.
2. ASMUSSEN, S. (1987). Applied Probability and Queues. Wiley, New York.
3. BARLOW, R. E. and PROSCHAN, F. (1975). Statistical Theory of Reliability and Life Testing. Holt Rinehart and Winston, New York.
4. BAXTER, L. A. and LI, L. (1994). Non-parametric confidence intervals for the renewal function and the point availablity. Scand. J. Statist. 21 277-287.
5. BINGHAM, N. H. (1989). Tauberian theorems in probability theory. Lecture Notes in Math. 1379 6-20. Springer, New York.
6. DORADO, C., HOLLANDER, M. and SETHURAMAN, J. (1997). Nonparametric estimation for a general repair model. Ann. Statist. 25 1140-1160.
7. FELLER, W. (1971). An Introduction to Probability Theory and Its Applications 2, 2nd ed. Wiley, New York.
8. FREES, E. W. (1986). Nonparametric renewal function estimation. Ann. Statist. 14 1366-1378.
9. GELFAND, I., RAIKOV, D. and SHILOV, G. (1964). Commutative Normed Rings. Chelsea, New York.
10. GILL, R. D. (1989). Non- and semi-parametric maximum likelihood estimators and the von Mises method (Part I). Scand. J. Statist. 16 97-128.
11. GROENEBOOM, P. and WELLNER, J. A. (1992). Information Bounds and Nonparametric Maximum Likelihood Estimation. Birkhäuser, Boston.
12. GRUBEL, R. (1989). Stochastic models as functionals: some remarks on the renewal case. J. Appl. Probab. 26 296-303.
13. GRUBEL, R. and PITTS, S. M. (1993). Nonparametric estimation in renewal theory I: the empirical renewal function. Ann. Statist. 21 1431-1451.
14. GUT, A. (1988). Stopped Random Walks: Limit Theorems and Applications. Springer, New York.
15. HAREL, M., O'CINNEIDE, C. A. and SCHNEIDER, H. (1995). Asymptotics of the sample renewal function. J. Math. Anal. Appl. 189 240-255.
16. HINDERER, K. (1987). Remarks on directly Riemann integrable functions. Math. Nachr. 130 225-230.
17. KARLIN, S. (1955). On the renewal equation. Pacific J. Math. 5 229-257.
18. PITTS, S. M. (1991). The single server queue: a functional approach. Ph.D. dissertation, Imperial College, London.
19. PITTS, S. M. (1994a). Nonparametric estimation of the stationary waiting time distribution for the GI/G/1 queue. Ann. Statist. 22 1428-1446.
20. PITTS, S. M. (1994b). Nonparametric estimation of compound distributions with applications in insurance. Ann. Inst. Statist. Math. 46 537-555.
21. PITTS, S. M., GRÜBEL, R. and EMBRECHTS, P. (1996). Confidence bounds for the adjustment coefficient. Adv. Appl. Probab. 28 802-827.
22. POLITIS, K. (1997). Approximations and nonparametric estimation for solutions of renewal-type equations. Ph.D. dissertation, Univ. Cambridge.
23. POLITIS, K. and PITTS, S. M. (1998). Approximations for solutions of renewal-type equations. Stochast. Process. Appl. 78 195-216.
24. POLLARD, D. (1984). Convergence of Stochastic Processes. Springer, New York.
25. ROGOZIN, B. A. (1976). Asymptotics of renewal functions. Theory Probab. Appl. 21 669-686.
26. RUDIN, W. (1987). Real and Complex Analysis, 3rd ed. McGraw-Hill, New York.
27. RUDIN, W. (1991). Functional Analysis, 2nd ed. McGraw-Hill, New York.
28. SHORACK, G. R. (1982). Bootstrapping robust regression. Comm. Statist. Theoret. Methods 11 961-972.
29. SHORACK, G. R. and WELLNER, J. (1986). Empirical Processes with Applications to Statistics. Wiley, New York.
30. SMITH, W. L. (1960). Infinitesimal renewal processes. In Contributions to Probability and Statistics (I. Olkin, S. G. Ghurye, W. Hoeffding, W. G. Madow and H. B. Mann, eds.) 396-413. Stanford Univ. Press.
31. SMITH, W. L. (1961). On some general renewal theorems for nonidentically distributed random variables. Proc. Fourth Berkeley Symp. Math. Statist. Probab. 467-514. Univ. California Press, Berkeley.
32. STONE, C. (1966). On absolutely continuous components and renewal theory. Ann. Math. Statist. 37 271-275.
33. VAN DER VAART, A. W. (1988). Statistical Estimation in Large Parameter Spaces. CWI Tract 44. Centrum voor Wiskunde en Informatica, Amsterdam.
34. VAN DER VAART, A. W. (1991). Efficiency and Hadamard differentiability. Scand. J. Statist. 18 63-75.