5. Random Right Censoring
6.1. Right Censoring Case
This subsection presents some simulations for right censoring case discussed inSection 5. A sample is generated from exp1and an exponential censoring scheme is used; the censoring distribution is taken to be exp1/9that the proportion of censoring is 10%. To study the robustness properties of our estimators, 20% of the observations are contaminated by exp5.
The DφDE’s αφθ are calculated for samples of sizes 25, 50, 100, and 150, and the hole procedure is repeated 500 times. We can see fromTable 9that the DφDEs perform well under the model in terms of MSE and are an attractive alternative to the AMLE.
Table 10 shows the variation in coverage of nominal 95% asymptotic confidence intervals according to the sample size. Clearly, under coverage of the confidence intervals, the DφDEs have poor coverage probabilities due to the censoring effect. However, for small-and moderate-sized samples, the DφDEs associated toγ2 outperform the AMLE.
Under contamination the performances of our estimators decrease considerably. Such findings are evidences for the need for more adequate procedures for right-censored data Tables11and12.
Remark 6.1. In order to extract methodological recommendations for the use of an appropriate divergence, it will be interesting to conduct extensive Monte Carlo experiments for several divergences or investigate theoretically the problem of the choice of the divergence which leads to an “optimal”in some senseestimate in terms of efficiency and robustness, which would go well beyond the scope of the present paper. Another challenging task is how to choose the bootstrap weights for a given divergence in order to obtain, for example, an efficient estimator.
Table 6: Empirical coverage probabilities for the normal distribution,B1000.
γ n25 n50 n75 n100 n150 n200
−1 0.87 0.90 0.93 0.92 0.93 0.96
0 0.91 0.94 0.94 0.93 0.94 0.96
0.5 0.93 0.93 0.95 0.93 0.94 0.96
1 0.46 0.45 0.48 0.46 0.45 0.50
2 0.96 0.97 0.96 0.95 0.96 0.96
Table 7: Empirical coverage probabilities for the exponential distribution,B500.
γ n25 n50 n75 n100 n150 n200
−1 0.67 0.83 0.87 0.91 0.93 0.92
0 0.73 0.87 0.91 0.93 0.96 0.93
0.5 0.76 0.88 0.91 0.94 0.96 0.93
1 0.76 0.88 0.90 0.95 0.97 0.93
2 0.76 0.89 0.91 0.96 0.96 0.94
Appendix
This section is devoted to the proofs of our results. The previously defined notation continues to be used in the following.
Proof ofTheorem 3.1. Proceeding as 34 in their proof of the Argmax theorem, that is, Corollary 3.2.3, it is straightforward to show the consistency of the bootstrapped estimates
α∗φθ.
Remark A.1. Note that the proof techniques of Theorem 3.3are largely inspired by that of Cheng and Huang6and changes have been made in order to adapt them to our purpose.
Proof ofTheorem 3.3. Keep in mind the following definitions:
Gn:√
nPn−Pθ0,
G∗n:√
nP∗n−Pn.
A.1
In view of the fact thatPθ0∂/∂αhθ, θ0 0, then a little calculation shows that
G∗n ∂
∂αhθ, θ0 Gn
∂
∂αhθ, θ0
√nPθ0 ∂
∂αh
θ,α∗φθ
− ∂
∂αhθ, θ0
G∗n ∂
∂αhθ, θ0− ∂
∂αh
θ,α∗φθ Gn
∂
∂αhθ, θ0− ∂
∂αh
θ,α∗φθ √
nP∗n ∂
∂αh
θ,α∗φθ .
A.2
Table 8: Empirical coverage probabilities for the exponential distribution,B1000.
γ n25 n50 n75 n100 n150 n200
−1 0.70 0.79 0.90 0.91 0.92 0.91
0 0.76 0.84 0.91 0.92 0.93 0.92
0.5 0.78 0.85 0.93 0.94 0.94 0.93
1 0.78 0.87 0.94 0.94 0.95 0.94
2 0.78 0.88 0.95 0.95 0.96 0.95
Table 9: MSE of the estimates for the exponential distribution under right censoring.
γ n25 n50 n100 n150
−1 0.1088 0.0877 0.0706 0.0563
0 0.1060 0.0843 0.0679 0.0538
0.5 0.1080 0.0860 0.0689 0.0544
1 0.1150 0.0914 0.0724 0.0567
2 0.1535 0.1276 0.1019 0.0787
Table 10: Empirical coverage probabilities for the exponential distribution under right censoring.
γ n25 n50 n100 n150
−1 0.55 0.63 0.63 0.64
0 0.59 0.66 0.64 0.64
0.5 0.61 0.66 0.64 0.65
1 0.63 0.67 0.66 0.66
2 0.64 0.70 0.68 0.67
Table 11: MSE of the estimates for the exponential distribution under right censoring, 20% of contamina-tion.
γ n25 n50 n100 n150
−1 0.1448 0.1510 0.1561 0.1591
0 0.1482 0.1436 0.1409 0.1405
0.5 0.1457 0.1402 0.1360 0.1342
1 0.1462 0.1389 0.1332 0.1300
2 0.1572 0.1442 0.1338 0.1266
Table 12: Empirical coverage probabilities for the exponential distribution under right censoring, 20% of contamination.
γ n25 n50 n100 n150
−1 0.44 0.49 0.54 0.57
0 0.46 0.49 0.53 0.57
0.5 0.46 0.49 0.53 0.57
1 0.45 0.49 0.53 0.57
2 0.45 0.49 0.52 0.53
Consequently, we have the following inequality:
According to 23, Theorem 2.2 under condition A.4, we have G1 OoP
W1 in Pθ0 -probability. In view of the CLT, we haveG2 OPθ01. By applying a Taylor series expansion, we have
W1inPθ0-probability. An analogous argument yields that -probability. Finally,G50 based on3.3. In summary,A.3can be rewritten as follows:
√
inPθ0-probability. On the other hand, by a Taylor series expansion, we can write Pθ0
Clearly it is straightforward to combineA.7withA.6, to infer the following:
√nSα∗φθ−θ0≤OoP
in Pθ0-probability, by considering again the consistency ofα∗φθand condition A.3and making use ofA.8to complete the proof of3.23. We next prove3.24. Introduce
H1:−G∗n
By some algebra, we obtain
√nPθ0
Therefore, we have established
√nPθ0
inPθ0-probability. To analyze the left-hand side ofA.11, we rewrite it as
√nPθ0
By a Taylor expansion, we obtain
√nS in Pθ0-probability. Keep in mind that, under conditionA.3, the matrix S is nonsingular.
Multiply both sides of A.13 byS−1 to obtain3.24. An application of 23, Lemma 4.6,
under the bootstrap weight conditions, thus implies3.25. Using1, Theorem 3.2and37, Lemma 2.11, it easily follows that
sup
x∈Rd
Pθ0
√n
αφθ−θ0
≤x
−PN0,Σ≤xoPθ01. A.14
By combining3.25andA.14, we readily obtain the desired conclusion3.27.
Acknowledgments
The authors are grateful to the referees, whose insightful comments helped to improve an early draft of this paper greatly. The authors are indebted to Amor Keziou for careful reading and fruitful discussions on the subject. They would like to thank the associate editor for comments which helped in the completion of this work.
References
1 M. Broniatowski and A. Keziou, “Parametric estimation and tests through divergences and the duality technique,” Journal of Multivariate Analysis, vol. 100, no. 1, pp. 16–36, 2009.
2 L. Pardo, Statistical inference based on divergence measures, vol. 185 of Statistics: Textbooks and Monographs, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2006.
3 F. Liese and I. Vajda, “On divergences and informations in statistics and information theory,” IEEE Transactions on Information Theory, vol. 52, no. 10, pp. 4394–4412, 2006.
4 F. Liese and I. Vajda, Convex Statistical Distances, vol. 95 of Teubner-Texte zur Mathematik, BSB B. G.
Teubner Verlagsgesellschaft, Leipzig, Germany, 1987.
5 A. Basu, H. Shioya, and C. Park, Statistical Inference: The Minimum Distance Approach, vol. 120 of Monographs on Statistics & Applied Probability, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2011.
6 G. Cheng and J. Z. Huang, “Bootstrap consistency for general semiparametric M-estimation,” The Annals of Statistics, vol. 38, no. 5, pp. 2884–2915, 2010.
7 B. Efron, “Bootstrap methods: another look at the jackknife,” The Annals of Statistics, vol. 7, no. 1, pp.
1–26, 1979.
8 B. Efron and R. J. Tibshirani, An Introduction to the Bootstrap, vol. 57 of Monographs on Statistics and Applied Probability, Chapman and Hall, New York, NY, USA, 1993.
9 D. B. Rubin, “The Bayesian bootstrap,” The Annals of Statistics, vol. 9, no. 1, pp. 130–134, 1981.
10 P. J. Bickel and D. A. Freedman, “Some asymptotic theory for the bootstrap,” The Annals of Statistics, vol. 9, no. 6, pp. 1196–1217, 1981.
11 A. Y. Lo, “A Bayesian bootstrap for censored data,” The Annals of Statistics, vol. 21, no. 1, pp. 100–123, 1993.
12 C.-S. Weng, “On a second-order asymptotic property of the Bayesian bootstrap mean,” The Annals of Statistics, vol. 17, no. 2, pp. 705–710, 1989.
13 Z. G. Zheng and D. S. Tu, “Random weighting methods in regression models,” Scientia Sinica. Series A, vol. 31, no. 12, pp. 1442–1459, 1988.
14 R. Beran, “Bootstrap methods in statistics,” Jahresbericht der Deutschen Mathematiker-Vereinigung, vol.
86, no. 1, pp. 14–30, 1984.
15 R. Beran and P. W. Millar, “Confidence sets for a multivariate distribution,” The Annals of Statistics, vol. 14, no. 2, pp. 431–443, 1986.
16 R. J. Beran, L. Le Cam, and P. W. Millar, “Convergence of stochastic empirical measures,” Journal of Multivariate Analysis, vol. 23, no. 1, pp. 159–168, 1987.
17 P. Gaenssler, “Confidence bands for probability distributions on Vapnik-Chervonenkis classes of sets in arbitrary sample spaces using the bootstrap,” in Bootstrapping and related techniques (Trier, 1990), vol.
376 of Lecture Notes in Econom. and Math. Systems, pp. 57–61, Springer, Berlin, Germany, 1992.
18 K. Lohse, “Consistency of the bootstrap,” Statistics & Decisions, vol. 5, no. 3-4, pp. 353–366, 1987.
19 S. Cs ¨org˝o and D. M. Mason, “Bootstrapping empirical functions,” The Annals of Statistics, vol. 17, no.
4, pp. 1447–1471, 1989.
20 E. Gin´e and J. Zinn, “Necessary conditions for the bootstrap of the mean,” The Annals of Statistics, vol.
17, no. 2, pp. 684–691, 1989.
21 E. Gin´e and J. Zinn, “Bootstrapping general empirical measures,” The Annals of Probability, vol. 18, no.
2, pp. 851–869, 1990.
22 D. M. Mason and M. A. Newton, “A rank statistics approach to the consistency of a general bootstrap,” The Annals of Statistics, vol. 20, no. 3, pp. 1611–1624, 1992.
23 J. Præstgaard and J. A. Wellner, “Exchangeably weighted bootstraps of the general empirical process,”
The Annals of Probability, vol. 21, no. 4, pp. 2053–2086, 1993.
24 E. del Barrio and C. Matr´an, “The weighted bootstrap mean for heavy-tailed distributions,” Journal of Theoretical Probability, vol. 13, no. 2, pp. 547–569, 2000.
25 S. Holmes and G. Reinert, “Stein’s method for the bootstrap,” in Stein’s Method: Expository Lectures and Applications, vol. 46 of IMS Lecture Notes Monogr. Ser., pp. 95–136, Inst. Math. Statist., Beachwood, Ohio, USA, 2004.
26 S. Chatterjee and A. Bose, “Generalized bootstrap for estimating equations,” The Annals of Statistics, vol. 33, no. 1, pp. 414–436, 2005.
27 P. Barbe and P. Bertail, The Weighted Bootstrap, vol. 98 of Lecture Notes in Statistics, Springer, New York, NY, USA, 1995.
28 A. Keziou, “Dual representation ofφ-divergences and applications,” Comptes Rendus Math´ematique.
Acad´emie des Sciences. Paris, vol. 336, no. 10, pp. 857–862, 2003.
29 N. Cressie and T. R. C. Read, “Multinomial goodness-of-fit tests,” Journal of the Royal Statistical Society.
Series B, vol. 46, no. 3, pp. 440–464, 1984.
30 A. R´enyi, “On measures of entropy and information,” in Proc. 4th Berkeley Sympos. Math. Statist. and Prob., Vol. I, pp. 547–561, Univ. California Press, Berkeley, Calif, USA, 1961.
31 M. Broniatowski and A. Keziou, “Minimization ofφ-divergences on sets of signed measures,” Studia Scientiarum Mathematicarum Hungarica, vol. 43, no. 4, pp. 403–442, 2006.
32 R. Jim´enez and Y. Shao, “On robustness and efficiency of minimum divergence estimators,” Test, vol.
10, no. 2, pp. 241–248, 2001.
33 A. Toma and M. Broniatowski, “Dual divergence estimators and tests: robustness results,” Journal of Multivariate Analysis, vol. 102, no. 1, pp. 20–36, 2011.
34 A. W. van der Vaart and J. A. Wellner, Weak Convergence and Empirical Processes, Springer Series in Statistics, Springer, New York, NY, USA, 1996.
35 M. R. Kosorok, Introduction to Empirical Processes and Semiparametric Inference, Springer Series in Statistics, Springer, New York, NY, USA, 2008.
36 J. A. Wellner and Y. Zhan, “Bootstrapping Z-estimators,” Tech. Rep. 308, 1996.
37 A. W. van der Vaart, Asymptotic Statistics, vol. 3 of Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press, Cambridge, UK, 1998.
38 L. F. James, “A study of a class of weighted bootstraps for censored data,” The Annals of Statistics, vol.
25, no. 4, pp. 1595–1621, 1997.
39 M. Broniatowski, “Minimum divergence estimators, maximum likelihood and exponential families,”
http://arxiv.org/abs/1108.0772.
40 M. Cherfi, “Dualϕ-divergences estimation in normal models,”http://arxiv.org/abs/1108.2999.
41 S. Bouzebda and A. Keziou, “New estimates and tests of independence in semiparametric copula models,” Kybernetika, vol. 46, no. 1, pp. 178–201, 2010.
42 S. Bouzebda and A. Keziou, “A new test procedure of independence in copula models via χ2 -divergence,” Communications in Statistics. Theory and Methods, vol. 39, no. 1-2, pp. 1–20, 2010.
43 M. Broniatowski and I. Vajda, “Several applications of divergence criteria in continuous families,”
Tech. Rep. 2257, Academy of Sciences of the Czech Republic, Institute of Information Theory and Automation, 2009.
44 S. Bouzebda and A. Keziou, “A test of independence in some copula models,” Mathematical Methods of Statistics, vol. 17, no. 2, pp. 123–137, 2008.
45 M. Broniatowski and S. Leorato, “An estimation method for the Neyman chi-square divergence with application to test of hypotheses,” Journal of Multivariate Analysis, vol. 97, no. 6, pp. 1409–1436, 2006.
46 A. Basu, I. R. Harris, N. L. Hjort, and M. C. Jones, “Robust and efficient estimation by minimising a density power divergence,” Biometrika, vol. 85, no. 3, pp. 549–559, 1998.
47 S. Basu, A. Basu, and M. C. Jones, “Robust and efficient parametric estimation for censored survival data,” Annals of the Institute of Statistical Mathematics, vol. 58, no. 2, pp. 341–355, 2006.
48 C. Hong and Y. Kim, “Automatic selection of the tuning parameter in the minimum density power divergence estimation,” Journal of the Korean Statistical Society, vol. 30, no. 3, pp. 453–465, 2001.
49 J. Warwick and M. C. Jones, “Choosing a robustness tuning parameter,” Journal of Statistical Computation and Simulation, vol. 75, no. 7, pp. 581–588, 2005.
50 H. Tsukahara, “Semiparametric estimation in copula models,” The Canadian Journal of Statistics, vol.
33, no. 3, pp. 357–375, 2005.
51 E. L. Kaplan and P. Meier, “Nonparametric estimation from incomplete observations,” Journal of the American Statistical Association, vol. 53, pp. 457–481, 1958.
52 W. Stute, “The statistical analysis of Kaplan-Meier integrals,” in Analysis of Censored Data (Pune, 1994/1995), vol. 27 of IMS Lecture Notes Monogr. Ser., pp. 231–254, Inst. Math. Statist., Hayward, Calif, USA, 1995.
53 D. Oakes, “An approximate likelihood procedure for censored data,” Biometrics. Journal of the Biometric Society, vol. 42, no. 1, pp. 177–182, 1986.
54 M. Cherfi, “Dual divergences estimation for censored survival data,”http://arxiv.org/abs/1106 .2627.
Submit your manuscripts at http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com Volume 2014