zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Large sample theory for semiparametric regression models with two-phase, outcome dependent sampling. (English) Zbl 1105.62335
Summary: Outcome-dependent, two-phase sampling designs can dramatically reduce the costs of observational studies by judicious selection of the most informative subjects for purposes of detailed covariate measurement. Here we derive asymptotic information bounds and the form of the efficient score and influence functions for the semiparametric regression models studied by Lawless, Kalbfleisch and Wild (1999) under two-phase sampling designs. We show that the maximum likelihood estimators for both the parametric and nonparametric parts of the model are asymptotically normal and efficient. The efficient influence function for the parametric part agrees with the more general information bound calculations of Robins, Hsieh and Newey (1995). By verifying the conditions of Murphy and van der Vaart (2000) for a least favorable parametric submodel, we provide asymptotic justification for statistical inference based on profile likelihood.

62G08Nonparametric regression
62D05Statistical sampling theory, sample surveys
Full Text: DOI Euclid
[1] BEGUN, J. M., HALL, W. J., HUANG, W.-M. and WELLNER, J. A. (1983). Information and asy mptotic efficiency in parametric-nonparametric models. Ann. Statist. 11 432-452. · Zbl 0526.62045 · doi:10.1214/aos/1176346151
[2] BICKEL, P. J., KLAASSEN, C. A. J., RITOV, Y. and WELLNER, J. A. (1993). Efficient and Adaptive Estimation for Semiparametric Models. Johns Hopkins Univ. Press, Baltimore, MA. · Zbl 0786.62001
[3] BRESLOW, N. E. and CHATTERJEE, N. (1999). Design and analysis of two-phase studies with binary outcomes applied to Wilms tumour prognosis. Appl. Statist. 48 457-468. · Zbl 0957.62091 · doi:10.1111/1467-9876.00165
[4] BRESLOW, N. E. and HOLUBKOV, R. (1997). Maximum likelihood estimation of logistic regression parameters under two-phase, outcome-dependent sampling. J. Roy. Statist. Soc. Ser. B 59 447-461. JSTOR: · Zbl 0886.62071 · doi:10.1111/1467-9868.00078 · http://links.jstor.org/sici?sici=0035-9246%281997%2959%3A2%3C447%3AMLEOLR%3E2.0.CO%3B2-S&origin=euclid
[5] BRESLOW, N. E., MCNENEY, B. and WELLNER, J. A. (2000). Large sample theory for semiparametric regression models with two-phase, outcome dependent sampling. Technical Report 381, Dept. Statistics, Univ. Washington. · Zbl 1105.62335
[6] BRESLOW, N. E., ROBINS, J. M. and WELLNER, J. A. (2000). On the semi-parametric efficiency of logistic regression under case-control sampling. Bernoulli 6 447-455. · Zbl 0965.62033 · doi:10.2307/3318670
[7] CHATTERJEE, N., CHEN, Y. H. and BRESLOW, N. E. (2003). A pseudoscore estimator for regression problems with two-phase sampling. J. Amer. Statist. Assoc. 98 158-168. · Zbl 1047.62031 · doi:10.1198/016214503388619184
[8] HUBER, P. J. (1967). The behavior of maximum likelihood estimates under nonstandard conditions. Proc. Fifth Berkeley Sy mp. Math. Statist. Probab. 1 221-233. Univ. California Press. · Zbl 0212.21504
[9] LAWLESS, J. F., KALBFLEISCH, J. D. and WILD, C. J. (1999). Semiparametric methods for response-selective and missing data problems in regression. J. R. Stat. Soc. Ser. B Stat. Methodol. 61 413-438. JSTOR: · Zbl 0915.62030 · doi:10.1111/1467-9868.00185 · http://links.jstor.org/sici?sici=1369-7412%281999%2961%3A2%3C413%3ASMFRAM%3E2.0.CO%3B2-2&origin=euclid
[10] MCCULLAGH, P. and NELDER, J. A. (1989). Generalized Linear Models, 2nd ed. Chapman and Hall, London. · Zbl 0744.62098
[11] MCNENEY, B. (1998). Asy mptotic efficiency in semiparametric models with non-i.i.d. data. Ph.D. dissertation, Univ. Washington.
[12] MURPHY, S. A. and VAN DER VAART, A. W. (1997). Semiparametric likelihood ratio inference. Ann. Statist. 25 1471-1509. · Zbl 0928.62036 · doi:10.1214/aos/1031594729
[13] MURPHY, S. A. and VAN DER VAART, A. W. (1999). Observed information in semi-parametric models. Bernoulli 5 381-412. · Zbl 0954.62036 · doi:10.2307/3318710
[14] MURPHY, S. A. and VAN DER VAART, A. W. (2000). On profile likelihood (with discussion). J. Amer. Statist. Assoc. 95 449-485. JSTOR: · Zbl 0995.62033 · doi:10.2307/2669386 · http://links.jstor.org/sici?sici=0162-1459%28200006%2995%3A450%3C449%3AOPL%3E2.0.CO%3B2-%23&origin=euclid
[15] NAN, B., EMOND, M. and WELLNER, J. A. (2000). Information bounds for regression models with missing data. Technical Report 378, Dept. Statistics, Univ. Washington.
[16] POLLARD, D. (1985). New way s to prove central limit theorems. Econometric Theory 1 295-314.
[17] PRENTICE, R. L. and Py KE, R. (1979). Logistic disease incidence models and case-control studies. Biometrika 66 403-411. JSTOR: · Zbl 0428.62078 · doi:10.1093/biomet/66.3.403 · http://links.jstor.org/sici?sici=0006-3444%28197912%2966%3A3%3C403%3ALDIMAC%3E2.0.CO%3B2-M&origin=euclid
[18] ROBINS, J. M., HSIEH, F. and NEWEY, W. (1995). Semiparametric efficient estimation of a conditional density with missing or mismeasured covariates. J. Roy. Statist. Soc. Ser. B 57 409-424. JSTOR: · Zbl 0813.62029 · http://links.jstor.org/sici?sici=0035-9246%281995%2957%3A2%3C409%3ASEEOAC%3E2.0.CO%3B2-6&origin=euclid
[19] ROBINS, J. M., ROTNITZKY, A. and ZHAO, L. P. (1994). Estimation of regression coefficients when some regressors are not alway s observed. J. Amer. Statist. Assoc. 89 846-866. JSTOR: · Zbl 0815.62043 · doi:10.2307/2290910 · http://links.jstor.org/sici?sici=0162-1459%28199409%2989%3A427%3C846%3AEORCWS%3E2.0.CO%3B2-A&origin=euclid
[20] SCOTT, A. J. and WILD, C. J. (1997). Fitting regression models to case-control data by maximum likelihood. Biometrika 84 57-71. JSTOR: · Zbl 1058.62505 · doi:10.1093/biomet/84.1.57 · http://links.jstor.org/sici?sici=0006-3444%28199703%2984%3A1%3C57%3AFRMTCD%3E2.0.CO%3B2-A&origin=euclid
[21] SCOTT, A. J. and WILD, C. J. (2000). Maximum likelihood for generalised case-control studies. Statistical design of medical experiments. II. J. Statist. Plann. Inference 96 3-27. · doi:10.1016/S0378-3758(00)00317-7
[22] SELF, S. G. and PRENTICE, R. L. (1988). Asy mptotic distribution theory and efficiency results for case-cohort studies. Ann. Statist. 16 64-81. · Zbl 0666.62108 · doi:10.1214/aos/1176350691
[23] VAN DER VAART, A. W. (1995). Efficiency of infinite-dimensional M-estimators. Statist. Neerlandica 49 9-30. · Zbl 0830.62029 · doi:10.1111/j.1467-9574.1995.tb01452.x
[24] VAN DER VAART, A. W. (1998). Asy mptotic Statistics. Cambridge Univ. Press. · Zbl 0910.62001 · doi:10.1017/CBO9780511802256
[25] VAN DER VAART, A. W. and WELLNER, J. A. (1992). Existence and consistency of maximum likelihood in upgraded mixture models. J. Multivariate Anal. 43 133-146. · Zbl 0752.62026 · doi:10.1016/0047-259X(92)90113-T
[26] VAN DER VAART, A. W. and WELLNER, J. A. (1996). Weak Convergence and Empirical Processes. Springer, New York. · Zbl 0862.60002
[27] VAN DER VAART, A. W. and WELLNER, J. A. (2000). Preservation theorems for Glivenko-Cantelli and uniform Glivenko-Cantelli classes. In High Dimensional Probability II (E. Giné, D. M. Mason and J. A. Wellner, eds.) 113-132. Birkhäuser, Boston. · Zbl 0967.60037
[28] VAN DER VAART, A. W. and WELLNER, J. A. (2001). Consistency of semiparametric maximum likelihood estimators for two-phase sampling. Canad. J. Statist. 29 269-288. JSTOR: · Zbl 0976.62017 · doi:10.2307/3316077 · http://links.jstor.org/sici?sici=0319-5724%28200106%2929%3A2%3C269%3ACOSMLE%3E2.0.CO%3B2-K&origin=euclid
[29] WELLNER, J. A. and ZHAN, Y. (1997). Bootstrapping Z-estimators. Technical Report 308, Dept. Statistics, Univ. Washington.
[31] SEATTLE, WASHINGTON 98195-4322 E-MAIL: jaw@stat.washington.edu