Frontiers of Mathematics in China >
Probability density estimation with surrogate data and validation sample
Received date: 21 Aug 2012
Accepted date: 19 Nov 2012
Published date: 01 Jun 2013
Copyright
The probability density estimation problem with surrogate data and validation sample is considered. A regression calibration kernel density estimator is defined to incorporate the information contained in both surrogate variates and validation sample. Also, we define two weighted estimators which have less asymptotic variances but have bigger biases than the regression calibration kernel density estimator. All the proposed estimators are proved to be asymptotically normal. And the asymptotic representations for the mean squared error and mean integrated square error of the proposed estimators are established, respectively. A simulation study is conducted to compare the finite sample behaviors of the proposed estimators.
Key words: Measurement error; asymptotic normality; convergent rate
Qihua WANG , Wenquan CUI . Probability density estimation with surrogate data and validation sample[J]. Frontiers of Mathematics in China, 2013 , 8(3) : 665 -694 . DOI: 10.1007/s11464-013-0267-0
1 |
Bowman A. An alternative method of cross-validation for the smoothing of density estimates. Biometrika, 1984, 71: 353-360
|
2 |
Carroll R J, Hall P. Optimal rates of convergence for deconvolving a density. J Amer Statist Assoc, 1988, 83: 1184-1186
|
3 |
Carroll R J, Wand M P. Semiparametric estimation in logistic measure error models. J Roy Statist Soc, Ser B, 1991, 53: 539-572
|
4 |
Chen Y-H. Cox regression in cohort studies with validation sampling. J Roy Statist Soc, Ser B, 2002, 64: 51-62
|
5 |
Devroye L, Györfi L. The equivalence of weak, strong and complete convergence in L1 for kernel density estimates. Ann Statist, 1983, 11: 896-904
|
6 |
Devroye L, Györfi L. Nonparametric Density Estimation. New York: John Wiley & Sons, 1985
|
7 |
Duncan G, Hill D. An investigations of the extent and consequences of measurement error in labor-economics survey data. J Labor Economics, 1985, 3: 508-532
|
8 |
Fan J. On the optimal rates of convergence for nonparametric deconvolution problems. Ann Statist, 1991, 19: 1257-1272
|
9 |
Parzen E. On estimation of a probability density function and mode. Ann Statist, 1962, 33: 1065-1076
|
10 |
Pepe M S. Inference using surrogate outcome data and a validation sample. Biometrika, 1992, 79: 355-365
|
11 |
Pepe M S, Fleming T R. A general nonparametric method for dealing with errors in missing or surrogate covariate data. J Amer Statist Assoc, 1991, 86: 108-113
|
12 |
Pepe M S, Reilly M, Fleming T R. Auxiliary outcome data and the mean score method. J Statist Plann Inference, 1994, 42: 137-160
|
13 |
Reilly M, Pepe M S. A mean score method for missing and auxiliary covariate data in regression models. Biometrika, 1995, 82: 299-314
|
14 |
Rosenblatt M. Remarks on some nonparametric estimates of a density function. Ann Math Stat, 1956, 27: 832-837
|
15 |
Rosner B, Willett W C, Spiegelman D. Correction of logistic regression relative risk estimates and confidence intervals for systematic within-person measurement error. Stat Med, 1989, 8: 1075-1093
|
16 |
Rudemo M. Empirical choice of histograms and kernel density estimators. Scand J Stat, 1982, 9: 65-78
|
17 |
Sepanski J H, Lee L F. Semiparametric Estimation of nonlinear error-in-variables models with validation study. J Nonparametr Stat, 1995, 4: 365-394
|
18 |
Silverman B W. On the estimation of a probability density function by the maximum penalized likelihood method. Ann Statist, 1982, 10: 795-810
|
19 |
Stefanski L A. Rates of convergence of some estimators in a class of deconvolution problems. Statist Probab Lett, 1990, 9: 229-235
|
20 |
Stefanski L A, Carroll R J. Conditional scores and optimal scores for generalized linear measurement error models. Biometrika, 1987, 74: 703-716
|
21 |
Stefanski L A, Carroll R J. Deconvoluting kernel density estimators. Statistics, 1990, 21: 169-184
|
22 |
Wahba G. Optimal convergence properties of variable knot, kernel and orthogonal series methods for density estimation. Ann Statist, 1975, 3: 15-29
|
23 |
Wang Q H. Estimation of partial linear error-in-variables model with validation data. J Multivariate Anal, 1999, 69: 30-64
|
24 |
Wang Q H. Estimation of linear error-in-covariables models with validation data under random censorship. J Multivariate Anal, 2000, 74: 245-266
|
25 |
Wang Q H, Härdle W. Empirical likelihood-based dimension reduction inference for linear error-in-responses models with validation study. Sci China Ser A, 2004, 47: 921-939
|
26 |
Wang Q H, Rao J N K. Empirical likelihood-based in linear errors-in-covariables models with validation data. 2003, 89: 345-358
|
27 |
Wittes J, Lakatos E, Probstfied J. Surrogate endpoints in clinical trails: Cardiovascular diseases. Stat Med, 1989, 8: 415-425
|
/
〈 | 〉 |