An overview of nonparametric contributions to the problem of functional estimation from biased data

Test

Volumn 10, Issue 2, 2001, Pages 309-332

  Cristóbal, José A ^a   Alcalá, José T ^a  

^a UNIVERSITY OF ZARAGOZA   (Spain)

Author keywords

Biased sampling; Density estimation; Length biased data; Nonparametric inference; Regression estimation; Semiparametric models; Weighted distributions

Indexed keywords

EID: 0347771988     PISSN: 11330686     EISSN: None     Source Type: Journal    
DOI: 10.1007/BF02595700     Document Type: Review

References (84)

1. Ahmad, I.A. (1995). On multivariate kernel estimation for samples from weighted distributions. Statistics and Probability Letters, 22, 121-129.
2. Alcalá, J.T., J.A. Cristóbal and J. Ojeda (2000). Nonparametric regression estimators in biased sampling models. In Statistical Modelling (V. Núñez-Antón and E. Ferreira, eds.) Universidad del País Vasco, Bilbao, 131-136.
3. Bayarri, M.J. and M.H. DeGroot (1992). A "BAD" view of weighted distributions and selection models. In Bayesian statistics, 4, 17-33. Oxford University Press, New York.
4. Bhattacharyya, B.B., L.A. Franklin and G.D. Richardson (1988). A comparison of nonparametric unweighted and length-biased density estimation of fibres. Communications in Statistics, Theory and Methods, 17, 3629-3644.
5. Bickel, P.J., C.A. Klaassen, Y. Ritov and J.A. Wellner (1993). Efficient and Adaptive Estimation for Semiparametric Models. Johns Hopkins University Press, Baltimore.
6. Bickel, P.J. and M. Rosenblatt (1973). On some global measures of the deviations of density function estimates. Annals of Statistics, 1, 1071-1095.
7. Breslow, N.E. (1996). Statistics in epidemiololgy: the case-control study. Journal of the American Statistical Association, 91, 14-28.
8. Breslow, E., J.M. Robins and J.A. Wellner (2000). On the semi-parametric efficiency of logistic regression under case-control sampling. Bernoulli, 6, 447-455.
9. Burnham, K.P., D.R. Anderson, and J.L. Laake (1980). Estimation of density from line transect sampling of biological populations. Wildlife Monograph, 72, supp. to Journal of Wildlife Management, 44.
10. Cao, R., P. Janssen and N. Veraverbeke (2000). Relative density estimation with censored data. Canadian Journal of Statistics, 28, 97-111.
11. Coleman, R. (1972). Sampling procedures for the lengths of random straight lines. Biometrika, 59, 415-426.
12. Cook, R.C. and F.B. Martin (1974). A model for quadrat sampling with visibility bias. Journal of the American Statistical Association, 69, 345-349.
13. Cosslett, S.R. (1981). Efficient estimation of discrete choice models. In Structural Analysis of Discrete Data with Econometric Applications, 51-111 (C.F. Mansky and D. McFadden, eds.) MIT Press, Cambridge.
14. Cox, D.R. (1969). Some sampling problems in technology. In New Developments in Survey Sampling, 506-527 (N. L. Johnson and H. Smith, eds.) John Wiley, New York.
15. Cristóbal, J.A. and J.T. Alcalá (2000). Nonparametric regression estimators for length biased data. Journal of Statistical Planning and Inference, 89, 145-168.
16. Cristóbal, J.A., J. Ojeda and J.T. Alcalá (2001). Confidence bands in nonparametric regression with length biased data. Seminario Matemático García de Galdeano II, 1, 1-28.
17. Ćwik, J. and J. Mielniczuk (1993). Data-dependent bandwidth choice for a grade density kernel estimate. Statistics and Probability Letters, 16, 397-405.
18. de Uña-Álvarez, J. (2000a) Product-limit estimation for length-biased censored data. Unpublished manuscript.
19. de Uña-Álvarez, J. (2000b) Large sample results under length-biased sampling when covariables are present. Unpublished manuscript.
20. Drummer, T.D. and L.L. McDonald (1987). Size bias in line transect sampling. Biometrics, 43, 13-21.
21. El Barmi, H. and M.D. Rothmann (1998). Nonparametric estimation in selection biased models in the presence of estimating equations. Nonparametric Statistics, 9, 381-399.
22. El Barmi, H. and M.D. Rothmann (1999). Estimation of weighted multinomial probabilities under log-convex constraints. Journal of Statistical Planning and Inference, 81, 1-11.
23. El Barmi, H. and J.S. Simonoff (2000). Transformation-based density estimation for weighted distributions. Nonparametric Statistics, 12, 861-878.
24. Fan, J. and I. Gijbels (1996). Local Polynomial Modelling and its Applications. Chapman and Hall, London.
25. Feller, W. (1966). Introduction to Probability Theory and Applications, 2. John Wiley, New York.
26. Fisher, R.A. (1934). The effects of methods of ascertainment upon the estimation of frequencies. Annals of Eugenics, 6, 13-25.
27. Gilbert, P.B. (2000). Large sample theory of maximum likelihood estimates in semiparametric biased sampling models. Annals of Statistics, 28, 151-194.
28. Gilbert, P.B., S.R. Lele and Y. Vardi (1999). Maximum likelihood estimation in semiparametric selection bias models with application to AIDS vaccine trials. Biometrika, 86, 27-43.
29. Gill, R.D., Y. Vardi and J.A. Wellner (1988). Large sample theory of empirical distributions in biased sampling models. Annals of Statistics, 16, 1069-1112.
30. Guillamón, A., J. Navarro and J.M. Ruiz (1998). Kernel density estimation using weighted data. Communications in Statistics, Theory and Methods, 27, 2123-2135.
31. Hanin, L.G., S.T. Rachev, A.D. Tsodikov and Y. Yakovlev (1997). A stochastic model of carcinogenesis and tumor size at detection. Advances in Applied Probability, 29, 607-628.
32. Holt, D., T.M.F. Smith and P.D. Winter (1980). Regression analysis of data from complex surveys. Journal of the Royal Statistical Society, A, 143, 474-487.
33. Horowitz, J.L. (1996). Semiparametric estimation of a regression model with an unknown transformation of the dependent variable. Econometrica, 64, 103-137.
34. Horvàth, L. (1985). Estimation from a length-biased distribution. Statistics and Decisions, 3, 91-113.
35. Jewell, N.P. (1985). Least squares regression with data arising from stratified samples of the dependent variable. Biometrika, 72, 11-21.
36. Jones, M.C. (1991). Kernel density estimation for length biased data. Biometrika, 78, 511-519.
37. Jones, M.C. and R.J. Kaunamuni (1997). Fourier series estimation for length biased data. Australian Journal of Statistics, 39, 57-68.
38. Kay, R. and S. Little (1987). Transformation of the explanatory variables in the logistic regression model for binary data. Biometrika, 74, 495-501.
39. Klein, R. and R. Sherman (1997). Estimating new product demand from biased survey data. Journal of Econometrics, 76, 53-76.
40. Li, G. (1995). Nonparametric likelihood ratio estimation of probabilities for truncated data. Journal of the American Statistical Association, 90, 997-1003.
41. Li, G. and J. Qin (1998). Semiparametric likelihood-based inference for biased and truncated data when the total sample size is known. Journal of the Royal Statistical Society, B, 60, 243-254.
42. Li, G., J. Qin and C. Tiwari (1997). Semiparametric likelihood ratio-based inferences for truncated data. Journal of the American Statistical Association, 92, 236-245.
43. Lloyd, J. and M.C. Jones (2000). Nonparametric density estimation from biased data with unknown biasing function. Journal of the American Statistical Association, 95, 865-876.
44. Manski, C.F. and S. Lerman (1977). The estimation of choice probabilities from choice-based samples. Econometrica, 45, 1977-1988.
45. Moøller, J. (1988). Stereological analysis of particles of varying ellipsoidal shape. Journal of Applied Probability, 25, 322-335.
46. Nair, V.N. and P.C.C. Wang (1989). Maximum likelihood estimation under a successive sampling discovery model. Technometrics, 31, 423-436.
47. Nathan, G. and D. Holt (1980). The effect of survey design on regression analysis. Journal of the Royal Statistical Society, B, 42, 377-386.
48. Nicoll, J.F. and I.E. Segal (1982). Spatial homogeneity and redshift-distance law. Proceedings of the National Academy of Sciences, 79, 3913-3917.
49. Owen, A. (1988). Empirical likelihood ratio confidence intervals for a single functional. Biometrika, 75, 237-249.
50. Patil, G.P. (1984). Studies in statistical ecology involving weighted distributions. In Statistics: Applications and New Directions. Indian Statistical Institute, 478-503.
51. Patil, G.P. and C.R. Rao (1977). The weighted distributions: a survey of their applications. In Applications of Statistics, 383-405 (R.R. Krishnaiah, ed.). North-Holland.
52. Patil, G.P. and C.R. Rao (1978). Weighted distributions. In Encyclopaedia of Statistical Sciences, 9, 565-571 (S. Kotz and N. L. Johnson, eds.) John Wiley.
53. Qin, J. (1993). Empirical likelihood in biased sample problem. Annals of Statistics, 21, 1182-1196.
54. Qin, J. (1998). Inferences for case-control and semiparametric two-sample density ratio models. Biometrika, 85, 619-630.
55. Qin, J. and J.F. Lawless (1994). Empirical likelihood and general estimating equations. Annals of Statistics, 22, 300-325.
56. Quang, P.X. (1991). A nonparametric approach to size-biased line transect sampling. Biometrics, 47, 269-279.
57. Quesenberry, C.P. and N.P. Jewell (1986). Regression analysis based on stratified samples. Biometrika, 73, 605-614.
58. Rao, C.R. (1965). On discrete distributions arising out of methods of ascertainment. In Classical and Contagious Discrete Distributions, 320-332 (G.P. Patil, ed.) Pergamon Press and Statistical Publishing Society, Calcutta.
59. Rao, C.R. (1977). A natural example of a weighted binomial distribution. American Statistician, 31, 24-26.
60. Rao, C.R. (1989). Statistics and truth. World Scientific, Singapore.
61. Richardson, G.D., M.K. Kazempour and B.B. Bhattacharyya (1991). Length biased density estimation of fibres. Nonparametric Statistics, 1, 127-141.
62. Robins, J.M., A. Rotnizky and L.P. Zhao (1994). Estimation of regression coefficients when some regressors are not always observed. Journal of the American Statistical Association, 89, 846-866.
63. Sen, P.K. (1984). On asymptotic representations for reduced quantiles in sampling from a length-biased distribution. Calcutta Statistical Association Bulletin, 33, 59-67.
64. Sen, P.K. (1997). On estimators of bundle-strength in length-biased sampling schemes. In Probability and its Applications, 120-134 (M.C. Bhattacharjee and S.K. Basu, eds.) Oxford University Press. New Delhi.
65. Sköld, M. (1999). Kernel regression in the presence of size-bias. Nonparametric Statistics, 12, 41-51.
66. Smart, R.J. (1963). Alcoholism, birth order, and family size. Journal Abnorm. Society of Psychology, 66, 17-23.
67. Sprott, D.A. (1964). Use of chi square. Journal Abnorm. Society of Psychology, 69, 101-103.
68. Suh, M.W., B.B. Bhattacharya and A.H.E. Grandage (1970). On the distribution and moments of the strength of a bundle of filaments. Journal of Applied Probability, 7, 712-720.
69. Sun, J. and M. Woodroofe (1997). Semi-parametric estimates under biased sampling. Statistica Sinica, 7, 545-575.
70. Tsui, K.L., N.P. Jewell and C.F.J. Wu (1988). A nonparametric approach to the truncated regression problem. Journal of the American Statistical Association, 83, 785-792.
71. Turner, E.L. (1979). Statistics of the Hubble diagram I: Determination of q0 and luminosity evolution with application to Quasars. Astrophysical Journal, 230, 291-303.
72. Vardi, Y. (1982). Nonparametric estimation in the presence of length bias. Annals of Statistics, 10, 616-620.
73. Vardi, Y. (1985). Empirical distributions in selection bias models. Annals of Statistics, 13, 178-205.
74. Vardi, Y. (1988). Statistical models for intercepted data. Journal of the American Statistical Association, 83, 183-197.
75. Wang, M.C. (1989). A semiparametric model for randomly truncated data. Journal of the American Statistical Association, 84, 742-748.
76. Weinberg, C.R. and D.P. Sandler (1991). Randomized recruitment in case-control studies. American Journal of Epidemiology, 134, 421-432.
77. Winter, B.B. and A. Földes (1988). A product-limit estimator for use with length-biased data. Canadian Journal of Statistics, 16, 337-355.
78. Woodroofe, M. (1985). Estimating a distribution function with truncated data. Annals of Statistics, 13, 163-177.
79. Wu, C.O. (1996). Kernel smoothing of the nonparametric maximum likelihood estimates for biased sampling models. Mathematical Methods in Statistics, 5, 275-298.
80. Wu, C.O. (1997). A cross-validation bandwidth choice for kernel density estimates with selection biased data. Journal of Multivariate Analysis, 61, 38-60.
81. Wu, C.O. (2000). Local Polynomial regression with selection biased data. Statistics Sinica, 10, 789-817.
82. Wu, C.O. and A.Q. Mao (1996). Minimax kernels for density estimation with biased data. Annals of the Institute of Statistical Mathematics, 48, 451-467.
83. Zhang, B. (2000a). M-estimation under a two-sample semiparametric model. Scandinavian Journal of Statistics, 27, 263-280.
84. Zhang, B. (2000b). Quantile estimation under a two-sample semi-parametric model. Bernoulli, 6, 491-511.