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Approximating conditional density functions using dimension reduction. (English) Zbl 1176.62030

Summary: We propose to approximate the conditional density function of a random variable \(Y\) given a dependent random \(d\)-vector \(\mathbf X\) by that of \(Y\) given \(\pmb{\theta} ^{\tau }\mathbf{X}\), where the unit vector \(\pmb{\theta}\) is selected such that the average Kullback-Leibler discrepancy distance between the two conditional density functions obtains the minimum. Our approach is nonparametric as far as the estimation of the conditional density functions is concerned. We have shown that this nonparametric estimator is asymptotically adaptive to the unknown index \(\pmb{\theta}\) in the sense that the first order asymptotic mean squared error of the estimator is the same as that when \(\pmb{\theta}\) was known. The proposed method is illustrated using both simulated and real-data examples.

MSC:

62G07 Density estimation
62E17 Approximations to statistical distributions (nonasymptotic)
62G20 Asymptotic properties of nonparametric inference
62H12 Estimation in multivariate analysis
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[1] Amemiya, T. Advanced Econometrics. Harvard University Press, Harvard, Boston, 1985
[2] Bashtannyk, D.M., Hyndman, R.J. Bandwidth selection for kernel conditional density estimation. Computat. Statist. Data Anal., 36: 279–298 (2001) · Zbl 1038.62034 · doi:10.1016/S0167-9473(00)00046-3
[3] Bhattacharya, P.K., Gangopadhyay, A.K. Kernel and nearest-neighbor estimation of a conditional quantile. Ann. Statist., 18: 1400–1415 (1990) · Zbl 0706.62040 · doi:10.1214/aos/1176347757
[4] Bond, S.A. A review of asymmetric conditional density functions in autoregressive conditional heteroscedasticity models. A preprint, Department of Land Economy, University Cambridge, 2000
[5] Cai, Z. Regression quantiles for time series. Econometric Theory, 18: 169–192 (2002) · Zbl 1181.62124 · doi:10.1017/S0266466602181096
[6] Chen, X., Linton, O., Robinson, P.M. The estimation of conditional densities. In Asymptotics in Statistics and Probability, Festschrift for George Roussas, ed. M.L. Puri. VSP International Science Publishers, Netherlands, 2001
[7] Engle, R.F. Financial econometrics a new discipline with new methods. J. Econometrics, 100: 53–56 (2001) · Zbl 1099.62548 · doi:10.1016/S0304-4076(00)00053-1
[8] Fan, J., Gijbels, I. Local polynomial modelling and its applications. Chapman and Hall, London, 1996
[9] Fan, J., Hu, T.C., Truong, Y.K. Robust nonparametric function estimation. Scand. J. Statist., 21: 433–446 (1994) · Zbl 0810.62038
[10] Fan, J. and Yao, Q. Nonlinear Time Series. Springer, New York. 2003 · Zbl 1014.62103
[11] Fan, J., Yao, Q., Tong, H. Estimation of conditional densities and sensitivity measures in nonlinear dynamical systems. Biometrika, 83: 189–206 (1996) · Zbl 0865.62026 · doi:10.1093/biomet/83.1.189
[12] Fan, J., Yim, T.H. A data-driven method for estimating conditional densities. Biometrika, 91: 819–834 (2004) · Zbl 1078.62032 · doi:10.1093/biomet/91.4.819
[13] Hall, P., Wolff, R.C.L., Yao, Q. Methods for estimating a conditional distribution function. J. Amer. Statist. Assoc., 94: 154–163 (1998) · Zbl 1072.62558 · doi:10.2307/2669691
[14] Hall, P., Yao, Q. Conditional distribution function approximation, and prediction, using dimension reduction. Ann. Stats., 33: 1404–1421 (2005) · Zbl 1072.62008 · doi:10.1214/009053604000001282
[15] Hyndman, R.J. Highest density forecast regions for non-linear and non-normal time series models. J. Forecasting, 14: 431–441 (1995) · Zbl 04529628 · doi:10.1002/for.3980140503
[16] Hyndman, R.J., Bashtannyk, D.M., Grunwald, G.K. Estimating and visualizing conditional densities. J. Comp. Graph. Statist., 5: 315–336. (1996) · doi:10.2307/1390887
[17] Hyndman, R.J., Yao, Q. Nonparametric estimation and symmetry tests for conditional density functions. J. Nonparametric Statist., 14: 259–278 (2002) · Zbl 1013.62040 · doi:10.1080/10485250212374
[18] Li, F., Tkacz, G. A consistent bootstrap test for conditional density functions with time-dependent data. Working paper #01-21, Band of Canada, 2001
[19] Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P. Numerical Recipes in C. Cambridge University Press, Cambridge, 1992 · Zbl 0778.65003
[20] Polonik, W., Yao, Q. Set-indexed conditional empirical and quantile processes based on dependent data. J. Ameri. Statist. Assoc., 80: 234–255 (2002) · Zbl 0992.62094
[21] Rosenblatt, M. Conditional probability density and regression estimators. In Multivariate Analysis II, ed. P. Krishnaiah. Academic, New York, 1969
[22] Roussas, G. Nonparametric estimation in Markov processes. Ann. Inst. Statist. Math., 21: 73–87 (1967) · Zbl 0181.45804 · doi:10.1007/BF02532233
[23] Roussas, G. Nonparametric estimation of the transition distribution function of a Markov process. Ann. Math. Statist., 40: 1386–1400 (1969) · Zbl 0188.50501 · doi:10.1214/aoms/1177697510
[24] Sheather, S.J., Marron, J.S. Kernel quantile estimators. J. Amer. Statist. Assoc., 85: 410–416 (1990) · Zbl 0705.62042 · doi:10.2307/2289777
[25] Silverman, B. Density Estimation for Statistics and Data Analysis. Chapman and Hall, London, 1996 · Zbl 0617.62042
[26] Stute, W. The oscillation behavior of empirical processes: the multivariate case. Ann. Prob., 12: 361–379 (1984) · Zbl 0533.62037 · doi:10.1214/aop/1176993295
[27] Tiao, G.C., Tsay, R.S. Some advances in nonlinear and adaptive modeling in time series. J. Forecasting, 13: 109–131 (1994) · Zbl 04518315 · doi:10.1002/for.3980130206
[28] van der Vaart, A.W. Asymptotic Statistics. Cambridge University Press, Cambridge, 1998 · Zbl 0910.62001
[29] Yao, Q., Tong, H. On prediction and chaos in stochastic systems. Philosophical Transaction of the Royal Society, (London) A, 348: 357–369 (1994) · Zbl 0859.62086 · doi:10.1098/rsta.1994.0096
[30] Yu, K., Jones, M.C. Local linear quantile regression. J. Amer. Statist. Assoc., 93: 228–237 (1998) · Zbl 0906.62038 · doi:10.2307/2669619
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