×
Hall, Peter; Yao, Qiwei

Approximating conditional distribution functions using dimension reduction. (English) Zbl 1072.62008

Ann. Stat. 33, No. 3, 1404-1421 (2005).
Summary: Motivated by applications to prediction and forecasting, we suggest methods for approximating the conditional distribution function of a random variable \(Y\) given a dependent random \(d\)-vector \(X\). The idea is to estimate not the distribution of \(Y\mid X\), but that of \(Y\mid\theta^TX\), where the unit vector \(\theta\) is selected so that the approximation is optimal under a least-squares criterion. We show that \(\theta\) may be estimated root-\(n\) consistently. Furthermore, estimation of the conditional distribution function of \(Y\), given \(\theta^T X\), has the same first-order asymptotic properties that it would enjoy if \(\theta\) were known.
The proposed method is illustrated using both simulated and real-data examples, showing its effectiveness for both independent datasets and data from time series. Numerical work corroborates the theoretical result that \(\theta\) can be estimated particularly accurately.

Cited in 27 Documents

MSC:

62E17 Approximations to statistical distributions (nonasymptotic)
62G05 Nonparametric estimation
65C60 Computational problems in statistics (MSC2010)
62G20 Asymptotic properties of nonparametric inference
62G08 Nonparametric regression and quantile regression

Keywords:

cross validation; dimension reduction; kernel methods; leave-one-out method; local linear regression; root-\(n\) consistency; time series analysis; prediction
Cite Review PDF
Full Text: DOI arXiv

References:

[1] Adali, T., Liu, X. and Sonmez, M. K. (1997). Conditional distribution learning with neural networks and its application to channel equalization. IEEE Trans. Signal Processing 45 1051–1064.
[2] Bashtannyk, D. M. and Hyndman, R. J. (2001). Bandwidth selection for kernel conditional density estimation. Comput. Statist. Data Anal. 36 279–298. · Zbl 1038.62034 · doi:10.1016/S0167-9473(00)00046-3
[3] Bhattacharya, P. K. and Gangopadhyay, A. K. (1990). Kernel and nearest-neighbor estimation of a conditional quantile. Ann. Statist. 18 1400–1415. JSTOR: · Zbl 0706.62040 · doi:10.1214/aos/1176347757
[4] Bond, S. A. and Patel, K. (2000). The conditional distribution of real estate returns: Relating time variation in higher moments to downside risk measurement. Technical report, Dept. Land Economy, Univ. Cambridge.
[5] Cai, Z. (2002). Regression quantiles for time series. Econometric Theory 18 169–192. · Zbl 1181.62124 · doi:10.1017/S0266466602181096
[6] Fan, J., Heckman, N. E. and Wand, M. P. (1995). Local polynomial kernel regression for generalized linear models and quasi-likelihood functions. J. Amer. Statist. Assoc. 90 141–150. · Zbl 0818.62036 · doi:10.2307/2291137
[7] Fan, J. and Yao, Q. (2003). Nonlinear Time Series : Nonparametric and Parametric Methods . Springer, New York. · Zbl 1014.62103
[8] Fan, J., Yao, Q. and Tong, H. (1996). Estimation of conditional densities and sensitivity measures in nonlinear dynamical systems. Biometrika 83 189–206. · Zbl 0865.62026 · doi:10.1093/biomet/83.1.189
[9] Foresi, S. and Paracchi, F. (1992). The conditional distribution of excess returns: An empirical analysis. Working Paper 92-49, C. V. Starr Center, New York Univ.
[10] Friedman, J. H. (1987). Exploratory projection pursuit. J. Amer. Statist. Assoc. 82 249–266. · Zbl 0664.62060 · doi:10.2307/2289161
[11] Friedman, J. H. and Stuetzle, W. (1981). Projection pursuit regression. J. Amer. Statist. Assoc. 76 817–823.
[12] Friedman, J. H., Stuetzle, W. and Schroeder, A. (1984). Projection pursuit density estimation. J. Amer. Statist. Assoc. 79 599–608.
[13] Hall, P. and Heyde, C. C. (1980). Martingale Limit Theory and Its Application. Academic Press, New York. · Zbl 0462.60045
[14] Hall, P., Wolff, R. C. L. and Yao, Q. (1999). Methods for estimating a conditional distribution function. J. Amer. Statist. Assoc. 94 154–163. · Zbl 1072.62558 · doi:10.2307/2669691
[15] Hall, P. and Yao, Q. (2002). Estimating conditional distribution functions using dimension reduction. Research Report 87, Dept. Statistics, London School of Economics. Available at www.lse.ac.uk/collections/statistics/documents/researchreport87.pdf.
[16] Härdle, W., Hall, P. and Ichimura, H. (1993). Optimal smoothing in single-index models. Ann. Statist. 21 157–178. JSTOR: · Zbl 0770.62049 · doi:10.1214/aos/1176349020
[17] Huber, P. J. (1985). Projection pursuit (with discussion). Ann. Statist. 13 435–525. JSTOR: · Zbl 0595.62059 · doi:10.1214/aos/1176349519
[18] Hyndman, R. J., Bashtannyk, D. M. and Grunwald, G. K. (1996). Estimating and visualizing conditional densities. J. Comput. Graph. Statist. 5 315–336.
[19] Hyndman, R. J. and Yao, Q. (2002). Nonparametric estimation and symmetry tests for conditional density functions. J. Nonparametr. Statist. 14 259–278. · Zbl 1013.62040 · doi:10.1080/10485250212374
[20] Ichimura, H. (1993). Semiparametric least squares (SLS) and weighted SLS estimation of single-index models. J. Econometrics 58 71–120. · Zbl 0816.62079 · doi:10.1016/0304-4076(93)90114-K
[21] Jones, M. C. and Sibson, R. (1987). What is projection pursuit? (with discussion). J. Roy. Statist. Soc. Ser. A 150 1–36. · Zbl 0632.62059 · doi:10.2307/2981662
[22] Klein, R. and Spady, R. (1993). An efficient semiparametric estimator for binary response models. Econometrica 61 387–422. · Zbl 0783.62100 · doi:10.2307/2951556
[23] Li, K.-C. (1991). Sliced inverse regression for dimension reduction (with discussion). J. Amer. Statist. Assoc. 86 316–342. · Zbl 0742.62044 · doi:10.2307/2290563
[24] Moran, P. A. P. (1953). The statistical analysis of the Canadian lynx cycle. I. Structure and prediction. Australian J. Zoology 1 163–173.
[25] Posse, C. (1995). Projection pursuit exploratory data analysis. Comput. Statist. Data Anal. 20 669–687. · Zbl 0875.62206 · doi:10.1016/0167-9473(95)00002-8
[26] Powell, J. L., Stock, J. H. and Stoker, T. M. (1989). Semiparametric estimation of index coefficients. Econometrica 57 1403–1430. · Zbl 0683.62070 · doi:10.2307/1913713
[27] Press, W. H., Teukolsky, S. A., Vetterling, W. T. and Flannery, B. P. (1992). Numerical Recipes in C. The Art of Scientific Computing , 2nd ed. Cambridge Univ. Press. · Zbl 0845.65001
[28] Rosenblatt, M. (1969). Conditional probability density and regression estimators. In Multivariate Analysis II (P. Krishnaiah, ed.) 25–31. Academic Press, New York.
[29] Sheather, S. J. and Marron, J. S. (1990). Kernel quantile estimators. J. Amer. Statist. Assoc. 85 410–416. · Zbl 0705.62042 · doi:10.2307/2289777
[30] Tiao, G. C. and Tsay, R. S. (1994). Some advances in nonlinear and adaptive modeling in time series. J. Forecasting 13 109–131.
[31] Watanabe, T. (2000). Excess kurtosis of conditional distribution for daily stock returns: The case of Japan. Applied Economics Letters 7 353–355.
[32] Yin, X. and Cook, R. D. (2002). Dimension reduction for the conditional \(k\)th moment in regression. J. R. Stat. Soc. Ser. B Stat. Methodol. 64 159–175. · Zbl 1067.62042 · doi:10.1111/1467-9868.00330
[33] Yu, K. and Jones, M. C. (1998). Local linear quantile regression. J. Amer. Statist. Assoc. 93 228–237. · Zbl 0906.62038 · doi:10.2307/2669619
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.