The Asymptotic Variance of the Continuous-Time Kernel Estimator with Applications to Bandwidth Selection

M. Sköld1
1Mathematical Statistics, Centre for Mathematical Sciences, Lund University, Sweden

Tóm tắt

We derive simple expressions for the asymptotic variance of the kernel-density estimator of a stationary continuous-time process in one and d dimensions and relate convergence rates to sample path smoothness. Important applications include methods for selecting optimal smoothing parameters and construction of confidence bands for testing hypotheses about the density. In a simulation study the results are applied to bandwidth selection for discrete-time processes that can be modelled as continuous-time processes sampled at a high rate.

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Tài liệu tham khảo

Aït-Sahalia, Y.: Nonparametric pricing of interest rate derivative securities, Econometrica 64 (1996), 527–560.

Aït-Sahalia, Y.: Testing continuous-time models of the spot interest rate, Rev. of Financ. Stud. 2 (1996), 385–426.

Blanke, D. and Bosq, D.: Accurate rates of density estimators for continuous-time processes, Statist. Probab. Lett. 33 (1997), 185–191.

Blanke, D. and Bosq, D.: A family of minimax rates for density estimation in continuous time, Stoch. Anal. Appl. 18 (2000), 871–900.

Bosq, D.: Nonparametric Statistics for Stochastic Processes: Estimation and Prediction. Springer Lecture Notes in Statistics 110, Springer, 1996.

Bosq, D. and Davydov, Y.: Local time and density estimation in continuous time, Math. Methods Statist. 8 (1999), 22–45.

Bosq, D. and Merlevede, F.: Asymptotic normality for density kernel estimators in discrete and continuous time, J. Multivariate Anal. 68 (1999), 78–95.

Bowman, A.: An alternative method of cross-validation for the smoothing of density estimates, Biometrika 71 (1984), 353–360.

Cacoullos, T.: Estimation of a multivariate density, Ann. Inst. Statist. Math. 18 (1966), 179–189.

Castellana, J.V. and Leadbetter, M.R.: On smoothed probability density estimation for stationary processes, Stochastic Process. Appl. 21 (1986), 179–193.

Hall, P., Lahiri, S.N. and Truong, Y.K.: On bandwidth choice for density estimation with dependent data, Ann. Statist. 23 (1995), 2241–2263.

Hart, J.D.: Some automated methods of smoothing time-dependent data, J. Nonparametric Statist. 6 (1996), 115–142.

Hart, J.D. and Vieu, P.: Data-driven bandwidth choice for density estimation based on dependent data, Ann. Statist. 18 (1990), 873–890.

Hart, J.D. and Wehrly, T.: Kernel regression estimation using repeated measurements data, J. Amer. Statist. Assoc. 18 (1986), 1080–1088.

Kim, T.Y. and Cox, D.D.: A study on bandwidth selection in density estimation under dependence, J. Multivariate Anal. 62 (1997), 190–203.

Kutoyants, Yu.: Some problems of nonparametric estimation by observations of ergodic diffusion process, Statist. Probab. Lett. 32 (1997), 311–320.

Leadbetter, M.R.: On crossings of levels and curves by a wide class of stochastic processes, Ann. Math. Statist. 37 (1965), 260–267.

Müller, H.-G.: Empirical bandwidth choice for nonparametric kernel regression by the mean of pilot estimators, Statist. Decisions 2 (1985), 193–206.

Pritsker, M.: Nonparametric density estimation and tests of continuous time interest rate models, Rev. Finan. Stud. 11 (1998), 449–487.

Rudemo, M.: Empirical choice of histograms and kernel density estimates, Scand. J. Statist. 9(1982), 65–78.

Rychlik I. and Lindgren, G.: Wave analysis toolbox — a tutorial, Technical Report 1995:1, Department of Mathematical Statistics, Lund University, 1995.

Scott, D.W.: Multivariate Density Estimation: Theory Practise and Visualization, Wiley, New York, 1992.

Silverman, B.W.: Density Estimation of Statistics and Data Analysis, Chapman and Hall, 1986.

Sköld, M.: Kernel intensity estimation for marks and crossing of differentiable stochastic processes, Theory Stochastic Process. 2 (1–2) (1996), 273-284.

Sköld, M.: Continuous-Time Models in Kernel Smoothing, Doctoral Theses in Mathematical Sciences 1995:5, Lund University, 1999.

Sköld, M.: A bias-correction for cross-validation bandwidth selection when a kernel estimate is based on dependent data, J. Time Series Anal (2001), (In press).

Sköld, M. and Hössjer, O.: On the asymptotic variance of the continuous-time kernel density estimator. Statist. Probab. Lett. 44 (1999), 97–106.

Staniswalis, J.G.: Local bandwidth selection of kernel estimates, J. Amer. Statist. Assoc. 84 (1989), 284–288.

Stanton, R.: A nonparametric model of term structure dynamics and the market price of interest rate risk, J. Finan. 52 (1997), 1973–2002.

Tran, L.T.: On multivariate variable-kernel density estimates for time series, Canad. J. Statist. 19 (1991), 371–387.

Wand, M.P. and Jones M.C.: Kernel Smoothing, Chapman and Hall, 1995.