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

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Abstract

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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References

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

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

    Google Scholar 

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

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

    Google Scholar 

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

  26. 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).

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

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

    M. Sköld

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  1. M. Sköld
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Sköld, M. The Asymptotic Variance of the Continuous-Time Kernel Estimator with Applications to Bandwidth Selection. Statistical Inference for Stochastic Processes 4, 99–117 (2001). https://doi.org/10.1023/A:1017520326698

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