Spectral and Probability Density Estimation From Irregularly Observed Data

Conference paper
Part of the Lecture Notes in Statistics book series (LNS, volume 25)


Let X = {X(t), − ∞ < t < ∞} be a stationary stochastic process with univariate probability density f(x), covariance function R(t), and spectral density φ(λ). The nonparametric estimation of f, R, and φ on the basis of irregularly-spaced observations \(\left\{ {{\text{X}}\left( {{\text{t}}_{\text{k}} } \right){\text{,t}}_{\text{k}} } \right\}_{{\text{k = 1}}}^{\text{n}}\) is considered. The sampling schemes {tk} which allow the consistent estimation of f, R, and φ, as the sample size n → ∞, are identified and the asymptotic statistical properties of the corresponding estimates \({\hat f_n}\left( x \right),{\hat f_n}\left( t \right)\), and \(\hat \varphi _{\text{n}} \left( \lambda \right)\) are presented.


Point Process Sampling Instant Stationary Stochastic Process Probability Density Estimation Random Sampling Scheme 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [1]
    Banon, G. (1978). “Nonparametric identification for diffusion processes.” SIAM J. Control and Optimization. 16: 380–395.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    Beutler, F.J. and Leneman, O.A.Z. (1966). “The theory of stationary point processes.” Acta Math. 116: 159–197.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    Beutler, F.J. (1970). “Alias free randomly timed sampling of stochastic processes.” IEEE Trans. Information Theory. 16: 147–152.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    Brillinger, D.R. (1972). “The spectral analysis of stationary interval functions.” In Proc. Sixth Berkeley Symp. Prob. Statist. Eds. L. Lecam, J. Neyman, and E.L. Scott, pp. 483–513. Berkeley: Univ. of Calif. Press.Google Scholar
  5. [5]
    Brillinger, D.R. (1975). Time Series Data Analysis and Theory, New York: Holt, Rinehart and Winston.zbMATHGoogle Scholar
  6. [6]
    Castellana, J.V. (1982). “Nonparametric density estimation for stationary stochastic processes.” Ph.D. dissertation, Statistics Department, University of North Carolina, Chapel Hill, NC.Google Scholar
  7. [7]
    Daley, D.J. and Vere-Jones, D. (1972). “A summary of the theory of point processes.” In Stochastic Point Processes, Ed. P.A.W. Lewis, New York: Wiley.Google Scholar
  8. [8]
    Delecroix, M. (1980). “Sur l’estimation des densities d’un processus stationaire a temps continu.” Publ. Institute Stat. Univ. Paris. 25: 17–39.MathSciNetzbMATHGoogle Scholar
  9. [9]
    Durranti, T.S. and Greated, C.A. (1977). Laser Systems in Flow Measurements, New York: Plenum.CrossRefGoogle Scholar
  10. [10]
    Fryer, M.J. (1977). “A review of some non-parametric methods of density estimation.” J. Inst. Math. Applics. 20: 335–354.MathSciNetzbMATHCrossRefGoogle Scholar
  11. [11]
    Gyorfi, L. (1981). “Strong consistent density estimate from ergodic sample.” J. Multivariate Analysis. 11: 81–84.MathSciNetCrossRefGoogle Scholar
  12. [12]
    Jenkins, G.M. and Watts, D.G. (1968). Spectral Analysis and its Applications, San Francisco: Holden-Day.zbMATHGoogle Scholar
  13. [13]
    Karlin, S. and Taylor, H.M. (1975). A First Course in Stochastic Processes, New York: Academic Press.zbMATHGoogle Scholar
  14. [14]
    Masry, E. (1978a). “Poisson sampling and spectral estimation of continuous-time processes.” IEEE Trans. Information Theory. 24: 173–183.MathSciNetzbMATHCrossRefGoogle Scholar
  15. [15]
    Masry, E. (1978b). “Alias-free sampling: An alternative conceptualization and its applications.” IEEE Trans. Information Theory. 24: 317–324.MathSciNetzbMATHCrossRefGoogle Scholar
  16. [16]
    Masry, E. (1980). “Discrete-time spectral estimation of continuous-time processes—the orthogonal series method.” Annals Statist. 8: 1100–1109.MathSciNetzbMATHCrossRefGoogle Scholar
  17. [17]
    Masry, E. (1983). “Nonparametric covariance estimation from irregularly-spaced data.” Advances in Appl. Prob. 15: 113–132.MathSciNetzbMATHCrossRefGoogle Scholar
  18. [18]
    Masry, E. (1983). “Probability density estimation from sampled data.” IEEE Trans. Information Theory. 29, to appear.Google Scholar
  19. [19]
    Nguyen, H.T. (1979). “Density estimation in a continuous time stationary Markov process.” Ann. Statist. 7: 341–348.MathSciNetzbMATHCrossRefGoogle Scholar
  20. [20]
    Parzen, E. (1957). “On consistent estimates of the spectrum of a stationary time series.” Annals Math. Statist. 28: 329–348.MathSciNetzbMATHCrossRefGoogle Scholar
  21. [21]
    Parzen, E. (1962). “On estimation of a probability density function and mode.” Annals Math. Statist. 33: 1065–1076.MathSciNetzbMATHCrossRefGoogle Scholar
  22. [22]
    Robinson, P.M. “Nonparametric estimators for time series.” J. Time Series Analysis. To appear.Google Scholar
  23. [23]
    Rosenblatt, M. (1970). “Density estimates and Markov sequences.” In Nonparametric Techniques in Statistical Inference, Ed., M. Puri, Cambridge: Cambridge University Press.Google Scholar
  24. [24]
    Roussas, G.G. (1969). “Non-parametric estimation of the transition distribution of a Markov process.” Ann. Instit. Statist. Math. 21: 73–87.MathSciNetzbMATHCrossRefGoogle Scholar
  25. [25]
    Schwartz, S. (1967). “Estimation of density function by orthogonal series.” Annals Math. Statist. 38: 1261–1265.zbMATHCrossRefGoogle Scholar
  26. [26]
    Shapiro, H.S. and Silverman, R.A. (1960). “Alias-free sampling of random noise.” J. Soc. Indust. Appl. Math. 8: 225–248.MathSciNetzbMATHCrossRefGoogle Scholar
  27. [27]
    Wahba, G. (1975). “Interpolation spline methods for density estimation I. Equi-spaced knots.” Annals Statist. 3: 30–48.MathSciNetzbMATHCrossRefGoogle Scholar
  28. [28]
    Wegman, E.J. (1972). “Nonparametric probability density estimation: I. A summary of available methods.” Technometrics. 14: 533–546.zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1984

Authors and Affiliations

  1. 1.Department of Electrical Engineering and Computer SciencesUniversity of CaliforniaSan Diego, La JollaUSA

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