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Estimating a density on the positive half line by the method of orthogonal series

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Abstract

The kernel method of density estimation is not so attractive when the density has its support confined to (0, ∞), particularly when the density is unsmooth at the origin. In this situation the method of orthogonal series is competitive. We consider three essentially different orthogonal series—those based on the even and odd Hermite functions, respectively, and that based on Laguerre functions—and compare them from the point of view of mean integrated square error.

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References

  1. Abramowitz, M. and Stegun, I. A., (1965).Handbook of Mathematical Functions, Dover, New York.

    MATH  Google Scholar 

  2. Cencov, N. N., (1962). Evaluation of an unknown distribution density from observations,Soviet Math. Dokl.,3, 1559–1562.

    Google Scholar 

  3. Kronmal, R. and Tarter, M., (1968). The estimation of probability densities and cumulatives by Fourier series methods,J. Amer. Statist. Ass.,63, 925–952.

    MathSciNet  MATH  Google Scholar 

  4. Magnus, W., Oberhettinger, F. and Soni, R. P. (1966).Formulas and Theorems for the Special Functions of Mathematical Physics, Springer, Berlin.

    Book  Google Scholar 

  5. Parzen, E. (1962). On the estimation of a probability density function and mode,Ann. Math. Statist.,33, 1065–1076.

    Article  MathSciNet  Google Scholar 

  6. Rosenblatt, M., (1956). Remarks on some non-parametric estimates of a density function,Ann. Math. Statist.,27, 832–837.

    Article  MathSciNet  Google Scholar 

  7. Rosenblatt, M., (1971). Curve estimates,Ann. Math. Statist.,42, 1815–1841.

    Article  MathSciNet  Google Scholar 

  8. Sansone, G., (1959).Orthogonal Functions, Interscience, New York.

    MATH  Google Scholar 

  9. Schwartz, S. C., (1967). Estimation of a probability density by an orthogonal series,Ann. Math. Statist.,38, 1262–1265.

    MathSciNet  Google Scholar 

  10. Szegö, G. (1939).Orthogonal Polynomials, Amer. Math. Soc. Coll. Publ.,32, Providence.

  11. Walter, G. G., (1977). Properties of Hermite series estimation of probability density,Ann. Statist.,5, 1258–1264.

    Article  MathSciNet  Google Scholar 

  12. Watson, G. S., (1969). Density estimation by orthogonal series,Ann. Math. Statist.,38, 1262–1265.

    MathSciNet  Google Scholar 

  13. Watson, G. S. and Leadbetter, M. R., (1963). On the estimation of the probability density, I,Ann. Math. Statist.,34, 480–491.

    Article  MathSciNet  Google Scholar 

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  1. Australian National University, Australia

    Peter Hall

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  1. Peter Hall
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Hall, P. Estimating a density on the positive half line by the method of orthogonal series. Ann Inst Stat Math 32, 351–362 (1980). https://doi.org/10.1007/BF02480339

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  • DOI: https://doi.org/10.1007/BF02480339

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