Page 315 - Introduction to Statistical Pattern Recognition

6 Nonparametric Density Estimation 297

9. A density function is given in the figure. Find the Hermite expansion up
to the fourth term and show how closely the expansion approximates the
density function.
p(x)
t

10. A density function of three binary inputs is given in the table.
(a) Show the Walsh expansion. (b) Show the Bahadur expansion.

-1 -1

+I +I +!

References

1. M. Rosenblatt, Remarks on some nonparametric estimates of a density
function, Ann. Math. Stat., 27, pp. 832-837, 1956.
2. E. Parzen, On the estimation of a probability density function and the
mode, Ann. Math. Stat., 33, pp. 1065-1076, 1962.
3. T. Cacoullos, Estimation of a multivariate density, Ann. Inst. Sfat. Math..
18, pp. 178-189, 1966.
4. L. Devroye and L. Gyorfi, “Nonparametric Density Estimation: The L,
View,” John Wiley, New York, 1985.