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SPEAKER: |
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TITLE: |
"Density Estimates Based on Orthogonal Polynomials" |
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DAY: |
Wednesday, April 4, 2001 |
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TIME: |
3:30 p.m. [Tea & cookies in BSB-202 at 3:00 p.m.] |
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PLACE: |
BSB-108 |
It is often the case that the moments of a distribution can be readily determined, while its exact density function either does not lend itself to numerical evaluation or proves to be mathematically intractable. We will show that probability mass functions and probability density functions can be estimated from the moments of distributions having compact supports by solving linear systems involving respectively Vandermonde and Hilbert matrices. For the latter case, the density estimates can also be expressed in terms of Legendre orthogonal polynomials or obtained from an explicit representation of the elements of the inverse of a Hilbert matrix; interestingly, these polynomial estimates can be expressed as kernel density estimates. We will demonstrate via a kernel analysis that, on extending the range of a sample, one can obtain estimates that are smoother and exhibit less fluctuation in the tails areas. This will be illustrated with the "Buffalo snowfall" data set. Another example involves approximating the distribution of the distance between two random points in a cube. We shall also discuss the use of moments in conjunction with Laguerre and Hermite orthogonal polynomials for the purpose of approximating density functions whose supports are respectively the positive half-line and the entire real line. An application to the distribution of quadratic forms, which is based on a recursive formula for evaluating their moments, will be presented.
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