Theory of Statistics - George Mason University

746 0 Statistical Mathematics
These Hermite polynomials are useful in probability and statistics. The
Gram-Charlier series and the Edgeworth series for asymptotic approximations
are based on these polynomials. See Section 1.2, beginning on page 65.
0.1.9.10 Multivariate Orthogonal Polynomials
Multivariate orthogonal polynomials can be formed easily as tensor products
of univariate orthogonal polynomials. The tensor product of the functions
f(x) over Dx and g(y) over Dy is a function of the arguments x and y over
Dx × Dy:
h(x, y) = f(x)g(y).
If {q1,k(x1)} and {q2,l(x2)} are sequences of univariate orthogonal polynomials,
a sequence of bivariate orthogonal polynomials can be formed as
qkl(x1, x2) = q1,k(x1)q2,l(x2). (0.1.97)
These polynomials are orthogonal in the same sense as in equation (0.1.84),
where the integration is over the two-dimensional domain. Similarly as in
equation (0.1.86), a bivariate function can be expressed as
f(x1, x2) =
∞
k=0 l=0
∞
cklqkl(x1, x2), (0.1.98)
with the coefficients being determined by integrating over both dimensions.
Although obviously such product polynomials, or radial polynomials,
would emphasize features along coordinate axes, they can nevertheless be
useful for representing general multivariate functions. Often, it is useful to
apply a rotation of the coordinate axes.
The weight functions, such as those for the Jacobi polynomials, that have
various shapes controlled by parameters can also often be used in a mixture
model of the function of interest. The weight function for the Hermite polynomials
can be generalized by a linear transformation (resulting in a normal
weight with mean µ and variance σ 2 ), and the function of interest may be
represented as a mixture of general normals.
0.1.10 Distribution Function Spaces
In probability and statistics, one of the most important kinds of function is a
cumulative distribution function, or CDF, defined on page 14 both in terms
of a probability distribution and in terms of four characterizing properties.
A set of CDFs cannot constitute a linear space, because of the restrictions
on the functions. Instead, we will define a distribution function space that
has similar properties. If P is a set of CDFs such that for any w ∈ [0, 1] and
P1, P2 ∈ P, (1 − w)P1 + wP2 ∈ P, then P is a distribution function space.
Theory of Statistics c○2000–2013 James E. Gentle