The Refined Positive Definite and Unimodal Regions for the Gram-Charlier and Edgeworth Series Expansion

Gram-Charlier and Edgeworth Series Expansions are used in the field of statistics to approximate probability density functions. The expansions have proven useful but have experienced limitations due to the values of the moments that admit a proper probability density function. An alternative approach in developing the boundary conditions for the boundary of the positive region for both series expansions is investigated using Sturm’s theorem. The result provides a more accurate representation of the positive region developed by others.


Introduction
The Gram-Charlier and Edgeworth Series Expansions are frequently used in statistics to approximate probability density functions.Shenton 1 investigated the efficiency of the Gram-Charlier Type A distribution, while Draper and Tierney 2 provided exact formulae for the Edgeworth Expansion up to the 10th moment cumulants .Hald 3 provided a thorough historical review of the cumulants and the Gram-Charlier Series.Hall 4 used the Edgeworth Expansion to investigate and develop properties of the bootstrap method.Chen and Sitter 5 derived the Edgeworth Expansion for stratified sampling without replacement from a finite population.Recently, the Edgeworth Expansion was used to investigate the robustness of various process capability indices 6, 7 .Berberan-Santos 8, 9 and Cohen 10 derive and illustrate the relationship between probability density functions and their respective cumulants using the Gram-Charlier and subsequently the Edgeworth expansions.

The Hermite Polynomial
Defining φ x 1/ √ 2π e − x 2 /2 for −∞ < x < ∞ and D d/dx, to be the pdf for the standard normal distribution and the differentiation operator, respectively, Hermite polynomials H r x are defined to be −D r φ x H r x φ x for r ≥ 0.

2.1
The first seven polynomials are

2.2
Two properties of the Hermite polynomials that will be used in the manuscript include see Stuart and Ord 15 DH r x rH r−1 x , for r ≥ 1, H r x − xH r−1 x r − 1 H r−2 x 0, for r ≥ 2. 2.3

Gram-Charlier Series: Type A
Kotz and Johnson 16 defined the Gram-Charlier Type A Series as follows: if f x is a pdf with cumulants κ 1 , κ 2 , . .., the function is with cumulants κ 1 ε 1 , κ 2 ε 2 , . ... In the case where f x is the normal pdf i.e., φ x , we have and λ r represents moment ratios defined as follows: Shenton 1 used the terms up to the 4th moment to represent the Gram-Charlier Type A Series.For λ 1 0 and λ 2 1, represents the Gram-Charlier approximation of a standardized pdf, with mean 0 and variance of 1.In the case of the normal pdf i.e., λ 3 0, λ 4 3 , the approximation is exactly the standard normal distribution.
It can be shown that in the case of the normal distribution, 3.5 is such that Using g x as defined in 3.5 and by changing the values of λ 3 and λ 4 one can examine their impact on the general shape of the resulting pdf. Figure 1 illustrates the impact on g x associated with changes in λ 3 and λ 4 .
In general as λ 3 increases, g x becomes more skewed, while as λ 4 increases, g x becomes more peaked and multimodal.This is why λ 3 is often used as a measure of skewness and λ 4 as a measure of peakedness kurtosis .For several combinations of λ 3 and λ 4 , g x will pass through the x-axis to produce an improper probability density function.For example, when λ 3 1, λ 4 7, g x crosses the x-axis at resulting in a portion of the pdf not being positive definite see Figure 1 .
There are many combinations of λ i 's that result in g x < 0. In order for the approximation to be valid, the polynomial must be nonnegative for all x.It is sufficient to say that the above is true when 3.8 has no real root i.e., it does not touch the x-axis and the coefficient of x 4 is positive.Shenton 1 obtained the solution analytically using the theory of equations.He stated that for There are also values of λ 3 and λ 4 where the Gram-Charlier and Edgeworth Expansion produce a multimodal pdf.In order to determine the regions, we will use the approach developed by Barton and Dennis 11 where by letting then in order for g x to be unimodal, g x can only have one real root.We will provide an alternative approach in obtaining the boundary values using Sturm's theorem and compare the results with those values obtained by Barton

Sturm's Theorem
Let p x represent a polynomial in x and define p 0 x , p 1 x , . . ., p r x to be

4.1
The resulting p 0 x , p 1 x , . . ., pr x is said to represent Sturm's sequence.Sturm's theorem states that if p x 0 is an algebraic equation with real coefficients and without multiple roots and if a and b are real numbers, a < b, and neither are a root of the given equation, then the number of real roots of p x 0 between a and b is equal to ν a − ν b where ν a and ν b are the variations of sign in Sturm's sequence at a and b.
Using Sturm's theorem, one can then determine the region where a polynomial is positive for a specific range of x.If we set a −∞, b ∞, we can determine how many real roots there are for g x .If the polynomial is always positive, it implies that ν a − ν b 0. Using Mathematica 14 we can tabulate the boundaries of the positive regions and plot them.The tabulated values, along with the plot, provide all the necessary information needed to describe the combination of moment ratios that result in the Gram-Charlier and Edgeworth Expansion being positive definite.The plots alone may not have sufficient definition to describe those regions near the boundary conditions.Extensive tables with precise detail can be developed for those regions where accuracy is important using this technique.The source code for determining the tabulated values has been included in the Appendices.
Figure 2 illustrates the boundaries for the positive and unimodal regions for values combination of λ 3 and λ 4 for both the Gram-Charlier Series and Edgeworth Expansions.Table 1 includes the boundary values of λ 3 , λ 4 where the Gram-Charlier series changes from positive definite to non-positive definite.If we use additional terms for the Gram-Charlier Series, then

4.2
If the coefficient of x 5 is nonzero, the polynomial will always cross the x-axis, rendering the associated pdf g x invalid.When the polynomial is as defined in 3.5 .Therefore, if we include λ 5 in the series, we are restricted to the condition λ 5 10λ 3 in order for g x to result in a proper pdf.
If we introduce one more term in the Gram-Charlier Series, we get

4.4
Since it is a 6-degree polynomial, it is possible to find the ranges of λ 3 , λ 4 , λ 5 , and λ 6 that produce a positive definite region.Applying Sturm's theorem, tables of values can be tabulated.If we can assume that the moments of the approximation match up to and including the 4th moment for the standard normal distribution i.e., λ 1 0, λ 2 1, λ 3 0, λ 4 3 , we can find the values of λ 5 , λ 6 that result in a positive definite pdf.
For the unimodal problem, since and we only want g x to have one real root, setting ν a − ν b 1 will calculate the boundary values of λ 3 , λ 4 for unimodality.The results have been tabulated in Table 2 and illustrated in Figure 2.

Edgeworth Expansion
The Edgeworth Expansion can be defined as g x ∞ j 0 c j H j x φ x where

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Barton and Dennis 11 examined the positive definite regions using the terms up to and including the 4th moment, defining

5.2
and provided the parametric equation for the combinations that satisfied the positive definite criteria.In order to ensure that g x is positive definite, the following must be true: and since we want g x to have exactly one real root, we set ν a − ν b 1 and can tabulate the boundary combinations to ensure that g x is unimodal see Table 4 .The results are depicted in Figure 2. Draper and Tierney 2 pointed out that the original unimodal graph from Barton and Dennis 11 , that used a numerical optimization nonlinear least square method to find the condition for the positive definite region, was incorrect.This is readily confirmed from Figure 2.
With the inclusion of the 5th moment, the Edgeworth Expansion becomes a polynomial of degree nine: A 9-degree polynomial will always have a real root; hence the pdf approximation will always have some combination of λ 3 and λ 4 that results in negative coordinates.Draper and Tierney 2 developed an algorithm to produce Edgeworth Expansions up to the 10th moment.

5.6
We see that g x is the same as the Gram-Charlier Series, and hence similar conditions for λ 3 and λ 4 apply.

Comments
We have developed a general procedure to find the positive definite and unimodal regions for the series expansions due to Gram-Charlier and Edgeworth.The method can be used to provide accurate results and boundaries for the region of λ 3 and λ 4 .With the technology available, we can easily produce the results accurately.Also the Mathematica functions are developed for evaluating whether the moment ratio combinations produce a proper pdf.It can be shown that expansions can also be obtained in terms of derivatives of the Gamma via Laguerre polynomials and Beta distributions via Jacobi polynomials .The paper also illustrates the fact that the probability density function is valid only for certain ranges of the λ i 's.If we are outside these ranges, we cannot safely apply the Edgeworth Expansion or the Gram-Charlier Series approximations.For example, if we use this method to investigate

Figure 2 :
Figure 2: Positive definite and unimodal regions for g x .