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

Authors: Fred Spiring
Journal: Advances in Decision Sciences
 (2011)

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## Abstract

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.


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## Body

## 1. 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.Although the expansions are useful, there are limitations on the values of the moments (or cumulants) that admit a proper probability density function (pdf). For example, if the first four moments are used, in both the Gram-Charlier and the Edgeworth Expansions there are regions where the resulting pdf is negative and not unimodal. The negative values of the pdf violate a basic condition associated with proper pdfs. Barton and Dennis [11] and Draper and Tierney [12] provided numerical solutions to the conditions where the Edgeworth Expansion and Gram-Charlier series are nonnegative and unimodal. Balitskaya and Zolotuhina [13] determined the positive region analytically for the Edgeworth Expansion up to the 4th moment. However the plot provided in Barton and Dennis [11] is not precise enough for practical use. We will use an alternative approach to develop the boundary of the positive and unimodal regions for both series expansions.  Mathematica [14] code is provided that allows verification of the moment ratio combinations that result in a proper pdf. This method will accurately determine the positive and unimodal regions of the expansions. Tabulated values are provided and plots included illustrate the proposed method. The results can then be compared with the regions developed by Barton and Dennis [11].


## 2. The Hermite Polynomial

Definingϕ(x)=(1/2π)e-(x2/2) for -∞<x<∞ and D=d/dx, to be the pdf for the standard normal distribution and the differentiation operator, respectively, Hermite polynomials Hr(x) are defined to be
(2.1)(-D)rϕ(x)=Hr(x)ϕ(x)forr≥0.
The first seven polynomials are
(2.2)H0(x)=1,H1(x)=x,H2(x)=x2-1,H3(x)=x3-3x,H4(x)=x4-6x2+3,H5(x)=x5-10x3+15x,H6(x)=x6-15x4+45x2-15,H7(x)=x7-21x5+105x3-105x.
Two properties of the Hermite polynomials that will be used in the manuscript include (see Stuart and Ord  [15])
(2.3)DHr(x)=rHr-1(x),forr≥1,Hr(x)-xHr-1(x)+(r-1)Hr-2(x)=0,forr≥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 κ1,κ2,…, the function is
(3.1)g(x)=exp⁡[∑j=1∞εj{(-D)jj!}]f(x)=f(x)-ε1Df(x)+12(ε12+ε2)D2f(x)-16(ε13+3ε1ε2+ε3)D3f(x)+124(ε14+6ε12ε2+4ε1ε3+ε4)D4f(x)+⋯=∑r=0∞cr(-1)-rDrϕ(x)
with cumulants κ1+ε1,κ2+ε2,…. In the case where f(x) is the normal pdf (i.e.,ϕ(x)), we have
(3.2)Drϕ(x)=(-1)rHr(x)ϕ(x),g(x)=∑r=0∞crHr(x)ϕ(x)=(c0H0(x)+c1H1(x)+c2H2(x)+c3H3(x)+c4H4(x)⋯)ϕ(x),
where the cr’s are defined to be
(3.3)c0=1,c1=0,c2=12(λ2-1),c3=16λ3,c4=124(λ4-6λ2+3),c5=1120(λ5-10λ3),c6=1720(λ6-15λ4+45λ2-15)
and λr represents moment ratios defined as follows:
(3.4)λr(x)=μr{μ2}-r/2whereμr=E((X-E(X))r).
Shenton [1] used the terms up to the 4th moment to represent the Gram-Charlier Type A Series.  For λ1=0 and λ2=1,(3.5)g(x)=(c0H0(x)+c1H1(x)+c2H2(x)+c3H3(x)+c4H4(x))12πe-x2/2=(1+λ36H3(x)+(λ4-324)H4(x))12πe-x2/2=(1+λ36(x3-3x)+(λ4-324)(x4-6x2+3))12πe-x2/2
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.,λ3=0, λ4=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(3.6)∫-∞∞g(x)dx=1.
Using g(x) as defined in (3.5) and by changing the values of λ3 and λ4 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 λ3 and λ4.Figure 1
ϕ(x) and g(x) for λ3=1 and λ4=7.In general asλ3 increases, g(x) becomes more skewed, while as λ4 increases, g(x) becomes more peaked and multimodal. This is why λ3 is often used as a measure of skewness and λ4 as a measure of peakedness (kurtosis). For several combinations of λ3 and λ4, g(x) will pass through the x-axis to produce an improper probability density function. For example, when λ3=1, λ4=7, g(x) crosses the x-axis at
(3.7)x=-3,3,12(-1-13),12(-1+13),
resulting in a portion of the pdf not being positive definite (see Figure 1).There are many combinations ofλi’s that result in g(x)<0.  In order for the approximation to be valid, the polynomial
(3.8)1+λ36(x3-3x)+(λ4-324)(x4-6x2+3)
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 x4 is positive.  Shenton [1] obtained the solution analytically using the theory of equations. He stated that for
(3.9)B(x)=24+4a4H3(x)+a4H4(x)>0,
the condition of a positive polynomial for -∞<x<∞ is 4a43-a44+4a3a42-3a32a42+4a34>0.There are also values ofλ3 and λ4 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
(3.10)dg(x)dx=ddx∑r=0∞crHr(x)ϕ(x)=ϕ(x){ddx[1+∑r=1ncrHr(x)]-[1+∑r=1ncrHr(x)]x}⟹ϕ(x){H1+∑r=1ncrHr+1(x)},
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 and Dennis [11] and Draper and Tierney [2].


## 4. Sturm's Theorem

Letp(x) represent a polynomial in x and define p0(x),p1(x),…,pr(x) to be
(4.1)p0(x)=p(x),p1(x)=p′(x),⋮pr(x)=-remainder(pr-2(x),pr-1(x))forr≥2.
The resulting p0(x),p1(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 νa-νb where νa and νb 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 ofx. 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 νa-νb=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.Figure2 illustrates the boundaries for the positive and unimodal regions for values combination of λ3 and λ4 for both the Gram-Charlier Series and Edgeworth Expansions. Table 1 includes the boundary values of  λ3, λ4 where the Gram-Charlier series changes from positive definite to non-positive definite.Table 1
Positive region boundary(λ4,λ3) for Gram-Charlier Series.
3.005, 0.0183.58, 0.5414.155, 0.81054.73, 0.97355.305, 1.0465.88, 1.0176.455, 0.8363.01, 0.033.585, 0.5444.16, 0.8124.735, 0.97455.31, 1.0465.885, 1.0166.46, 0.83353.015, 0.04053.59, 0.5474.165, 0.8144.74, 0.97555.315, 1.0465.89, 1.01556.465, 0.83053.02, 0.053.595, 0.54954.17, 0.8164.745, 0.97655.32, 1.04655.895, 1.01456.47, 0.8283.025, 0.0593.6, 0.55254.175, 0.81754.75, 0.97755.325, 1.04655.9, 1.01356.475, 0.8253.03, 0.06753.605, 0.55554.18, 0.81954.755, 0.97855.33, 1.0475.905, 1.0136.48, 0.82253.035, 0.07553.61, 0.55854.185, 0.82154.76, 0.97955.335, 1.0475.91, 1.0126.485, 0.81953.04, 0.08353.615, 0.56154.19, 0.8234.765, 0.98055.34, 1.0475.915, 1.0116.49, 0.8173.045, 0.0913.62, 0.5644.195, 0.8254.77, 0.98155.345, 1.04755.92, 1.01056.495, 0.8143.05, 0.0983.625, 0.5674.2, 0.82654.775, 0.98255.35, 1.04755.925, 1.00956.5, 0.8113.055, 0.1053.63, 0.574.205, 0.82854.78, 0.98355.355, 1.04755.93, 1.00856.505, 0.80853.06, 0.1123.635, 0.57254.21, 0.834.785, 0.98455.36, 1.0485.935, 1.00756.51, 0.80553.065, 0.1193.64, 0.57554.215, 0.8324.79, 0.98555.365, 1.0485.94, 1.00656.515, 0.80253.07, 0.12553.645, 0.5784.22, 0.83354.795, 0.9865.37, 1.0485.945, 1.0066.52, 0.79953.075, 0.1323.65, 0.5814.225, 0.83554.8, 0.9875.375, 1.0485.95, 1.0056.525, 0.79653.08, 0.1383.655, 0.5844.23, 0.8374.805, 0.9885.38, 1.04855.955, 1.0046.53, 0.79353.085, 0.1443.66, 0.58654.235, 0.83854.81, 0.9895.385, 1.04855.96, 1.0036.535, 0.79053.09, 0.15053.665, 0.58954.24, 0.84054.815, 0.995.39, 1.04855.965, 1.0026.54, 0.7873.095, 0.1563.67, 0.5924.245, 0.8424.82, 0.9915.395, 1.04855.97, 1.0016.545, 0.7843.1, 0.1623.675, 0.5954.25, 0.8444.825, 0.99155.4, 1.04855.975, 16.55, 0.7813.105, 0.1683.68, 0.59754.255, 0.84554.83, 0.99255.405, 1.04855.98, 0.9996.555, 0.77753.11, 0.17353.685, 0.64.26, 0.8474.835, 0.99355.41, 1.0495.985, 0.9986.56, 0.77453.115, 0.1793.69, 0.6034.265, 0.8494.84, 0.99455.415, 1.0495.99, 0.9976.565, 0.7713.12, 0.18453.695, 0.60554.27, 0.85054.845, 0.9955.42, 1.0495.995, 0.9966.57, 0.7683.125, 0.193.7, 0.6084.275, 0.8524.85, 0.9965.425, 1.0496, 0.9956.575, 0.76453.13, 0.19553.705, 0.6114.28, 0.8544.855, 0.9975.43, 1.0496.005, 0.9946.58, 0.7613.135, 0.2013.71, 0.61354.285, 0.85554.86, 0.9985.435, 1.0496.01, 0.99256.585, 0.7583.14, 0.2063.715, 0.6164.29, 0.8574.865, 0.99855.44, 1.0496.015, 0.99156.59, 0.75453.145, 0.2113.72, 0.6194.295, 0.8594.87, 0.99955.445, 1.0496.02, 0.99056.595, 0.7513.15, 0.21653.725, 0.62154.3, 0.86054.875, 1.00055.45, 1.0496.025, 0.98956.6, 0.74753.155, 0.22153.73, 0.6244.305, 0.8624.88, 1.0015.455, 1.0496.03, 0.98856.605, 0.7443.16, 0.22653.735, 0.62654.31, 0.86354.885, 1.0025.46, 1.0496.035, 0.9876.61, 0.743.165, 0.23153.74, 0.6294.315, 0.8654.89, 1.00255.465, 1.0496.04, 0.9866.615, 0.73653.17, 0.23653.745, 0.63154.32, 0.8674.895, 1.00355.47, 1.0496.045, 0.9856.62, 0.7333.175, 0.2413.75, 0.63454.325, 0.86854.9, 1.00455.475, 1.0496.05, 0.98356.625, 0.7293.18, 0.2463.755, 0.6374.33, 0.874.905, 1.0055.48, 1.0496.055, 0.98256.63, 0.72553.185, 0.25053.76, 0.63954.335, 0.87154.91, 1.0065.485, 1.0496.06, 0.98156.635, 0.72153.19, 0.25553.765, 0.6424.34, 0.8734.915, 1.00655.49, 1.0496.065, 0.986.64, 0.71753.195, 0.263.77, 0.64454.345, 0.87454.92, 1.00755.495, 1.04856.07, 0.9796.645, 0.7143.2, 0.2653.775, 0.6474.35, 0.8764.925, 1.0085.5, 1.04856.075, 0.97756.65, 0.713.205, 0.26953.78, 0.64954.355, 0.87754.93, 1.0095.505, 1.04856.08, 0.97656.655, 0.7063.21, 0.2743.785, 0.6524.36, 0.87954.935, 1.00955.51, 1.04856.085, 0.9756.66, 0.7023.215, 0.27853.79, 0.65454.365, 0.8814.94, 1.01055.515, 1.04856.09, 0.9746.665, 0.69753.22, 0.2833.795, 0.6574.37, 0.88254.945, 1.0115.52, 1.0486.095, 0.97256.67, 0.69353.225, 0.28753.8, 0.65954.375, 0.8844.95, 1.0125.525, 1.0486.1, 0.9716.675, 0.68953.23, 0.2923.805, 0.66154.38, 0.88554.955, 1.01255.53, 1.0486.105, 0.976.68, 0.6853.235, 0.2963.81, 0.6644.385, 0.8874.96, 1.01355.535, 1.0486.11, 0.96856.685, 0.6813.24, 0.30053.815, 0.66654.39, 0.88854.965, 1.0145.54, 1.04756.115, 0.9676.69, 0.67653.245, 0.3053.82, 0.6694.395, 0.894.97, 1.0155.545, 1.04756.12, 0.9666.695, 0.6723.25, 0.3093.825, 0.67154.4, 0.89154.975, 1.01555.55, 1.04756.125, 0.96456.7, 0.66753.255, 0.31353.83, 0.6744.405, 0.89254.98, 1.0165.555, 1.0476.13, 0.9636.705, 0.6633.26, 0.31753.835, 0.6764.41, 0.8944.985, 1.0175.56, 1.0476.135, 0.96156.71, 0.65853.265, 0.32153.84, 0.67854.415, 0.89554.99, 1.01755.565, 1.0476.14, 0.966.715, 0.6543.27, 0.3263.845, 0.6814.42, 0.8974.995, 1.0185.57, 1.04656.145, 0.95856.72, 0.6493.275, 0.333.85, 0.68354.425, 0.89855, 1.0195.575, 1.04656.15, 0.95756.725, 0.64453.28, 0.3343.855, 0.68554.43, 0.95.005, 1.01955.58, 1.04656.155, 0.9566.73, 0.63953.285, 0.3383.86, 0.6884.435, 0.90155.01, 1.025.585, 1.0466.16, 0.95456.735, 0.63453.29, 0.3423.865, 0.69054.44, 0.9035.015, 1.0215.59, 1.0466.165, 0.9536.74, 0.62953.295, 0.3463.87, 0.69254.445, 0.9045.02, 1.02155.595, 1.04556.17, 0.95156.745, 0.62453.3, 0.353.875, 0.6954.45, 0.90555.025, 1.0225.6, 1.04556.175, 0.956.75, 0.61953.305, 0.3543.88, 0.6974.455, 0.9075.03, 1.02255.605, 1.0456.18, 0.94856.755, 0.6143.31, 0.3583.885, 0.69954.46, 0.90855.035, 1.02355.61, 1.0456.185, 0.94656.76, 0.6093.315, 0.3623.89, 0.7024.465, 0.915.04, 1.0245.615, 1.04456.19, 0.9456.765, 0.60353.32, 0.3663.895, 0.7044.47, 0.9115.045, 1.02455.62, 1.04456.195, 0.94356.77, 0.5983.325, 0.36953.9, 0.70654.475, 0.91255.05, 1.0255.625, 1.0446.2, 0.9426.775, 0.59253.33, 0.37353.905, 0.70854.48, 0.9145.055, 1.02555.63, 1.04356.205, 0.94056.78, 0.5873.335, 0.37753.91, 0.7114.485, 0.91555.06, 1.0265.635, 1.04356.21, 0.93856.785, 0.58153.34, 0.3813.915, 0.7134.49, 0.91655.065, 1.0275.64, 1.0436.215, 0.9376.79, 0.57553.345, 0.3853.92, 0.71554.495, 0.9185.07, 1.02755.645, 1.0436.22, 0.93556.795, 0.56953.35, 0.38853.925, 0.71754.5, 0.91955.075, 1.0285.65, 1.04256.225, 0.93356.8, 0.56353.355, 0.39253.93, 0.71954.505, 0.92055.08, 1.02855.655, 1.0426.23, 0.9326.805, 0.55753.36, 0.3963.935, 0.7224.51, 0.9225.085, 1.0295.66, 1.04156.235, 0.93056.81, 0.55153.365, 0.39953.94, 0.7244.515, 0.92355.09, 1.02955.665, 1.04156.24, 0.92856.815, 0.5453.37, 0.40353.945, 0.72654.52, 0.92455.095, 1.035.67, 1.0416.245, 0.9276.82, 0.53853.375, 0.4073.95, 0.72854.525, 0.9265.1, 1.03055.675, 1.04056.25, 0.9256.825, 0.5323.38, 0.41053.955, 0.73054.53, 0.9275.105, 1.0315.68, 1.046.255, 0.9236.83, 0.52553.385, 0.4143.96, 0.7334.535, 0.92855.11, 1.03155.685, 1.046.26, 0.92156.835, 0.51853.39, 0.4183.965, 0.7354.54, 0.935.115, 1.0325.69, 1.03956.265, 0.91956.84, 0.51153.395, 0.42153.97, 0.7374.545, 0.9315.12, 1.03255.695, 1.0396.27, 0.9186.845, 0.50453.4, 0.4253.975, 0.7394.55, 0.93255.125, 1.0335.7, 1.03856.275, 0.9166.85, 0.4973.405, 0.42853.98, 0.74154.555, 0.93355.13, 1.03355.705, 1.0386.28, 0.9146.855, 0.48953.41, 0.4323.985, 0.74354.56, 0.9355.135, 1.0345.71, 1.03756.285, 0.9126.86, 0.4823.415, 0.43553.99, 0.74554.565, 0.9365.14, 1.03455.715, 1.03756.29, 0.91056.865, 0.47453.42, 0.4393.995, 0.74754.57, 0.93755.145, 1.0355.72, 1.0376.295, 0.90856.87, 0.46653.425, 0.44254, 0.74954.575, 0.93855.15, 1.03555.725, 1.03656.3, 0.90656.875, 0.45853.43, 0.44554.005, 0.7524.58, 0.945.155, 1.0365.73, 1.0366.305, 0.90456.88, 0.453.435, 0.4494.01, 0.7544.585, 0.9415.16, 1.03655.735, 1.03556.31, 0.90256.885, 0.44153.44, 0.45254.015, 0.7564.59, 0.9425.165, 1.03655.74, 1.0356.315, 0.90056.89, 0.43253.445, 0.4564.02, 0.7584.595, 0.94355.17, 1.0375.745, 1.03456.32, 0.89856.895, 0.42353.45, 0.4594.025, 0.764.6, 0.94455.175, 1.03755.75, 1.0346.325, 0.89656.9, 0.41453.455, 0.46254.03, 0.7624.605, 0.9465.18, 1.0385.755, 1.03356.33, 0.89456.905, 0.40453.46, 0.4664.035, 0.7644.61, 0.9475.185, 1.03855.76, 1.0336.335, 0.89256.91, 0.3953.465, 0.4694.04, 0.7664.615, 0.9485.19, 1.03855.765, 1.03256.34, 0.896.915, 0.38453.47, 0.47254.045, 0.7684.62, 0.94955.195, 1.0395.77, 1.03156.345, 0.8886.92, 0.3743.475, 0.4764.05, 0.774.625, 0.95055.2, 1.03955.775, 1.0316.35, 0.8866.925, 0.3633.48, 0.4794.055, 0.7724.63, 0.95155.205, 1.045.78, 1.03056.355, 0.8846.93, 0.35153.485, 0.4824.06, 0.7744.635, 0.9535.21, 1.04055.785, 1.036.36, 0.88156.935, 0.33953.49, 0.48554.065, 0.7764.64, 0.9545.215, 1.04055.79, 1.02956.365, 0.87956.94, 0.3273.495, 0.48854.07, 0.7784.645, 0.9555.22, 1.0415.795, 1.0296.37, 0.8776.945, 0.3143.5, 0.4924.075, 0.784.65, 0.95655.225, 1.04155.8, 1.0286.375, 0.8756.95, 0.33.505, 0.4954.08, 0.7824.655, 0.95755.23, 1.04155.805, 1.02756.38, 0.87256.955, 0.28553.51, 0.4984.085, 0.7844.66, 0.95855.235, 1.0425.81, 1.0276.385, 0.87056.96, 0.273.515, 0.50154.09, 0.7864.665, 0.95955.24, 1.04255.815, 1.02656.39, 0.8686.965, 0.25353.52, 0.50454.095, 0.7884.67, 0.96055.245, 1.04255.82, 1.02556.395, 0.86556.97, 0.23553.525, 0.50754.1, 0.794.675, 0.9625.25, 1.0435.825, 1.0256.4, 0.86356.975, 0.21553.53, 0.51054.105, 0.79154.68, 0.9635.255, 1.0435.83, 1.02456.405, 0.8616.98, 0.19353.535, 0.5144.11, 0.79354.685, 0.9645.26, 1.04355.835, 1.02356.41, 0.85856.985, 0.16853.54, 0.5174.115, 0.79554.69, 0.9655.265, 1.0445.84, 1.0236.415, 0.8566.99, 0.1383.545, 0.524.12, 0.79754.695, 0.9665.27, 1.0445.845, 1.0226.42, 0.85356.995, 0.09853.55, 0.5234.125, 0.79954.7, 0.9675.275, 1.04455.85, 1.02156.425, 0.85153.555, 0.5264.13, 0.8014.705, 0.96855.28, 1.04455.855, 1.02056.43, 0.8493.56, 0.5294.135, 0.8034.71, 0.96955.285, 1.0455.86, 1.026.435, 0.84653.565, 0.5324.14, 0.8054.715, 0.97055.29, 1.0455.865, 1.0196.44, 0.84353.57, 0.5354.145, 0.80654.72, 0.97155.295, 1.04555.87, 1.01856.445, 0.8413.575, 0.5384.15, 0.80854.725, 0.97255.3, 1.04555.875, 1.01756.45, 0.8385Figure 2
Positive definite and unimodal regions forg(x).If we use additional terms for the Gram-Charlier Series, then(4.2)g(x)=(c0H0(x)+c1H1(x)+c2H2(x)+c3H3(x)+c4H4(x)+c5H5(x))12πe-x2/2=(1+λ36H3(x)+(λ4-324)H4(x)+(λ5-10λ3120)H5(x))12πe-x2/2=(1+λ36(x3-3x)+(λ4-324)(x4-6x2+3)+(λ5-10λ3120)(x5-10x3+15x))×12πe-x2/2.
If the coefficient of x5 is nonzero, the polynomial will always cross the x-axis, rendering the associated pdf g(x) invalid.  When
(4.3)λ5-10λ3120=0,
the polynomial is as defined in (3.5).  Therefore, if we include λ5 in the series, we are restricted to the condition λ5=10λ3   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)g(x)=(c0H0(x)+c1H1(x)+c2H2(x)+c3H3(x)+c4H4(x)+c5H5(x)+c6H6(x))12πe-x2/2=(1+λ36H3(x)+(λ4-324)H4(x)+(λ5-10λ3120)H5(x)+(λ6-15λ4+45λ2-15720)H6(x))12πe-x2/2=(1+λ36(x3-3x)+(λ4-324)(x4-6x2+3)+(λ5-10λ3120)(x5-10x3+15x)+(λ6-15λ4+45λ2-15720)(x6-15x4+45x2-15))12πe-x2/2.
Since it is a 6-degree polynomial, it is possible to find the ranges of λ3, λ4, λ5, andλ6 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., λ1=0, λ2=1, λ3=0, λ4=3), we can find the values of λ5, λ6 that result in a positive definite pdf.For the unimodal problem, since(4.5)dg(x)dx=ddx∑r=0∞crHr(x)ϕ(x)=ϕ(x){H1+∑r=1ncrHr+1(x)}=x+λ36(x4-6x2+3)+λ4-324(x5-10x3+15x)
and we only want g′(x) to have one real root, setting νa-νb=1 will calculate the boundary values of  λ3, λ4 for unimodality.  The results have been tabulated in Table 2 and illustrated in Figure 2.Table 2
Unimodal region boundary(λ4,λ3) for Gram-Charlier Series.
3.01, 0.033.355, 0.3883.7, 0.5914.045, 0.72754.39, 0.8084.735, 0.82855.08, 0.5073.015, 0.04053.36, 0.39153.705, 0.59354.05, 0.7294.395, 0.8094.74, 0.8275.085, 0.50153.02, 0.053.365, 0.3953.71, 0.5964.055, 0.73054.4, 0.80954.745, 0.8235.09, 0.4963.025, 0.0593.37, 0.39853.715, 0.59854.06, 0.73254.405, 0.81054.75, 0.8195.095, 0.493.03, 0.06753.375, 0.4023.72, 0.60054.065, 0.7344.41, 0.8114.755, 0.8155.1, 0.48453.035, 0.07553.38, 0.40553.725, 0.6034.07, 0.73554.415, 0.8124.76, 0.8115.105, 0.47853.04, 0.08353.385, 0.4093.73, 0.60554.075, 0.7374.42, 0.81254.765, 0.8075.11, 0.47253.045, 0.0913.39, 0.41253.735, 0.60754.08, 0.73854.425, 0.8134.77, 0.80255.115, 0.4673.05, 0.0983.395, 0.4163.74, 0.614.085, 0.744.43, 0.8144.775, 0.79855.12, 0.4613.055, 0.1053.4, 0.41953.745, 0.61254.09, 0.74154.435, 0.81454.78, 0.79455.125, 0.4553.06, 0.1123.405, 0.4233.75, 0.61454.095, 0.74254.44, 0.8154.785, 0.795.13, 0.4493.065, 0.11853.41, 0.4263.755, 0.6174.1, 0.7444.445, 0.8164.79, 0.7865.135, 0.4433.07, 0.1253.415, 0.42953.76, 0.6194.105, 0.74554.45, 0.81654.795, 0.7825.14, 0.4373.075, 0.13153.42, 0.4333.765, 0.62154.11, 0.7474.455, 0.8174.8, 0.77755.145, 0.4313.08, 0.1383.425, 0.4363.77, 0.62354.115, 0.74854.46, 0.81754.805, 0.77355.15, 0.42453.085, 0.1443.43, 0.43953.775, 0.6264.12, 0.754.465, 0.8184.81, 0.7695.155, 0.41853.09, 0.153.435, 0.44253.78, 0.6284.125, 0.7514.47, 0.8194.815, 0.7655.16, 0.4123.095, 0.1563.44, 0.4463.785, 0.634.13, 0.75254.475, 0.81954.82, 0.76055.165, 0.4063.1, 0.16153.445, 0.4493.79, 0.63254.135, 0.7544.48, 0.824.825, 0.75655.17, 0.39953.105, 0.16753.45, 0.45253.795, 0.63454.14, 0.75554.485, 0.82054.83, 0.7525.175, 0.3933.11, 0.1733.455, 0.45553.8, 0.63654.145, 0.75654.49, 0.8214.835, 0.74755.18, 0.38653.115, 0.17853.46, 0.45853.805, 0.6394.15, 0.7584.495, 0.82154.84, 0.74355.185, 0.383.12, 0.1843.465, 0.4623.81, 0.6414.155, 0.75954.5, 0.8224.845, 0.7395.19, 0.37353.125, 0.18953.47, 0.4653.815, 0.6434.16, 0.76054.505, 0.82254.85, 0.73455.195, 0.3673.13, 0.1953.475, 0.4683.82, 0.6454.165, 0.7624.51, 0.8234.855, 0.735.2, 0.363.135, 0.23.48, 0.4713.825, 0.64754.17, 0.7634.515, 0.8234.86, 0.72555.205, 0.35353.14, 0.20553.485, 0.4743.83, 0.64954.175, 0.76454.52, 0.82354.865, 0.7215.21, 0.34653.145, 0.21053.49, 0.4773.835, 0.65154.18, 0.7664.525, 0.8244.87, 0.71655.215, 0.33953.15, 0.21553.495, 0.48053.84, 0.65354.185, 0.7674.53, 0.82454.875, 0.7125.22, 0.33253.155, 0.22053.5, 0.48353.845, 0.65554.19, 0.76854.535, 0.8254.88, 0.70755.225, 0.32553.16, 0.22553.505, 0.48653.85, 0.65754.195, 0.76954.54, 0.8254.885, 0.7035.23, 0.31853.165, 0.23053.51, 0.48953.855, 0.65954.2, 0.77054.545, 0.82554.89, 0.69855.235, 0.31153.17, 0.2353.515, 0.4923.86, 0.66154.205, 0.7724.55, 0.8264.895, 0.6945.24, 0.3043.175, 0.243.52, 0.4953.865, 0.66354.21, 0.7734.555, 0.82654.9, 0.68955.245, 0.2973.18, 0.2453.525, 0.4983.87, 0.66554.215, 0.77454.56, 0.82654.905, 0.6855.25, 0.28953.185, 0.24953.53, 0.5013.875, 0.66754.22, 0.77554.565, 0.8274.91, 0.685.255, 0.2823.19, 0.2543.535, 0.5043.88, 0.66954.225, 0.77654.57, 0.8274.915, 0.67555.26, 0.27453.195, 0.25853.54, 0.5073.885, 0.67154.23, 0.7784.575, 0.82754.92, 0.6715.265, 0.26653.2, 0.26353.545, 0.50953.89, 0.67354.235, 0.7794.58, 0.82754.925, 0.6665.27, 0.2593.205, 0.2683.55, 0.51253.895, 0.67554.24, 0.784.585, 0.8284.93, 0.66155.275, 0.2513.21, 0.27253.555, 0.51553.9, 0.67754.245, 0.7814.59, 0.8284.935, 0.65655.28, 0.2433.215, 0.2773.56, 0.5183.905, 0.6794.25, 0.78254.595, 0.82854.94, 0.6525.285, 0.2353.22, 0.2813.565, 0.5213.91, 0.6814.255, 0.78354.6, 0.82854.945, 0.6475.29, 0.2273.225, 0.28553.57, 0.5243.915, 0.6834.26, 0.78454.605, 0.8294.95, 0.6425.295, 0.21853.23, 0.293.575, 0.52653.92, 0.6854.265, 0.78554.61, 0.8294.955, 0.63755.3, 0.213.235, 0.2943.58, 0.52953.925, 0.68654.27, 0.78654.615, 0.8294.96, 0.63255.305, 0.20153.24, 0.29853.585, 0.5323.93, 0.68854.275, 0.78754.62, 0.82954.965, 0.62755.31, 0.1933.245, 0.30253.59, 0.5353.935, 0.69054.28, 0.78854.625, 0.82954.97, 0.62255.315, 0.1843.25, 0.3073.595, 0.53753.94, 0.6924.285, 0.78954.63, 0.82954.975, 0.61755.32, 0.1753.255, 0.3113.6, 0.543.945, 0.6944.29, 0.79054.635, 0.82954.98, 0.61255.325, 0.1663.26, 0.3153.605, 0.5433.95, 0.69554.295, 0.79154.64, 0.82954.985, 0.60755.33, 0.15653.265, 0.3193.61, 0.54553.955, 0.69754.3, 0.79254.645, 0.834.99, 0.60255.335, 0.1473.27, 0.32353.615, 0.5483.96, 0.69954.305, 0.79354.65, 0.834.995, 0.59755.34, 0.13753.275, 0.32753.62, 0.5513.965, 0.7014.31, 0.79454.655, 0.835, 0.59255.345, 0.12753.28, 0.33153.625, 0.55353.97, 0.7034.315, 0.79554.66, 0.835.005, 0.58755.35, 0.11753.285, 0.33553.63, 0.5563.975, 0.70454.32, 0.79654.665, 0.835.01, 0.58255.355, 0.1073.29, 0.3393.635, 0.55853.98, 0.70654.325, 0.79754.67, 0.835.015, 0.5775.36, 0.09653.295, 0.3433.64, 0.56153.985, 0.7084.33, 0.79854.675, 0.835.02, 0.5725.365, 0.08553.3, 0.3473.645, 0.5643.99, 0.70954.335, 0.7994.68, 0.835.025, 0.56655.37, 0.07453.305, 0.3513.65, 0.56653.995, 0.71154.34, 0.84.685, 0.835.03, 0.56155.375, 0.0633.31, 0.35453.655, 0.5694, 0.7134.345, 0.8014.69, 0.82955.035, 0.5565.38, 0.05153.315, 0.35853.66, 0.57154.005, 0.71454.35, 0.8024.695, 0.82955.04, 0.5515.385, 0.0393.32, 0.36253.665, 0.5744.01, 0.71654.355, 0.80254.7, 0.82955.045, 0.54555.39, 0.02653.325, 0.3663.67, 0.57654.015, 0.7184.36, 0.80354.705, 0.82955.05, 0.545.395, 0.01353.33, 0.36953.675, 0.5794.02, 0.71954.365, 0.80454.71, 0.8295.055, 0.53453.335, 0.37353.68, 0.58154.025, 0.72154.37, 0.8054.715, 0.8295.06, 0.5293.34, 0.3773.685, 0.5844.03, 0.7234.375, 0.8064.72, 0.8295.065, 0.52353.345, 0.3813.69, 0.58654.035, 0.72454.38, 0.80654.725, 0.82855.07, 0.5183.35, 0.38453.695, 0.5894.04, 0.7264.385, 0.80754.73, 0.82855.075, 0.5125


## 5. Edgeworth Expansion

The Edgeworth Expansion can be defined asg(x)=∑j=0∞cjHj(x)ϕ(x) where
(5.1)c0=1,c1=c2=0,c3=16λ3,c4=124(λ4-3),c5=0,c6=172λ32.
Barton and Dennis [11] examined the positive definite regions using the terms up to and including the 4th moment, defining
(5.2)g(x)=(c0H0(x)+c1H1(x)+c2H2(x)+c3H3(x)+c4H4(x)+c5H5(x)+c6H6(x))12πe-x2/2=(1+λ36H3(x)+(λ4-324)H4(x)+(λ3272)H6(x))12πe-x2/2=(1+λ36(x3-3x)+(λ4-324)(x4-6x2+3)+λ3272(x6-15x4+45x2-15))×12πe-x2/2,
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:
(5.3)1+λ36(x3-3x)+(λ4-324)(x4-6x2+3)+λ3272(x6-15x4+45x2-15)>0.
Using Sturm’s theorem, we can obtain the boundaries of the positive region by fixing νa-νb=0.  The boundary values of λ3, λ4 have been tabulated (Table 3) and drawn in Figure 2.Table 3
Positive region boundary(λ4,λ3) for Edgeworth Expansion.
3, 0.13773.575, 0.47234.15, 0.58654.725, 0.65095.3, 0.68155.875, 0.67746.45, 0.61673.005, 0.15173.58, 0.47374.155, 0.58724.73, 0.65135.305, 0.68175.88, 0.67716.455, 0.61573.01, 0.16323.585, 0.4754.16, 0.5884.735, 0.65175.31, 0.68185.885, 0.67696.46, 0.61483.015, 0.17323.59, 0.47634.165, 0.58874.74, 0.65215.315, 0.68195.89, 0.67676.465, 0.61383.02, 0.1823.595, 0.47764.17, 0.58944.745, 0.65255.32, 0.6825.895, 0.67646.47, 0.61283.025, 0.19013.6, 0.47894.175, 0.59014.75, 0.65295.325, 0.68215.9, 0.67626.475, 0.61173.03, 0.19753.605, 0.48024.18, 0.59084.755, 0.65335.33, 0.68285.905, 0.67596.48, 0.61073.035, 0.20443.61, 0.48154.185, 0.59164.76, 0.65375.335, 0.68285.91, 0.67576.485, 0.60973.04, 0.21093.615, 0.48284.19, 0.59234.765, 0.65415.34, 0.68285.915, 0.67546.49, 0.60863.045, 0.2173.62, 0.48414.195, 0.5934.77, 0.65455.345, 0.68285.92, 0.67526.495, 0.60753.05, 0.22283.625, 0.48544.2, 0.59374.775, 0.65485.35, 0.68285.925, 0.67496.5, 0.60653.055, 0.22833.63, 0.48664.205, 0.59444.78, 0.65525.355, 0.68285.93, 0.67476.505, 0.60543.06, 0.23363.635, 0.48794.21, 0.59514.785, 0.65565.36, 0.68295.935, 0.67446.51, 0.60423.065, 0.23873.64, 0.48914.215, 0.59574.79, 0.6565.365, 0.68295.94, 0.67416.515, 0.60313.07, 0.24353.645, 0.49034.22, 0.59644.795, 0.65635.37, 0.68305.945, 0.67386.52, 0.6023.075, 0.24823.65, 0.49164.225, 0.59714.8, 0.65675.375, 0.68315.95, 0.67366.525, 0.60083.08, 0.25283.655, 0.49284.23, 0.59784.805, 0.65715.38, 0.68325.955, 0.67336.53, 0.59973.085, 0.25723.66, 0.4944.235, 0.59854.81, 0.65745.385, 0.68335.96, 0.6736.535, 0.59853.09, 0.26143.665, 0.49524.24, 0.59924.815, 0.65785.39, 0.68335.965, 0.67276.54, 0.59733.095, 0.26563.67, 0.49644.245, 0.59984.82, 0.65825.395, 0.68355.97, 0.67246.545, 0.59613.1, 0.26963.675, 0.49764.25, 0.60054.825, 0.65855.4, 0.68365.975, 0.67216.55, 0.59483.105, 0.27353.68, 0.49884.255, 0.60124.83, 0.65895.405, 0.68365.98, 0.67186.555, 0.59363.11, 0.27733.685, 0.54.26, 0.60184.835, 0.65925.41, 0.68375.985, 0.67156.56, 0.59233.115, 0.28113.69, 0.50114.265, 0.60254.84, 0.65965.415, 0.68375.99, 0.67116.565, 0.59113.12, 0.28473.695, 0.50234.27, 0.60314.845, 0.65995.42, 0.68385.995, 0.67086.57, 0.58983.125, 0.28823.7, 0.50354.275, 0.60384.85, 0.66035.425, 0.68396, 0.67056.575, 0.58843.13, 0.29173.705, 0.50464.28, 0.60444.855, 0.66065.43, 0.68396.005, 0.67026.58, 0.58713.135, 0.29513.71, 0.50584.285, 0.60514.86, 0.6615.435, 0.6846.01, 0.66986.585, 0.58583.14, 0.29843.715, 0.50694.29, 0.60574.865, 0.66135.44, 0.6846.015, 0.66956.59, 0.58443.145, 0.30173.72, 0.5084.295, 0.60644.87, 0.66165.445, 0.68416.02, 0.66926.595, 0.5833.15, 0.30493.725, 0.50924.3, 0.6074.875, 0.6625.45, 0.68416.025, 0.66886.6, 0.58163.155, 0.3083.73, 0.51034.305, 0.60764.88, 0.66235.455, 0.68426.03, 0.66856.605, 0.58023.16, 0.31113.735, 0.51144.31, 0.60834.885, 0.66265.46, 0.68426.035, 0.66816.61, 0.57873.165, 0.31413.74, 0.51254.315, 0.60894.89, 0.66295.465, 0.68436.04, 0.66776.615, 0.57733.17, 0.31713.745, 0.51364.32, 0.60954.895, 0.66335.47, 0.68436.045, 0.66746.62, 0.57583.175, 0.323.75, 0.51474.325, 0.61014.9, 0.66365.475, 0.68436.05, 0.6676.625, 0.57433.18, 0.32293.755, 0.51584.33, 0.61084.905, 0.66395.48, 0.68446.055, 0.66666.63, 0.57283.185, 0.32573.76, 0.51694.335, 0.61144.91, 0.66425.485, 0.68446.06, 0.66626.635, 0.57123.19, 0.32853.765, 0.5184.34, 0.6124.915, 0.66455.49, 0.68446.065, 0.66586.64, 0.56963.195, 0.33133.77, 0.5194.345, 0.61264.92, 0.66495.495, 0.68456.07, 0.66556.645, 0.5683.2, 0.3343.775, 0.52014.35, 0.61324.925, 0.66525.5, 0.68456.075, 0.66516.65, 0.56643.205, 0.33663.78, 0.52124.355, 0.61384.93, 0.66555.505, 0.68456.08, 0.66476.655, 0.56483.21, 0.33933.785, 0.52224.36, 0.61444.935, 0.66585.51, 0.68456.085, 0.66426.66, 0.56313.215, 0.34183.79, 0.52334.365, 0.6154.94, 0.66615.515, 0.68456.09, 0.66386.665, 0.56143.22, 0.34443.795, 0.52434.37, 0.61564.945, 0.66645.52, 0.68456.095, 0.66346.67, 0.55973.225, 0.34693.8, 0.52544.375, 0.61624.95, 0.66675.525, 0.68456.1, 0.6636.675, 0.55793.23, 0.34943.805, 0.52644.38, 0.61684.955, 0.6675.53, 0.68456.105, 0.66266.68, 0.55613.235, 0.35183.81, 0.52744.385, 0.61744.96, 0.66735.535, 0.68456.11, 0.66216.685, 0.55433.24, 0.35423.815, 0.52844.39, 0.6184.965, 0.66765.54, 0.68456.115, 0.66176.69, 0.55253.245, 0.35663.82, 0.52954.395, 0.61854.97, 0.66785.545, 0.68456.12, 0.66126.695, 0.55063.25, 0.3593.825, 0.53054.4, 0.61914.975, 0.66815.55, 0.68456.125, 0.66086.7, 0.54873.255, 0.36133.83, 0.53154.405, 0.61974.98, 0.66845.555, 0.68456.13, 0.66036.705, 0.54683.26, 0.36363.835, 0.53254.41, 0.62034.985, 0.66875.56, 0.68456.135, 0.65996.71, 0.54493.265, 0.36593.84, 0.53354.415, 0.62084.99, 0.6695.565, 0.68456.14, 0.65946.715, 0.54293.27, 0.36813.845, 0.53454.42, 0.62144.995, 0.66935.57, 0.68456.145, 0.65896.72, 0.54083.275, 0.37043.85, 0.53544.425, 0.6225, 0.66955.575, 0.68456.15, 0.65856.725, 0.53883.28, 0.37253.855, 0.53644.43, 0.62255.005, 0.66985.58, 0.68446.155, 0.6586.73, 0.53673.285, 0.37473.86, 0.53744.435, 0.62315.01, 0.67015.585, 0.68446.16, 0.65756.735, 0.53453.29, 0.37693.865, 0.53844.44, 0.62375.015, 0.67035.59, 0.68446.165, 0.6576.74, 0.53233.295, 0.3793.87, 0.53934.445, 0.62425.02, 0.67065.595, 0.68436.17, 0.65656.745, 0.53013.3, 0.38113.875, 0.54034.45, 0.62475.025, 0.67095.6, 0.68436.175, 0.6566.75, 0.52793.305, 0.38323.88, 0.54134.455, 0.62535.03, 0.67115.605, 0.68436.18, 0.65556.755, 0.52563.31, 0.38523.885, 0.54224.46, 0.62585.035, 0.67145.61, 0.68426.185, 0.65496.76, 0.52323.315, 0.38733.89, 0.54324.465, 0.62645.04, 0.67165.615, 0.68426.19, 0.65446.765, 0.52083.32, 0.38933.895, 0.54414.47, 0.62695.045, 0.67195.62, 0.68416.195, 0.65396.77, 0.51843.325, 0.39133.9, 0.5454.475, 0.62755.05, 0.67215.625, 0.68416.2, 0.65336.775, 0.51593.33, 0.39333.905, 0.5464.48, 0.6285.055, 0.67245.63, 0.6846.205, 0.65286.78, 0.51333.335, 0.39523.91, 0.54694.485, 0.62855.06, 0.67265.635, 0.6846.21, 0.65236.785, 0.51073.34, 0.39713.915, 0.54784.49, 0.6295.065, 0.67295.64, 0.68396.215, 0.65176.79, 0.50813.345, 0.39913.92, 0.54874.495, 0.62965.07, 0.67315.645, 0.68386.22, 0.65116.795, 0.50543.35, 0.4013.925, 0.54974.5, 0.63015.075, 0.67335.65, 0.68386.225, 0.65066.8, 0.50263.355, 0.40293.93, 0.55064.505, 0.63065.08, 0.67365.655, 0.68376.23, 0.656.805, 0.49973.36, 0.40473.935, 0.55154.51, 0.63115.085, 0.67385.66, 0.68366.235, 0.64946.81, 0.49683.365, 0.40663.94, 0.55244.515, 0.63165.09, 0.6745.665, 0.68366.24, 0.64886.815, 0.49393.37, 0.40843.945, 0.55334.52, 0.63225.095, 0.67435.67, 0.68356.245, 0.64826.82, 0.49083.375, 0.41023.95, 0.55424.525, 0.63275.1, 0.67455.675, 0.68346.25, 0.64766.825, 0.48773.38, 0.4123.955, 0.5554.53, 0.63325.105, 0.67475.68, 0.68336.255, 0.6476.83, 0.48453.385, 0.41383.96, 0.55594.535, 0.63375.11, 0.67495.685, 0.68326.26, 0.64646.835, 0.48123.39, 0.41563.965, 0.55684.54, 0.63425.115, 0.67525.69, 0.68316.265, 0.64576.84, 0.47793.395, 0.41733.97, 0.55774.545, 0.63475.12, 0.67545.695, 0.6836.27, 0.64516.845, 0.47443.4, 0.41913.975, 0.55854.55, 0.63525.125, 0.67565.7, 0.68296.275, 0.64456.85, 0.47083.405, 0.42083.98, 0.55944.555, 0.63575.13, 0.67585.705, 0.68286.28, 0.64386.855, 0.46723.41, 0.42253.985, 0.56034.56, 0.63625.135, 0.6765.71, 0.68276.285, 0.64326.86, 0.46343.415, 0.42423.99, 0.56114.565, 0.63665.14, 0.67625.715, 0.68266.29, 0.64256.865, 0.45953.42, 0.42593.995, 0.5624.57, 0.63715.145, 0.67645.72, 0.68256.295, 0.64186.87, 0.45553.425, 0.42764, 0.56284.575, 0.63765.15, 0.67665.725, 0.68246.3, 0.64126.875, 0.45143.43, 0.42924.005, 0.56374.58, 0.63815.155, 0.67685.73, 0.68236.305, 0.64056.88, 0.44713.435, 0.43094.01, 0.56454.585, 0.63865.16, 0.6775.735, 0.68226.31, 0.63986.885, 0.44263.44, 0.43254.015, 0.56544.59, 0.6395.165, 0.67725.74, 0.6826.315, 0.63916.89, 0.4383.445, 0.43414.02, 0.56624.595, 0.63955.17, 0.67745.745, 0.68196.32, 0.63846.895, 0.43333.45, 0.43574.025, 0.5674.6, 0.645.175, 0.67765.75, 0.68186.325, 0.63766.9, 0.42833.455, 0.43734.03, 0.56784.605, 0.64055.18, 0.67785.755, 0.68166.33, 0.63696.905, 0.42323.46, 0.43894.035, 0.56874.61, 0.64095.185, 0.6785.76, 0.68156.335, 0.63626.91, 0.41783.465, 0.44054.04, 0.56954.615, 0.64145.19, 0.67815.765, 0.68146.34, 0.63546.915, 0.41213.47, 0.4424.045, 0.57034.62, 0.64185.195, 0.67835.77, 0.68126.345, 0.63476.92, 0.40633.475, 0.44364.05, 0.57114.625, 0.64235.2, 0.67855.775, 0.68116.35, 0.63396.925, 0.40013.48, 0.44514.055, 0.57194.63, 0.64275.205, 0.67875.78, 0.68096.355, 0.63326.93, 0.39353.485, 0.44664.06, 0.57274.635, 0.64325.21, 0.67885.785, 0.68086.36, 0.63246.935, 0.38663.49, 0.44814.065, 0.57354.64, 0.64375.215, 0.6795.79, 0.68066.365, 0.63166.94, 0.37933.495, 0.44964.07, 0.57434.645, 0.64415.22, 0.67925.795, 0.68046.37, 0.63086.945, 0.37153.5, 0.45114.075, 0.57514.65, 0.64455.225, 0.67945.8, 0.68036.375, 0.636.95, 0.36313.505, 0.45264.08, 0.57594.655, 0.6455.23, 0.67955.805, 0.68016.38, 0.62926.955, 0.35413.51, 0.45414.085, 0.57674.66, 0.64545.235, 0.67975.81, 0.67996.385, 0.62846.96, 0.34433.515, 0.45564.09, 0.57744.665, 0.64595.24, 0.67985.815, 0.67986.39, 0.62756.965, 0.33363.52, 0.4574.095, 0.57824.67, 0.64635.245, 0.685.82, 0.67966.395, 0.62676.97, 0.32163.525, 0.45844.1, 0.5794.675, 0.64675.25, 0.68015.825, 0.67946.4, 0.62586.975, 0.30823.53, 0.45994.105, 0.57984.68, 0.64725.255, 0.68035.83, 0.67926.405, 0.6256.98, 0.29263.535, 0.46134.11, 0.58054.685, 0.64765.26, 0.68045.835, 0.6796.41, 0.62416.985, 0.27413.54, 0.46274.115, 0.58134.69, 0.6485.265, 0.68065.84, 0.67886.415, 0.62326.99, 0.25063.545, 0.46414.12, 0.5824.695, 0.64845.27, 0.68075.845, 0.67866.42, 0.62236.995, 0.21753.55, 0.46554.125, 0.58284.7, 0.64885.275, 0.68095.85, 0.67846.425, 0.62147, 0.1443.555, 0.46694.13, 0.58354.705, 0.64935.28, 0.6815.855, 0.67826.43, 0.62053.56, 0.46834.135, 0.58434.71, 0.64975.285, 0.68115.86, 0.6786.435, 0.61963.565, 0.46964.14, 0.5854.715, 0.65015.29, 0.68135.865, 0.67786.44, 0.61863.57, 0.4714.145, 0.58584.72, 0.65055.295, 0.68145.87, 0.67766.445, 0.6177For the unimodal problem, since(5.4)dg(x)dx=ddx∑r=0∞crHr(x)ϕ(x)=ϕ(x){H1+∑r=1ncrHr+1(x)}=x+λ36(x4-6x2+3)+λ4-324(x5-10x3+15x)+λ3272(x7-21x5+105x3-105x)
and since we want g′(x) to have exactly one real root, we set νa-νb=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.Table 4
Unimodal region boundary(λ4,λ3) for Edgeworth Expansion.
3, 0.12653.345, 0.36153.69, 0.44254.035, 0.48854.38, 0.51154.725, 0.51155.07, 0.483.005, 0.143.35, 0.3633.695, 0.44354.04, 0.4894.385, 0.51154.73, 0.51155.075, 0.47953.01, 0.1513.355, 0.36453.7, 0.44454.045, 0.48954.39, 0.51154.735, 0.5115.08, 0.47853.015, 0.163.36, 0.3663.705, 0.4454.05, 0.494.395, 0.5124.74, 0.5115.085, 0.47753.02, 0.16853.365, 0.36753.71, 0.4464.055, 0.49054.4, 0.5124.745, 0.5115.09, 0.4773.025, 0.1763.37, 0.3693.715, 0.4474.06, 0.4914.405, 0.5124.75, 0.51055.095, 0.4763.03, 0.1833.375, 0.37053.72, 0.4484.065, 0.49154.41, 0.5124.755, 0.51055.1, 0.4753.035, 0.18953.38, 0.3723.725, 0.44854.07, 0.4924.415, 0.51254.76, 0.515.105, 0.4743.04, 0.1953.385, 0.37353.73, 0.44954.075, 0.49254.42, 0.51254.765, 0.515.11, 0.4733.045, 0.2013.39, 0.3753.735, 0.454.08, 0.49254.425, 0.51254.77, 0.50955.115, 0.4723.05, 0.2063.395, 0.37653.74, 0.4514.085, 0.4934.43, 0.5134.775, 0.50955.12, 0.4713.055, 0.2113.4, 0.3783.745, 0.4524.09, 0.49354.435, 0.5134.78, 0.5095.125, 0.473.06, 0.2163.405, 0.37953.75, 0.45254.095, 0.4944.44, 0.5134.785, 0.5095.13, 0.4693.065, 0.22053.41, 0.3813.755, 0.45354.1, 0.49454.445, 0.5134.79, 0.50855.135, 0.4683.07, 0.2253.415, 0.3823.76, 0.4544.105, 0.4954.45, 0.5134.795, 0.50855.14, 0.4673.075, 0.22953.42, 0.38353.765, 0.4554.11, 0.49554.455, 0.51354.8, 0.5085.145, 0.4663.08, 0.23353.425, 0.3853.77, 0.45554.115, 0.49554.46, 0.51354.805, 0.50755.15, 0.4653.085, 0.23753.43, 0.38653.775, 0.45654.12, 0.4964.465, 0.51354.81, 0.50855.155, 0.4643.09, 0.24153.435, 0.38753.78, 0.4574.125, 0.49654.47, 0.51354.815, 0.50755.16, 0.46253.095, 0.2453.44, 0.3893.785, 0.4584.13, 0.4974.475, 0.51354.82, 0.5075.165, 0.46153.1, 0.24853.445, 0.393.79, 0.45854.135, 0.49754.48, 0.5144.825, 0.50655.17, 0.46053.105, 0.2523.45, 0.39153.795, 0.45954.14, 0.49754.485, 0.5144.83, 0.50655.175, 0.4593.11, 0.25553.455, 0.3933.8, 0.464.145, 0.4984.49, 0.5144.835, 0.5065.18, 0.4583.115, 0.2593.46, 0.3943.805, 0.4614.15, 0.49854.495, 0.5144.84, 0.50555.185, 0.45653.12, 0.2623.465, 0.39553.81, 0.46154.155, 0.4994.5, 0.5144.845, 0.50555.19, 0.45553.125, 0.26553.47, 0.39653.815, 0.46254.16, 0.49954.505, 0.5144.85, 0.5055.195, 0.4543.13, 0.26853.475, 0.3983.82, 0.46354.165, 0.49954.51, 0.5144.855, 0.50455.2, 0.4533.135, 0.27153.48, 0.3993.825, 0.4644.17, 0.54.515, 0.51454.86, 0.5045.205, 0.45153.14, 0.27453.485, 0.40053.83, 0.46454.175, 0.50054.52, 0.51454.865, 0.5045.21, 0.453.145, 0.2773.49, 0.40153.835, 0.4654.18, 0.50054.525, 0.51454.87, 0.50355.215, 0.44853.15, 0.283.495, 0.4033.84, 0.4664.185, 0.5014.53, 0.51454.875, 0.5035.22, 0.4473.155, 0.2833.5, 0.4043.845, 0.46654.19, 0.50154.535, 0.51454.88, 0.50255.225, 0.44553.16, 0.28553.505, 0.4053.85, 0.4674.195, 0.50154.54, 0.51454.885, 0.50255.23, 0.4443.165, 0.2883.51, 0.40653.855, 0.4684.2, 0.5024.545, 0.51454.89, 0.5025.235, 0.44253.17, 0.2913.515, 0.40753.86, 0.46854.205, 0.50254.55, 0.51454.895, 0.50155.24, 0.4413.175, 0.29353.52, 0.40853.865, 0.4694.21, 0.50254.555, 0.51454.9, 0.5015.245, 0.43953.18, 0.2963.525, 0.413.87, 0.474.215, 0.5034.56, 0.51454.905, 0.50055.25, 0.4383.185, 0.29853.53, 0.4113.875, 0.47054.22, 0.50354.565, 0.51454.91, 0.55.255, 0.4363.19, 0.3013.535, 0.4123.88, 0.4714.225, 0.50354.57, 0.51454.915, 0.49955.26, 0.43453.195, 0.3033.54, 0.4133.885, 0.4724.23, 0.5044.575, 0.51454.92, 0.4995.265, 0.43253.2, 0.30553.545, 0.41453.89, 0.47254.235, 0.50454.58, 0.51454.925, 0.49855.27, 0.4313.205, 0.3083.55, 0.41553.895, 0.4734.24, 0.50454.585, 0.51454.93, 0.4985.275, 0.4293.21, 0.313.555, 0.41653.9, 0.47354.245, 0.5054.59, 0.51454.935, 0.49755.28, 0.4273.215, 0.31253.56, 0.41753.905, 0.47454.25, 0.5054.595, 0.51454.94, 0.4975.285, 0.4253.22, 0.31453.565, 0.41853.91, 0.4754.255, 0.50554.6, 0.51454.945, 0.49655.29, 0.4233.225, 0.3173.57, 0.41953.915, 0.47554.26, 0.5064.605, 0.51454.95, 0.4965.295, 0.4213.23, 0.3193.575, 0.42053.92, 0.4764.265, 0.5064.61, 0.51454.955, 0.49555.3, 0.4193.235, 0.3213.58, 0.42153.925, 0.47654.27, 0.50654.615, 0.5144.96, 0.4955.305, 0.4173.24, 0.3233.585, 0.4233.93, 0.47754.275, 0.50654.62, 0.5144.965, 0.49455.31, 0.41453.245, 0.3253.59, 0.4243.935, 0.4784.28, 0.5074.625, 0.5144.97, 0.4945.315, 0.41253.25, 0.32753.595, 0.4253.94, 0.47854.285, 0.5074.63, 0.5144.975, 0.49355.32, 0.413.255, 0.32953.6, 0.4263.945, 0.4794.29, 0.50754.635, 0.5144.98, 0.4935.325, 0.40753.26, 0.3313.605, 0.4273.95, 0.47954.295, 0.50754.64, 0.5144.985, 0.4925.33, 0.4053.265, 0.3333.61, 0.4283.955, 0.484.3, 0.5084.645, 0.5144.99, 0.49155.335, 0.40253.27, 0.3353.615, 0.4293.96, 0.48054.305, 0.5084.65, 0.51354.995, 0.4915.34, 0.43.275, 0.3373.62, 0.433.965, 0.48154.31, 0.50854.655, 0.51355, 0.49055.345, 0.18753.28, 0.3393.625, 0.4313.97, 0.4824.315, 0.50854.66, 0.51355.005, 0.48955.35, 0.1633.285, 0.34053.63, 0.43153.975, 0.48254.32, 0.5094.665, 0.51355.01, 0.4895.355, 0.1423.29, 0.34253.635, 0.43253.98, 0.4834.325, 0.5094.67, 0.51355.015, 0.48855.36, 0.12253.295, 0.34453.64, 0.43353.985, 0.48354.33, 0.50954.675, 0.5135.02, 0.48755.365, 0.1053.3, 0.3463.645, 0.43453.99, 0.4844.335, 0.50954.68, 0.5135.025, 0.4875.37, 0.08853.305, 0.3483.65, 0.43553.995, 0.48454.34, 0.50954.685, 0.5135.03, 0.48655.375, 0.07253.31, 0.34953.655, 0.43654, 0.4854.345, 0.514.69, 0.51255.035, 0.48555.38, 0.0573.315, 0.35153.66, 0.43754.005, 0.48554.35, 0.514.695, 0.51255.04, 0.4855.385, 0.04253.32, 0.3533.665, 0.4384.01, 0.4864.355, 0.51054.7, 0.51255.045, 0.4845.39, 0.0283.325, 0.35453.67, 0.4394.015, 0.48654.36, 0.51054.705, 0.51255.05, 0.48355.395, 0.013.33, 0.35653.675, 0.444.02, 0.4874.365, 0.51054.71, 0.5125.055, 0.48253.335, 0.3583.68, 0.4414.025, 0.48754.37, 0.5114.715, 0.5125.06, 0.4823.34, 0.35953.685, 0.4424.03, 0.4884.375, 0.5114.72, 0.5125.065, 0.481With the inclusion of the 5th moment, the Edgeworth Expansion becomes a polynomial of degree nine:(5.5)g(x)=(1+λ36H3(x)+(λ4-324)H4(x)+(λ3272)H6(x)+λ5120H5(x)+λ3(λ4-3)144H7(x)+(λ331296)H9(x))12πe-x2/2.
A 9-degree polynomial will always have a real root; hence the pdf approximation will always have some combination of λ3 and λ4 that results in negative coordinates. Draper and Tierney [2] developed an algorithm to produce Edgeworth Expansions up to the 10th moment.Using the same approach developed previously and assuming that all moment up to the 4th moment match, (i.e.,λ1=0, λ2=1, λ3=0, λ4=3),
(5.6)g(x)=(1+λ5120H5(x)+(λ6-15720)H6(x))12πe-x2/2.
We see that g(x) is the same as the Gram-Charlier Series, and hence similar conditions for λ3 and λ4 apply.


## 6. 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 ofλ3 and λ4. 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 λi’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 the robustness of departures from normality, we must restrict our conclusions to specific values of  λ3 and λ4.  Similar limitations occur when examining the relationships between probability density functions and their cumulants.Algorithm 1:Mathematica source code.
f:= proc (b1, b2, x)1+sqrt(b1)/6 * (x∧3-3x) + (b2-3)/24 *(x∧4-6x∧2+3)end;readlib(sturm):for i from 3 by 0.001 to 7 dotest:=0for j from 0.0001 by 0.0001 while test=0 dotest:=sturm(sturmseq(f(j,i,x),x),x,-100,100):if test <> 0then lprint(evalf(i,4),evalf(j-0.0001,9));fi; od; od; quit;f:= proc (b1, b2, x)x+sqrt(b1)/6 *(x∧4-6*x∧2+3) + (b2-3)/24 * (x∧5 - 10*x∧3 + 15 * x)end;readlib(sturm):for i from 0.01 by 0.001 to 0.7 dofor j from 3 by 0.005 to 5.6 dotest:=sturm(sturmseq(f(i, j, x),x), x, -100, 100);if test = 1then lprint(evalf(i,4), evalf(j,4))fi; od; od;



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*Source: 463097-2011-05-26.xml*