Secant method

The Gram–Charlier A series (named in honor of Jørgen Pedersen Gram and Carl Charlier), and the Edgeworth series (named in honor of Francis Ysidro Edgeworth) are series that approximate a probability distribution in terms of its cumulants. The series are the same; but, the arrangement of terms (and thus the accuracy of truncating the series) differ.

Gram–Charlier A series

The key idea of these expansions is to write the characteristic function of the distribution whose probability density function is ${\mathit {F}}$ to be approximated in terms of the characteristic function of a distribution with known and suitable properties, and to recover ${\mathit {F}}$ through the inverse Fourier transform.

Let ${\mathit {f}}$ be the characteristic function of the distribution whose density function is ${\mathit {F}}$, and $\kappa _{r}$ its cumulants. We expand in terms of a known distribution with probability density function $\Psi$, characteristic function $\psi$, and cumulants $\gamma _{r}$. The density $\Psi$ is generally chosen to be that of the normal distribution, but other choices are possible as well. By the definition of the cumulants, we have the following formal identity:

$f(t)=\exp \left[\sum _{r=1}^{\infty }(\kappa _{r}-\gamma _{r}){\frac {(it)^{r}}{r!}}\right]\psi (t)\,.$

By the properties of the Fourier transform, (it)^rψ(t) is the Fourier transform of (−1)^r D^r $\Psi$(x), where D is the differential operator with respect to x. Thus, we find for F the formal expansion

$F(x)=\exp \left[\sum _{r=1}^{\infty }(\kappa _{r}-\gamma _{r}){\frac {(-D)^{r}}{r!}}\right]\Psi (x)\,.$

If $\Psi$ is chosen as the normal density with mean and variance as given by F, that is, mean μ = κ₁ and variance σ² = κ₂, then the expansion becomes

$F(x)=\exp \left[\sum _{r=3}^{\infty }\kappa _{r}{\frac {(-D)^{r}}{r!}}\right]{\frac {1}{{\sqrt {2\pi }}\sigma }}\exp \left[-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}\right]\,.$

By expanding the exponential and collecting terms according to the order of the derivatives, we arrive at the Gram–Charlier A series. If we include only the first two correction terms to the normal distribution, we obtain

$F(x)\approx {\frac {1}{{\sqrt {2\pi }}\sigma }}\exp \left[-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}\right]\left[1+{\frac {\kappa _{3}}{3!\sigma ^{3}}}H_{3}\left({\frac {x-\mu }{\sigma }}\right)+{\frac {\kappa _{4}}{4!\sigma ^{4}}}H_{4}\left({\frac {x-\mu }{\sigma }}\right)\right]\,,$

with H₃(x) = x³ − 3x and H₄(x) = x⁴ − 6x² + 3 (these are Hermite polynomials).

Note that this expression is not guaranteed to be positive, and is therefore not a valid probability distribution. The Gram–Charlier A series diverges in many cases of interest—it converges only if F(x) falls off faster than exp(−x²/4) at infinity (Cramér 1957). When it does not converge, the series is also not a true asymptotic expansion, because it is not possible to estimate the error of the expansion. For this reason, the Edgeworth series (see next section) is generally preferred over the Gram–Charlier A series.

Edgeworth series

Edgeworth developed a similar expansion as an improvement to the central limit theorem. The advantage of the Edgeworth series is that the error is controlled, so that it is a true asymptotic expansion.

Let {X_i} be a sequence of independent and identically distributed random variables with mean μ and variance σ², and let Y_n be their standardized sums:

$Y_{n}={\frac {1}{\sqrt {n}}}\sum _{i=1}^{n}{\frac {X_{i}-\mu }{\sigma }}.$

Let F_n denote the cumulative distribution functions of the variables Y_n. Then by the central limit theorem,

$\lim _{n\to \infty }F_{n}(x)=\Phi (x)\equiv \int _{-\infty }^{x}{\tfrac {1}{\sqrt {2\pi }}}e^{-{\frac {1}{2}}q^{2}}dq$

for every x, as long as the mean and variance are finite.

Now assume that the random variables X_i have mean μ, variance σ², and higher cumulants κ_r=σ^rλ_r. If we expand in terms of the standard normal distribution, that is, if we set

$\Psi (x)={\frac {1}{\sqrt {2\pi }}}\exp(-{\tfrac {1}{2}}x^{2})$

then the cumulant differences in the formal expression of the characteristic function f_n(t) of F_n are

$\kappa _{1}^{F(n)}-\gamma _{1}=0\,,$

$\kappa _{2}^{F(n)}-\gamma _{2}=0\,,$

$\kappa _{r}^{F(n)}-\gamma _{r}={\frac {\kappa _{r}}{\sigma ^{r}n^{r/2-1}}}={\frac {\lambda _{r}}{n^{r/2-1}}};\quad r\geq 3\,.$

The Edgeworth series is developed similarly to the Gram–Charlier A series, only that now terms are collected according to powers of n. Thus, we have

$f_{n}(t)=\left[1+\sum _{j=1}^{\infty }{\frac {P_{j}(it)}{n^{j/2}}}\right]\exp(-t^{2}/2)\,,$

where P_j(x) is a polynomial of degree 3j. Again, after inverse Fourier transform, the density function F_n follows as

$F_{n}(x)=\Phi (x)+\sum _{j=1}^{\infty }{\frac {P_{j}(-D)}{n^{j/2}}}\Phi (x)\,.$

The first five terms of the expansion are

${\begin{aligned}F_{n}(x)=&\Phi (x)\\&-{\frac {1}{n^{1/2}}}{\bigg (}{\tfrac {1}{6}}\lambda _{3}\,\Phi ^{(3)}(x){\bigg )}\\&+{\frac {1}{n}}{\bigg (}{\tfrac {1}{24}}\lambda _{4}\,\Phi ^{(4)}(x)+{\tfrac {1}{72}}\lambda _{3}^{2}\,\Phi ^{(6)}(x){\bigg )}\\&-{\frac {1}{n^{3/2}}}{\bigg (}{\tfrac {1}{120}}\lambda _{5}\,\Phi ^{(5)}(x)+{\tfrac {1}{144}}\lambda _{3}\lambda _{4}\,\Phi ^{(7)}(x)+{\tfrac {1}{1296}}\lambda _{3}^{3}\,\Phi ^{(9)}(x){\bigg )}\\&+{\frac {1}{n^{2}}}{\bigg (}{\tfrac {1}{720}}\lambda _{6}\,\Phi ^{(6)}(x)+{\big (}{\tfrac {1}{1152}}\lambda _{4}^{2}+{\tfrac {1}{720}}\lambda _{3}\lambda _{5}{\big )}\Phi ^{(8)}(x)\\&\qquad \quad +{\tfrac {1}{1728}}\lambda _{3}^{2}\lambda _{4}\,\Phi ^{(10)}(x)+{\tfrac {1}{31104}}\lambda _{3}^{4}\,\Phi ^{(12)}(x){\bigg )}\\&+O(n^{-5/2})\,.\end{aligned}}$

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Illustration: density of the sample mean of 3 Χ²

Density of the sample mean of three chi2 variables. The chart compares the true density, the normal approximation, and two edgeworth expansions

Take $X_{i}\sim \chi ^{2}(k=2)\qquad i=1,2,3$ and the sample mean ${\bar {X}}={\frac {1}{3}}\sum _{i=1}^{3}X_{i}$.

We can use several distributions for ${\bar {X}}$:

• The exact distribution, which follows a gamma distribution: ${\bar {X}}\sim \mathrm {Gamma} \left(\alpha =n\cdot k/2,\theta =2/n\right)$ = $\mathrm {Gamma} \left(\alpha =3,\theta =2/3\right)$
• The asymptotic normal distribution: ${\bar {X}}{\xrightarrow {n\to \infty }}N(k,2\cdot k/n)=N(2,4/3)$
• Two Edgeworth expansion, of degree 2 and 3

Disadvantages of the Edgeworth expansion

Edgeworth expansions can suffer from a few issues:

• They are not guaranteed to be a proper probability distribution as:
  • The integral of the density needs not integrate to 1
  • Probabilities can be negative
• They can be inaccurate, especially in the tails, due to mainly two reasons:
  • They are obtained under a Taylor series around the mean
  • They guarantee (asymptotically) an absolute error, not a relative one. This is an issue when one wants to approximate very small quantities, for which the absolute error might be small, but the relative error important.

See also

• Cornish–Fisher expansion

References

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Further reading

• H. Cramér. (1957). Mathematical Methods of Statistics. Princeton University Press, Princeton.
• D. L. Wallace. (1958). "Asymptotic approximations to distributions". Annals of Mathematical Statistics, 29: 635–654.
• M. Kendall & A. Stuart. (1977), The advanced theory of statistics, Vol 1: Distribution theory, 4th Edition, Macmillan, New York.
• P. McCullagh (1987). Tensor Methods in Statistics. Chapman and Hall, London.
• D. R. Cox and O. E. Barndorff-Nielsen (1989). Asymptotic Techniques for Use in Statistics. Chapman and Hall, London.
• P. Hall (1992). The Bootstrap and Edgeworth Expansion. Springer, New York.
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• S. Blinnikov and R. Moessner (1998). Expansions for nearly Gaussian distributions. Astronomy and astrophysics Supplement series, 130: 193–205.
• J. E. Kolassa (2006). Series Approximation Methods in Statistics (3rd ed.). (Lecture Notes in Statistics #88). Springer, New York.