Principles of Edgeworth Expansion

Abstract

In this chapter we define, develop, and discuss Edgeworth expansions as approximations to distributions of estimates θ̂ of unknown quantities θ ₀. We call θ ₀ a “parameter”, for want of a better term. Briefly, if θ̂ is constructed from a sample of size n, and if n ^1/2 (θ̂ — θ ₀) is asymptotically Normally distributed with zero mean and variance σ ², then in a great many cases of practical interest the distribution function of n ^1/2 (θ̂ — θ ₀) may be expanded as a power series in n ^-1/2,

$$ P\left\{ {{n^{1/2}}\left( {\hat \theta - {\theta _0}} \right)/\sigma \le x} \right\} = \Phi \left( x \right) + {n^{ - 1/2}}{p_1}\left( x \right)\phi \left( x \right) + \ldots + {n^{-j/2}}{p_j}\left( x \right)\phi \left( x \right) + \ldots , $$

((2.1))

where $\phi \left( x \right) = {\left( {2\pi } \right)^{ - 1/2}}{e^{ - {x^{2/2}}}}$ is the Standard Normal density function and

$$ \Phi \left( x \right) = \int_{ - \infty }^x \phi \left( u \right)du$$

is the Standard Normal distribution function. Formula (2.1) is termed an Edgeworth expansion. The functions p _j are polynomials with coefficients depending on cumulants of θ̂ — θ ₀. Conceptually, we think of the expansion (2.1) as starting after the Normal approximation Φ(x), so that n ^-1/2 p ₁ (x) ϕ(x) is the first term rather than the second term.