Edgeworth and Cornish-Fisher Expansion for the Non-normal t

다국어 초록 (Multilingual Abstract)

Let $X_i,...,X_n$ be a random sample from a distribution with cumulants $K_1, K_2,...$. The statistic $t = \frac{\sqrt{x}(\bar{X}-K_1)}{S}$ has the well-known 'student' distribution with $\nu = n-1$ degrees of freedom if the $X_i$ are normally distributed (i.e., $K_i = 0$ for $i \geq 3$). An Edgeworth series expansion for the distribution of t when the $X_i$ are not normally distributed is obtained. The form of this expansion is Prob $(t<x)=Prob (t^\circ < x)+f(x)\sum P_i(x)/\sqrt{\nu}^i$ where $t^0$ is student's t, $P_i(x)$ is a polynomial of degree $3i-1$ whose coefficients are functions of the first $i+2$ cumulants, and $f(x) = exp (-x^2/2)/\sqrt{2\pi}$. The Edgeworth series is inverted to yield the Cornish-Fisher expansion $t_p = t^{\circ}_p + \sub Q_i(x)/\sqrt{\nu}^i$, where $t_p,t^{\circ}_p$, and $x=s_p$ are the 100p percentile points of the non-normal t, "student" t and the unit normal, respectively, and $Q_i(x)$ is a polynomial of degree $i+1$ in x whose coefficients are functions of the first $i+2$ cumulants. Comparison between the values obtained by the computer simulation and by the approximate formula shows good agreement on each of the 100p percentile points.le points.