The Studentized Empirical Characteristic Function and Its Application to Test for the Shape of Distribution

Abstract

The empirical characteristic function is found to be effectively applied to test for the shape of distribution. The squared modulus of the studentized empirical characteristic function is suggested for testing the composite hypothesis that $\mu +\sigma X$ is subject to a known distribution for unknown constants $\mu$ and $\sigma$. It is shown that the studentized empirical characteristic function, if properly normalized, converges weakly to a complex Gaussian process. Asymptotic considerations as well as computer simulation reveal that the proposed statistic, when applied to test normality, is more efficient than or as efficient as the test by the sample kurtosis for certain types of alternatives.

This chapter was first published in Murota and Takeuchi (1981) Biometrika 68, 55–65.

Appendix

Proof of $\tilde{\rho }(t, s)\ge \rho (t, s)$

Since both $\tilde{\rho }$ and $\rho$ are positive, the desired inequality is equivalent to

$$\begin{aligned} \frac{\rho (t, s)}{\tilde{\rho }(t, s)} = \frac{g(ts)}{\{g(t^2)\}^{1/2} \{g(s^2)\}^{1/2}} \le 1, \end{aligned}$$

(9.26)

where

$$\begin{aligned} g(y) = \frac{\cosh (y)-1}{\cosh (y)-1-\frac{1}{2}y^2}. \end{aligned}$$

(9.27)

Now, it can be fairly easily shown that for functions $g(y)>0$ for $y>0$ the inequality

$$\begin{aligned} \frac{g(ts)}{\{g(t^2)\}^{1/2} \{g(s^2)\}^{1/2}} \le 1\quad (t>0,\ s>0) \end{aligned}$$

is equivalent to the convexity of $\log g(\mathrm {e}^x)\ (-\infty<x<\infty )$.

Since g(y) given by (9.27) is even, it suffices to establish (9.26) for $t>0$ and $s>0$. Thus we have to show the convexity of

$$\begin{aligned} \log \{\cosh (\mathrm {e}^x)-1\}-\log \Big \{\cosh (\mathrm {e}^x)-1-\frac{1}{2}\mathrm {e}^{2x}\Big \}. \end{aligned}$$

Successive differentiation proves this.