**Sankhya: The Indian Journal of Statistics**

1996, Volume 58, Series A, Pt. 1, pp. 8--24

**COMPLEX VALUED CHARACTERISTIC FUNCTIONS WITH (SOME) REAL POWERS**

By

B. L. S. PRAKASA RAO, *Indian Statistical Institute*

SUMMARY. Consider a stochastic process ${X_t, t \geq 0}$ whose distributions depend on an unknown parameter ($\gamma$, $\theta$). A locally asymptotically most powerful test, for testing the composite hypothesis $H_0: \gamma = \gamma_{0}$ against $H_1: \gamma \neq \gamma_{0}$ in the presence of a nuisance parameter $\theta$ is developed following the concept of $C(\alpha)$-tests introduced by Neyman. Results are illustrated by means of examples of process ${X_t, t \geq 0}$ satisfying the linear stochastic differential equation $dX(t)=(\gammaX(t)+\theta)dt+dW(t)$, $t \geq 0$.

*AMS (1991) subject classification.* 62M99, 62F03.

*Key words and phrases.* $C(/alpha)$- tests, optimal asymptotic test for composite hypotheses, linear stochastic differential equation.