The Annals of Mathematical Statistics

On a Theorem of Bahadur on the Rate of Convergence of Test Statistics

M. Raghavachari

Full-text: Open access


Let $x_1, x_2, cdots, x_n$ be $n$ independent and identically distributed random variables whose distribution depends on a parameter $\theta, \theta \in \Theta$. Let $\Theta_0$ be a subset of $\Theta$ and consider the test of the hypothesis that $\theta \in \Theta_0. L_n (x_1, \cdots, x_n)$ is the level attained by a test statistic $T_n(x_1, \cdots, x_n)$ in the sense that it is the maximum probability under the hypothesis of obtaining a value large or larger than $T_n$ where large values of $T_n$ are significant for the hypothesis. Under some assumptions Bahadur [3] showed that where a non-null $\theta$ obtains $L_n$ cannot tend to zero at a rate faster than $\lbrack\rho(\theta)\rbrack^n$ where $\rho$ is a function defined in terms Kullback-Liebler information numbers. In this paper this result has been shown to be true without any assumptions whatsoever (Theorem 1). Some aspects of the relationship between the rate of convergence of $L_n$ and rate of convergence of the size of the tests are also studied and an equivalence property is shown (Theorem 2).

Article information

Ann. Math. Statist., Volume 41, Number 5 (1970), 1695-1699.

First available in Project Euclid: 27 April 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier



Raghavachari, M. On a Theorem of Bahadur on the Rate of Convergence of Test Statistics. Ann. Math. Statist. 41 (1970), no. 5, 1695--1699. doi:10.1214/aoms/1177696813.

Export citation