Distance measures and asymptotic relative efficiency

The relationship between distance measures and asymptotic relative efficiency is discussed. It is shown that the ratio of the Bhattacharyya distance or J divergences of two test statistics is equivalent to asymptotic relative efficiency. Two-input systems are discussed as examples, and the performances of the polarity coincidence correlator (PCC) and the correlator are discussed in terms of the distance measures of reduced data. HE NOTION of a distance measure between two probability measures is widely used in statistics. Grettenberg [l], Kailath [2], and Kadota and Shepp [3] discuss the application of some of these measures to communication problems. Let pi(z) and pz(s) be density functions of probability measures P,(z) and Pz(x) defined over X, a space of observations x, under the hypotheses HI and H,, respectively. Let L(x) be the Radon-Nikodym derivative of P, with respect to P,, i.e., the likelihood ratio L(x) = Pz(X>lP1(X>* 0) Then many of the distance measures currently used can be written in the form [4] Manuscript received June 11, 1969; revised October 10, 19!9. This work was supported by the National Science Foundation under Grants GK-187 and GK-1439 and by the U.S. Army Research Office, Durham, N. C. under Contract DA-31-124-ARO-D-292. The author is on sabbatical leave from IBM Thomas J. Watson Research Center, Yorktown Heights, N. Y. He is presently with the School of Engineering and Applied Science, University of California, Los Angeles, Calif. 90024. (2) where $,( *) is a continuous convex function on (0, a), f( .) is an increasing real-valued function of a real variable, and E,[ -1 is the expectation under the probability measure P,. Typical examples are the following. J Divergence [5] J, = E,[(L(x) 1) In L(x)]. Bhattachuryya Distance (B Distance) [2], [B]