Entropy Testing is Ef cient

—This paper compares the power divergence statistics of orders > 1 with the information divergence statistic in the problem of testing the uniformity of a distribution. In this problem the information divergence statistic is equivalent to the entropy statistic. Extending some previously established results about information diagrams, it is proved that in this problem the information divergence statistic is more ef cient in the Bahadur sense than any power divergence statistic of order > 1: This means that the entropy provides in a certain sense the most ef cient way of characterizing the uniformity of a distribution. I. POWER DIVERGENCE STATISTICS Let M(k) denote the set of all discrete probability distributions of the form P = (p1; :::; pk) and M(kjn) the subset of possible types. One of the fundamental problems of mathematical statistics can be described as follows. Consider n balls distributed into boxes 1; :::; k independently according to an unknown probability law Pn 2M (k), which possibly depends on the number of balls n. This results in frequency counts Xn1; : : : ; Xnk the vector of which Xn = (Xn1; : : : ; Xnk) 2 f0; 1; : : :g is multinomially distributed with parameters k, n and Pn. The problem is to decide on the basis of observations Xn whether the unknown law Pn is equal to a given Q = (q1; :::; qk) 2M (k) or not. The observations Xn are represented by the empirical distribution P̂n = p̂n1 4 = Xn1=n; :::; p̂nk 4 = Xnk=n 2M(kjn) (1) and procedure T on accepting or rejecting a hypothesis based on P̂n is called a test. The test uses a statistic Tn(P̂n; Q) which characterizes the goodness-oft between the distributions P̂n and Q. The test T rejects the hypothesis Pn = Q if T = Tn(P̂n; Q) exceeds a certain level rn 2 R. The goodness-oft statistic is usually one of the power divergence statistics T = T ;n = 2nD (P̂n; Q); 2 R: (2) where D (P;Q) denotes the so-called -divergence (power divergence of order ) of distributions P;Q 2 M (k) de ned by