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X The Kullback-Leibler divergence is related to Shannon’s entropy. If we assume a finite support and the probability distribution Pθ 2 is the uniform distribution, we have DKull (θ1 , θ2 ) = H(Pθ 2 ) − H(Pθ 1 ). The infinite support case may be written in terms of limits. R´enyi (1961) was the first who presented a generalization of Shannon’s entropy, given by Z h i 1 1 Hr1 (θ) = log fθ (x)r dµ(x) = log Eθ fθ (X)r−1 , r > 0, r 6= 1. 1−r 1−r X Liese and Vajda (1987) extended R´enyi’s entropy for all r ∈ R− {0, 1} by means of the expression Z h i 1 1 1 Hr (θ) = log fθ (x)r dµ(x) = log Eθ fθ (X)r−1 , r 6= 0, 1.

If φ(t) = t1/2 , M(θ1 , θ2 ) is the Matusita’s distance (1964). Some applications of K-divergences in statistical problems can be seen in P´erez and Pardo, J. A. (2002, 2003a, 2003b, 2003c, 2004 and 2005). 3. Bregman’s Distances Bregman (1967) introduced a family of divergences in the following way, Z ¢ ¡ Bϕ (θ1 , θ2 ) = ϕ(fθ 1 (x)) − ϕ(fθ 2 (x)) − ϕ0 (fθ 2 (x))(fθ 1 (x) − fθ 2 (x)) dµ(x) X for any differentiable convex function ϕ : (0, ∞) → R with ϕ(0) = lim ϕ(t) ∈ t→0 (−∞, ∞). We observe that for ϕ(t) = t log t, Bϕ (θ1 , θ2 ) is the Kullback-Leibler divergence and for ϕ(t) = t2 and discrete probability distributions, the Euclidean distance.

D, are normal with mean µi and variance aii , then H (Xi ) = H(µi , aii ) = log (aii 2πe)1/2 . © 2006 by Taylor & Francis Group, LLC 44 Statistical Inference based on Divergence Measures On the other hand, 1 log (det (Σ) (2πe))d . 29) we have the stated result. , Xd ) = 10. , Yd ) be a d-variate random vector with multivariate normal distribution, with mean vector µ and nonsingular variance-covariance matrix Σ. , xd ) (x − µ)T Σ−1 (x − µ) dx. + 2 Rd Furthermore, since Σ−1 is a symmetric nonnegative definite matrix, there exists an orthogonal matrix L such LT Σ−1 L = Λ for some diagonal matrix Λ.

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