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Z ³ © ª´ (2π)−d/2 det (Σ1 )−1/2 exp − 12 (x − µ1 )T Σ−1 (x − µ ) 1 1 ³ Rd ´ © ª × (2π)−d/2 det (Σ2 )−1/2 exp − 12 (x − µ2 )T Σ−1 dx. 2 (x − µ2 ) A= © 2006 by Taylor & Francis Group, LLC 48 Statistical Inference based on Divergence Measures Therefore, ½ ¾ Z ¢ 1¡ T −1 T −1 exp − (x − µ1 ) Σ1 (x − µ1 ) + (x − µ1 ) Σ1 (x − µ2 ) A=L 2 Rd with L = (2π)−d det (Σ1 )−1/2 det (Σ2 )−1/2 . Now we shall write the expression T −1 (x − µ1 )T Σ−1 1 (x − µ1 ) + (x − µ2 ) Σ2 (x − µ2 ) as (x − µ∗ )T C −1 (x − µ∗ ) + B.
4. Let X be a random variable with probability density function f(x). Show Z 1 x2 f (x) dx ≥ exp (2H (X)) . 2πe R 5. f. f. gθ (y), if there exists a nonnegative function h on the product space X × Y for which the following relations are satisfied: Z h(x, y)fθ (x)dµ(x) i) gθ (y) = X Z Z ii) h(x, y) ≥ 0, h(x, y)dµ(x) = h(x, y)dµ(y) = 1. X Y Show that H(Y ) ≥ H(X). 6. Derive the expression of Shannon’s entropy for the following random variables: Beta, Cauchy, Chi-square, Erlang, Exponential, F-Snedecor, Gamma, Laplace, Logistic, Lognormal, Maxwell-Normal, Normal, Normal-generalized, Pareto, Rayleigh and T-Student.
This author also presented a simple measure of divergence: the J-divergence among k populations. , k} . , xk ) = − kj=1 xj j , aj ≥ 0 with Pk j=1 aj = 1 the f-dissimilarity is the negative aﬃnity introduced by Toussaint, (1974). More examples can be seen in Gyorfi and Nemetz (1978) and Zografos (1998a). The f-dissimilarity leads also to the Csiszar’s φ-divergence if f(x1 , x2 ) = x2 φ(x1 /x2 ). Other interesting families of divergence measures among k populations can be seen in Kapur (1988), Sahoo and Wong (1988), Rao (1982a), Toussaint (1978).