By Leandro Pardo
The belief of utilizing functionals of data idea, comparable to entropies or divergences, in statistical inference isn't really new. even though, despite the fact that divergence statistics became a superb substitute to the classical probability ratio attempt and the Pearson-type statistic in discrete versions, many statisticians stay blind to this robust approach.
Statistical Inference in accordance with Divergence Measures explores classical difficulties of statistical inference, resembling estimation and speculation trying out, at the foundation of measures of entropy and divergence. the 1st chapters shape an outline, from a statistical viewpoint, of an important measures of entropy and divergence and research their homes. the writer then examines the statistical research of discrete multivariate information with emphasis is on difficulties in contingency tables and loglinear versions utilizing phi-divergence try records in addition to minimal phi-divergence estimators. the ultimate bankruptcy appears to be like at checking out typically populations, proposing the attention-grabbing probability of introducing replacement try information to classical ones like Wald, Rao, and probability ratio. every one bankruptcy concludes with workouts that make clear the theoretical effects and current extra effects that supplement the most discussions.
Clear, accomplished, and logically constructed, this booklet bargains a distinct chance to achieve not just a brand new viewpoint on a few common records difficulties, however the instruments to place it into perform.
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Extra resources for Statistical Inference Based on Divergence Measures
Example text
D )T and nonsingular variance-covariance matrix Σ, and L is an orthogonal matrix. We have det(LT ΣL) = det(LT ) det (Σ) det(L) = det(L−1 ) det (Σ) det(L) = det (Σ) . Then, H(X) ≡H(µ, Σ) = H(Y ). 9. ,d . , d. , Xd ) ≤ d X H (Xi ) . , 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.
Determine the Bhattacharyya divergence, Z B (θ1 , θ2 ) = − log (fθ 1 (x) fθ 2 (x))1/2 dµ(x), X between two univariate normal distributions. 14. 1/2 ¶2 q (θ1 , θ2 ) = fθ 1 (x) − fθ 2 (x) dµ(x) X is a metric. Find its expression for two multivariate normal distributions. 15. Evaluate the R´enyi’s divergence as well as the Kullback-Leibler divergence for two Poisson populations. 16. , Yl ) and (X, Y ) random vectors with multivariate normal distribution and variance-covariance matrices given by A, B and C respectively.
Then the conditional Shannon entropy of X given Y =y is defined by Z f(x,y) f(x,y) H (X/Y =y) = − log dx f2 (y) Rn f2 (y) and the conditional Shannon’s entropy of X given Y , by Z Z f(x,y) H (X/Y ) = − f(x,y) log f2 (y)H (X/Y =y) dy, dxdy= f2 (y) Rn+m Rm assuming the existence of the previous entropy. The following properties are verified by Shannon’s entropy: a) The Shannon’s entropy of X can be negative. , ϕn ) be a smooth bijection on Rn and we assume that Y = ϕ(X). , ψn ) of ϕ. ,n . s. This inequality is called Gibbs’s lemma for continuous random vectors.