Che Tat Ng

Measures of distance between probability distributions

Abstract In statistical estimation problems measures between probability distributions play significant roles. Hellinger coefficient, Jeffreys distance, Chernoff coefficient, directed divergence, and its symmetrization J -divergence are examples of such measures. Here these and like measures are characterized through a composition law and the sum form they possess. The functional equations f ( pr , qs ) + f ( ps , qr ) = ( r + s ) f ( p , q ) + ( p + q ) f ( r , s ) and f ( pr , qs ) + f ( ps , qr ) = f ( p , q ) f ( r , s ) are instrumental in their deduction.

Jensen's functional equation on groups

Summary.A natural extension of Jensens functional equation on the real line is the equation

Measurable solutions of functional equations related to information theory

f(xy)+f(xy^{-1}) = 2f(x)

Functional Equations and Inequalities in ‘Rational Group Decision Making’

where f maps a group G into an abelian group H. When normalized, it defines semi-homomorphisms. It has been solved when G is free with up to two generators, and when

Recursive inset entropies of multiplicative type on open domains

G=GL_2(\Bbb Z)

A functional equation arising from ranked additive and separable utility

. Here, the results are extended to include all free groups and

Jensen's functional equation on groups, III

GL_n(\Bbb Z),\ n\geq 3

Functional equations arising in a theory of rank dependence and homogeneous joint receipts

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On Cauchy differences of all orders

Measurable solutions of functional equations connected with Shannons measure of entropy, directed divergence or information gain and inaccuracy are found.

Entropy-Related Measures of the Utility of Gambling

This paper consists of a reformulation and generalization of results in [2–4]. A problem of ‘rational group decision making’ is the following: A fixed amount s is to be allocated to a fixed number m of competing projects. Each member of a group of n decision makers makes recommendations, the jth allocating, say, xij to the ith project, in order to establish the ‘consensus’ allocation fi(x i) [we write x i = (xil,..., xin)]. We suppose only that fi(0) = 0 [‘consensus of rejection’; 0 = (0,...,0)] and that the allocations are non-negative. In the case m > 2, we prove that each fi is the same weighted arithmetic mean. There are other solutions for m = 2, even if f1 = f2 is supposed. We determine all of them. The solutions are also established for variable s.