We offer two noiseless codes for blocks of integers Xn = (X1, . . . , Xn). We provide explicit bounds on the relative redundancy that are valid for any distribution F in the class of memoryless sources with a possibly infinite alphabet whose marginal distribution is monotone. Specifically we show that the expected code length L(Xn) of our first universal code is dominated by a linear function o...

A minimax tree is similar to a Huffman tree except that, instead of minimizing the weighted average of the leaves’ depths, it minimizes the maximum of any leaf’s weight plus its depth. Golumbic (1976) introduced minimax trees and gave a Huffman-like, O(n log n)-time algorithm for building them. Drmota and Szpankowski (2002) gave another O(n log n)-time algorithm, which checks the Kraft Inequali...

We present a multidimensional optimization problem that is formulated and solved in the tropical mathematics setting. The problem consists in minimizing a nonlinear objective function defined on vectors in a finite-dimensional semimodule over an idempotent semifield by means of a conjugate transposition operator, subject to the constraints in the form of linear vector inequalities. A complete d...

Rényi divergence is related to Rényi entropy much like Kullback-Leibler divergence is related to Shannon’s entropy, and comes up in many settings. It was introduced by Rényi as a measure of information that satisfies almost the same axioms as Kullback-Leibler divergence, and depends on a parameter that is called its order. In particular, the Rényi divergence of order 1 equals the Kullback-Leibl...

We consider a sequential treatment problem with covariates. Given realization of the covariate vector, instead targeting highest conditional expectation, decision-maker targets which maximizes general functional potential outcome distribution, e.g., quantile, trimmed mean, or socioeconomic such as an inequality, welfare, poverty measure. develop expected regret lower bounds for this and constru...

 Minimax estimation problems with restricted parameter space reached increasing interest within the last two decades Some authors derived minimax and admissible estimators of bounded parameters under squared error loss and scale invariant squared error loss In some truncated estimation problems the most natural estimator to be considered is the truncated version of a classic...