n n n xvi CONTENTS 2.3.8 Distributions of Functions of Random Variables 2.4 Mathematical Expectation 2.4.1 Expected Value of a Random Variable 2.4.2 Expectation of a Function of a Single Random Variable 2.4.3 Expectations of Functions of Several Random Variables 2.4.4 Moments ' 2.4.

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JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. The MIT Press is collaborating with JSTOR to digitize, preserve and extend access to The Review of Economics and Statistics.Abstract-The problem of recovering the entries of a large matrix of expenditure, trade or income flows from limitedincomplete multisectoral economic data is considered. Making use of some consistency and adding up restrictions, the problem is cast as a pure inverse problem and specified within a nonlinear optimization framework. Estimates of the unknown entries are provided along with an overall measure of uncertainty for the complete matrix and a measure of uncertainty for the individual elements. Artificial and real data are used to illustrate how the procedures may be applied and interpreted and to gauge performance under entropy and squared error measures.

The focus of this article is on entropy and Markov processes. We study the properties of functionals which are invariant with respect to monotonic transformations and analyze two invariant “additivity” properties: (i) existence of a monotonic transformation which makes the functional additive with respect to the joining of independent systems and (ii) existence of a monotonic transformation which makes the functional additive with respect to the partitioning of the space of states. All Lyapunov functionals for Markov chains which have properties (i) and (ii) are derived. We describe the most general ordering of the distribution space, with respect to which all continuous-time Markov processes are monotonic (the Markov order). The solution differs significantly from the ordering given by the inequality of entropy growth. For inference, this approach results in a convex compact set of conditionally “most random” distributions

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