Bivariate least squares linear regression: towards a unified analytic formalism

Concerning bivariate least squares linear regression, the classical approach pursued for functional models in earlier attempts is reviewed using a new formalism in terms of deviation (matrix) traces. Within the framework of classical error models, the dependent variable relates to the independent variable according to the usual additive model. Linear models of regression lines are considered in the general case of correlated errors in X and in Y for heteroscedastic data. The special case of (C) generalized orthogonal regression is considered in detail together with well known subcases. In the limit of homoscedastic data, the results determined for functional models are compared with their counterparts related to extreme structural models. While regression line slope and intercept estimators for functional and structural models necessarily coincide, the contrary holds for related variance estimators even if the residuals obey a Gaussian distribution, with a single exception. An example of astronomical application is considered, concerning the [O/H]-[Fe/H] empirical relations deduced from five samples related to different stars and/or different methods of oxygen abundance determination. For selected samples and assigned methods, different regression models yield consistent results within the errors for both heteroscedastic and homoscedastic data. Conversely, samples related to different methods produce discrepant results, due to the presence of (still undetected) systematic errors, which implies no definitive statement can be made at present. A comparison is also made between different expressions of regression line slope and intercept variance estimators, where fractional discrepancies are found to be not exceeding a few percent, which grows up to about 20% in presence of large dispersion data.Comment: 56 pages, 2 tables, and 2 figures. New Astronomy, accepte

Shelf space strategy in long-tail markets

The Internet is known to have had a powerful impact on on-line retailer strategies in markets characterised by long-tail distribution of sales. Such retailers can exploit the long tail of the market, since they are effectively without physical limit on the number of choices on offer. Here we examine two extensions of this phenomenon. First, we introduce turnover into the long-tail distribution of sales. Although over any given period such as a week or a month, the distribution is right-skewed and often power law distributed, over time there is considerable turnover in the rankings of sales of individual products. Second, we establish some initial results on the implications for shelf-space strategy of physical retailers in such markets.Comment: 10 pages, 3 figure

Towards Machine Wald

The past century has seen a steady increase in the need of estimating and predicting complex systems and making (possibly critical) decisions with limited information. Although computers have made possible the numerical evaluation of sophisticated statistical models, these models are still designed \emph{by humans} because there is currently no known recipe or algorithm for dividing the design of a statistical model into a sequence of arithmetic operations. Indeed enabling computers to \emph{think} as \emph{humans} have the ability to do when faced with uncertainty is challenging in several major ways: (1) Finding optimal statistical models remains to be formulated as a well posed problem when information on the system of interest is incomplete and comes in the form of a complex combination of sample data, partial knowledge of constitutive relations and a limited description of the distribution of input random variables. (2) The space of admissible scenarios along with the space of relevant information, assumptions, and/or beliefs, tend to be infinite dimensional, whereas calculus on a computer is necessarily discrete and finite. With this purpose, this paper explores the foundations of a rigorous framework for the scientific computation of optimal statistical estimators/models and reviews their connections with Decision Theory, Machine Learning, Bayesian Inference, Stochastic Optimization, Robust Optimization, Optimal Uncertainty Quantification and Information Based Complexity.Comment: 37 page

Il contributo di La Volpe alla teoria dinamica dell'economia

The paper presents the dynamic theory proposed by La Volpe in 1936. This analysis has been innovative in many ways: general equilibrium is defined as temporary, the presence and the role of expectations are introduced, the intertemporal choice of the agents is determined in such a way as to anticipate the life-cycle theory, and some important problems that emerge in the dynamic analysis are addressed. The relevance of La Volpe's book led Michio Morishima to publish its English translation