139 research outputs found
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
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