Power-law (PL) formalism is known to provide an appropriate framework for canonical modeling of nonlinear systems. We estimated three stochastically distinct models of constant elasticity of substitution (CES) class functions as non-linear inverse problem and showed that these PL related functions should have a closed form. The first model is related to an aggregator production function, the second to an aggregator utility function (the Armington) and the third to an aggregator technical transformation function. A q-generalization of K-L information divergence criterion function with a priori consistency constraints is proposed. Related inferential statistical indices are computed. The approach leads to robust estimation and to new findings about the true stochastic nature of this class of nonlinear-up until now-analytically intractable functions. Outputs from traditional econometric techniques (Shannon entropy, NLLS, GMM, ML) are also presented.

The document proposes a new entropy-based approach for estimating the parameters of nonlinear and complex models, i.e. those whose no transformation renders linear in parameters. Presently, for estimating such class of functions, various iterative technics like the GaussNewton algorithm are applied and completed by the least square methods approaches. Due to conceptual nature of such methods, denitely estimated functions are dierent from the original nonlinear one and the estimated values of parameters are in most of cases far from the true values. The proposed approach, being related to the statistical theory of information, is very dierent from those so far applied for that class of functions. To apply the approach, we select a stochastic non-homogeneous constant elasticity of substitution aggregated production function of the 27 EU countries which we estimate maximizing a non-extensive entropy model under consistency restrictions related to the constant elasticity of substitution model plus regular normality conditions. The procedure might be seen as an attempt to generalize the recent works (e.g. Golan et al. 1996) on entropy econometrics in the case of ergodic systems, related to the GibbsShannon maximum entropy principle. Since this nonlinear constant elasticity of substitution estimated model contains four parameters in one equation and statistical observations are limited to twelve years, we have to deal with an inverse problem and the statistical distribution law of the data generating system is unknown. Because of the above reasons, our approach moves away from the normal Gaussian hypothesis to the more general Levy instable time (or space) processes characterized by long memory, complex correlation and by a convergence, in relative long range, to the attraction basin of the central theorem limit. In such a case, fractal properties may eventually exist and the q non extensive parameter could give us useful information. Thus, as already suggested, we will propose to solve for a stochastic inverse problem through the generalized minimum entropy divergence under the constant elasticity of substitution model and other normalization factor restrictions. At the end, an inferential condence interval for parameters is proposed. The output parameters from entropy formalism represent the long-run state of the system in equilibrium, and so, their interpretation is slightly dierent from the ceteris paribus interpretation related to the classical econometrical modeling. The approach seems to produce very ecient parameters in comparison to those obtained from the classical iterative nonlinear method which will be presented, too.

Air pollution is closely associated with the development of respiratory illness. The aim of the present study was to assess the relationship between long-term exposure to PM2.5, PM10, NO2, and SO2 pollution and the incidence of lung cancer in the squamous subtype in south-eastern Poland from the years 2004 to 2014. We collected data of 4237 patients with squamous cell lung cancer and the level of selected pollutants. To investigate the relationship between the level of concentrations of pollutants and the place of residence of patients with lung cancer in the squamous subtype, proprietary pollution maps were applied to the places of residence of patients. To analyze the data, the risk ratio was used as well as a number of statistical methods, i.e., the pollution model, inverse distance weighted interpolation, PCA, and ordered response model. Cancer in women and in men seems to depend in particular on the simultaneous inhalation of NO2 and PM10 (variable NO2PM10) and of NO2 and SO2 (variable NO2 SO2), respectively. Nitrogen dioxide exercises a synergistic leading effect, which once composed with the other elements it becomes more persistent in explaining higher odds in the appearance of cancers and could constitute the main cause of squamous cancer.

This paper presents the quantitative characteristics of correlations (and cross-correlations) of plant main ecofactors i.e. the ground and over-ground temperature, the wind speed, and the humidity. The study is based upon hourly data statistical observations collected in the region of Lublin, in Poland for the period 2001.05. 07-2009.04.10. This paper indicates that plant growth conditions constitute an emergent response to the above direct eco-factors. Then, the dynamics properties of each eco-factor is first analyzed alone for its multifractal structure. We apply the multifractal detrended correlation analysis and multifractal detrended cross-correlation analysis. We show that the widest multifractal spectrum is for over-ground temperature and the strongest power-law cross-correlations exist between ground and over-ground temperature. Next, an impulse response analysis is carried out to measure dynamical inter causalities within all the considered variables. As far as cross-impact between different eco-variables is concerned, one observes that the wind speed, the ground temperature and the air humidity dynamics are the most influenced, in terms of memory length time, by external temperature.

This paper proposes the non-extensive entropy econometric approach to predict regional cross-industry greenhouse emissions within a country, based on imperfect knowledge of industrial and regional aggregates. The solution of this stochastic inverse problem is applied to Poland. Non-extensive entropy should remain a valuable device for econometric modelling even in the case of low frequency series since outputs provided by the Gibbs-Shannon entropy approach correspond to the Tsallis entropy limiting case of the Gaussian law when the Tsallis q-parameter equals unity. We, therefore, set up a q-Tsallis-Kullback-Leibler entropy criterion function with a priori consistency constraints, including the environmental Kuznets econometric model and regular conditions. As in the case of Shannon-Gibbs-based entropy models, we found that the Tsallis entropy estimator also belongs to the family of Stein estimators, meaning that smaller probabilities are shrunk and higher probabilities dominate in the solution space. Fortunately, adding more pertinent data to the model priors will enhance parameter precision and then allow for the recovery of the real influence of smaller events. The q-Tsallis-Kullback-Leibler entropy index is computed for different scenarios of the Kuznets model. The model outputs continue to conform to empirical expectations. In spite of the close to unity q-Tsallis parameter, this Tsallis related approach reflects higher stability for parameter computation in comparison with the Shannon-Gibbs entropy econometrics technique.