A Quadratic and Linear Metric Characterizing the Sampling Design with Fixed Sample Size Considered From a Geometric Viewpoint

The first-order and second-order inclusion probabilities are chosen by the statistician. They are subjective probabilities. We innovatively define univariate and bivariate random quantities whose logically possible values are samples of a given size in order to obtain the first-order and second-order inclusion probabilities by means of their coherent previsions. We consider linear maps connected with univariate random quantities as well as bilinear maps connected with bivariate random quantities. The covariance of two univariate random quantities that are the components of a bivariate random quantity has been expressed by means of two bilinear maps. We show that a univariate random quantity denoted by S is complementary to the univariate Horvitz-Thompson estimator. We identify a quadratic and linear metric with regard to two univariate random quantities representing deviations that we innovatively define. We use the α-criterion of concordance introduced byGini in order to identify it. It is a statistical criterion that we innovatively apply to probability

Metric representations of a preference ordering

We prove that when we decompose the expected utility function inside of an mdimensional metric space we refer to a preference ordering based on the notion of distance. We prove that when we deal with a scale of measurable utilities we refer to a preference ordering based on the notion of distance. A contingent consumption plan is studied inside of an m-dimensional metric space because utility and probability are both subjective. The right closed structure in order to deal with utility and probability is a metric space in which we study coherent decisions under uncertainty having as their goal the maximization of the prevision of the utility associated with a contingent consumption plan

A reinterpretation of principal component analysis connected with linear manifolds identifying risky assets of a portfolio

We use the mean-variance model to study a portfolio problem characterized by an investment in two different types of asset. We consider m logically independent risky assets and a risk-free asset. We analyze m risky assets coinciding with m distributions of probability inside of a linear space. They generate a distribution of probability of a multivariate risky asset of order m. We show that an m-dimensional linear manifold is generated by m basic risky assets. They identify m finite partitions, where each of them is characterized by n incompatible and exhaustive elementary events. We suppose that it turns out to be n > m without loss of generality. Given m risky assets, we prove that all risky assets contained in an m-dimensional linear manifold are related. We prove that two any risky assets of them are conversely αorthogonal, so their covariance is equal to 0. We reinterpret principal component analysis by showing that the principal components are basic risky assets of an m-dimensional linear manifold. We consider a Bayesian adjustment of differences between prior distributions to posterior distributions existing with respect to a probabilistic and economic hypothesi

Aggregate Bound Choices about Random and Nonrandom Goods Studied via a Nonlinear Analysis

In this paper, bound choices are made after summarizing a finite number of alternatives. This means that each choice is always the barycenter of masses distributed over a finite set of alternatives. More than two marginal goods at a time are not handled. This is because a quadratic metric is used. In our models, two marginal goods give rise to a joint good, so aggregate bound choices are shown. The variability of choice for two marginal goods that are the components of a multiple good is studied. The weak axiom of revealed preference is checked and mean quadratic differences connected with multiple goods are proposed. In this paper, many differences from vast majority of current research about choices and preferences appear. First of all, conditions of certainty are viewed to be as an extreme simplification. In fact, in almost all circumstances, and at all times, we all find ourselves in a state of uncertainty. Secondly, the two notions, probability and utility, on which the correct criterion of decision-making depends, are treated inside linear spaces over R having a different dimension in accordance with the pure subjectivistic point of vie

The consumer’s demand functions defined to study contingent consumption plans. Summarized probability distributions: a mathematical application to contingent consumption choices

Given two probability distributions expressing returns on two single risky assets of a portfolio, we innovatively define two consumer’s demand functions connected with two contingent consumption plans. This thing is possible whenever we coherently summarize every probability distribution being chosen by the consumer. Since prevision choices are consumption choices being made by the consumer inside of a metric space, we show that prevision choices can be studied by means of the standard economic model of consumer behavior. Such a model implies that we consider all coherent previsions of a joint distribution. They are decomposed inside of a metric space. Such a space coincides with the consumer’s consumption space. In this paper, we do not consider a joint distribution only. It follows that we innovatively define a stand-alone and double risky asset. Different summary measures of it characterizing consumption choices being made by the consumer can then be studied inside of a linear space over ℝ. We show that it is possible to obtain different summary measures of probability distributions by using two different quadratic metrics. In this paper, our results are based on a particular approach to the origin of the variability of probability distributions. We realize that it is not standardized, but it always depends on the state of information and knowledge of the consumer

Tensors Associated with Mean Quadratic Differences Explaining the Riskiness of Portfolios of Financial Assets

Bound choices such as portfolio choices are studied in an aggregate fashion using an extension of the notion of barycenter of masses. This paper answers the question of whether such an extension is a natural fashion of studying bound choices or not. Given n risky assets, the question of why it is appropriate to treat only two risky assets at a time inside the budget set of the decision-maker is handled in this paper. Two risky assets are two goods. They are two marginal goods. The question of why they always give rise to a joint good inside the budget set of the decision-maker is addressed by this research work. A single risky asset is viewed as a double one using four nonparametric joint distributions of probability. The variability of a joint distribution of probability always depends on the state of information and knowledge associated with a given decision-maker. For this reason, two variability tensors are defined to identify the riskiness of the same risky asset. A multilinear version of the Sharpe ratio is shown. It is based on tensors. After computing the expected return on an n-risky asset portfolio, its riskiness is obtained using mean quadratic differences developed through tensor

Non-parametric probability distributions embedded inside of a linear space provided with a quadratic metric

There exist uncertain situations in which a random event is not a measurable set, but it is a point of a linear space inside of which it is possible to study different random quantities characterized by non-parametric probability distributions. We show that if an event is not a measurable set then it is contained in a closed structure which is not a σ-algebra but it is a linear space over R. We think of probability as being a mass. It is really a mass with respect to problems of statistical sampling. It is a mass with respect to problems of social sciences. In particular, it is a mass with regard to economic situations studied by means of the subjective notion of utility. We are able to decompose a random quantity meant as a geometric entity inside of a metric space. It is also possible to decompose its prevision and variance inside of it. We show a quadratic metric in order to obtain the variance of a random quantity. The origin of the notion of variability is not standardized within this context. It always depends on the state of information and knowledge of an individual. We study different intrinsic properties of non-parametric probability distributions as well as of probabilistic indices summarizing them. We define the notion of α-distance between two non-parametric probability distributio

R&D Subsidization effect and network centralization. Evidence from an agent-based micro-policy simulation

This paper presents an agent-based micro-policy simulation model assessing public R&D policy effect when R&D and non-R&D performing companies are located within a network. We set out by illustrating the behavioural structure and the computational logic of the proposed model; then, we provide a simulation experiment where the pattern of the total level of R&D activated by a fixed amount of public support is analysed as function of companies’ network topology. More specifically, the suggested simulation experiment shows that a larger “hubness” of the network is more likely accompanied with a decreasing median of the aggregated total R&D performance of the system. Since the aggregated firm idiosyncratic R&D (i.e., the part of total R&D independent of spillovers) is slightly increasing, we conclude that positive cross-firm spillover effects - in the presence of a given amount of support - have a sizeable impact within less centralized networks, where fewer hubs emerge. This may question the common wisdom suggesting that larger R&D externality effects should be more likely to arise when few central champions receive a support

On a Geometric Representation of Probability Laws and of a Coherent Prevision-Function According to Subjectivistic Conception of Probability

We distinguish the two extreme aspects of the logic of certainty by identifying their corresponding structures into a linear space. We extend probability laws P formally admissible in terms of coherence to random quantities. We give a geometric representation of these laws P and of a coherent prevision function P which we previously defined in an original way. We are the first in the world to do this kind of work: it is the foundation of our next and extensive study concerning the formulation of a geometric, wellorganized and original theory of random quantities

Postoperative acute kidney injury defined by RIFLE criteria predicts early health outcome and long-term survival in patients undergoing redo coronary artery bypass graft surgery

AbstractObjectiveTo investigate the impact of postoperative acute kidney injury (AKI) on early health outcome and on long-term survival in patients undergoing redo coronary artery bypass grafting (CABG).MethodsWe performed a Cox analysis with 398 consecutive patients undergoing redo CABG over a median follow-up of 7 years (interquartile range, 4-12.2 years). Renal function was assessed using baseline and peak postoperative levels of serum creatinine. AKI was defined according to the risk, injury, failure, loss, and end-stage (RIFLE) criteria. Health outcome measures included the rate of in-hospital AKI and all-cause 30-day and long-term mortality, using data from the United Kingdom's Office of National Statistics. Propensity score matching, as well as logistic regression analyses, were used. The impact of postoperative AKI at different time points was related to survival.ResultsIn patients with redo CABG, the occurrence of postoperative AKI was associated with in-hospital mortality (odds ratio [OR], 3.74; 95% confidence interval [CI], −1.3 to 10.5; P < .01], high Euroscore (OR, 1.27; 95% CI, 1.07-1.52; P < .01), use of IABP (OR, 6.9; 95% CI, 2.24-20.3; P < .01), and reduced long-term survival (hazard ratio [HR], 2.42; 95% CI, 1.63-3.6; P = .01). Overall survival at 5 and 10 years was lower in AKI patients with AKI compared with those without AKI (64% vs 85% at 5 years; 51% vs 68% at 10 years). On 1:1 propensity score matching analysis, postoperative AKI was independently associated with reduced long term survival (HR, 2.8; 95% CI, 1.15-6.7).ConclusionsIn patients undergoing redo CABG, the occurrence of postoperative AKI is associated with increased 30-day mortality and major complications and with reduced long-term survival