On a multiplication and a theory of integration for belief and plausibility functions

Belief and plausibility functions have been introduced as generalizations of probability measures, which abandon the axiom of additivity. It turns out that elementwise multiplication is a binary operation on the set of belief functions. If the set functions of the type considered here are defined on a locally compact and separable space X, a theorem by Choquet ensures that they can be represented by a probability measure on the space containing the closed subsets of X, the so-called basic probability assignment. This is basic for defining two new types of integrals. One of them may be used to measure the degree of non-additivity of the belief or plausibility function. The other one is a generalization of the Lebesgue integral. The latter is compared with Choquet's and Sugeno's integrals for non-additive set functions.     

Empirical likelihood method for linear transformation models

Empirical likelihood inferential procedure is proposed for right censored survival data under linear transformation models, which include the commonly used proportional hazards model as a special case. A log-empirical likelihood ratio test statistic for the regression coefficients is developed. We show that the proposed log-empirical likelihood ratio test statistic converges to a standard chi-squared distribution. The result can be used to make inference about the entire regression coefficients vector as well as any subset of it. The method is illustrated by extensive simulation studies and a real example.     

On sparse estimation for semiparametric linear transformation models

Semiparametric linear transformation models have received much attention due to their high flexibility in modeling survival data. A useful estimating equation procedure was recently proposed by Chen et al. (2002) [21] for linear transformation models to jointly estimate parametric and nonparametric terms. They showed that this procedure can yield a consistent and robust estimator. However, the problem of variable selection for linear transformation models has been less studied, partially because a convenient loss function is not readily available under this context. In this paper, we propose a simple yet powerful approach to achieve both sparse and consistent estimation for linear transformation models. The main idea is to derive a profiled score from the estimating equation of Chen et al. [21], construct a loss function based on the profile scored and its variance, and then minimize the loss subject to some shrinkage penalty. Under regularity conditions, we have shown that the resulting estimator is consistent for both model estimation and variable selection. Furthermore, the estimated parametric terms are asymptotically normal and can achieve a higher efficiency than that yielded from the estimation equations. For computation, we suggest a one-step approximation algorithm which can take advantage of the LARS and build the entire solution path efficiently. Performance of the new procedure is illustrated through numerous simulations and real examples including one microarray data.     

On translating one polyomino to tile the plane

Given a polyomino, we prove that we can decide whether translated copies of the polyomino can tile the plane. Copies that are rotated, for example, are not allowed in the tilings we consider. If such a tiling exists the polyomino is called anexact polyomino. Further, every such tiling of the plane by translated copies of the polyomino is half-periodic. Moreover, all the possible surroundings of an exact polyomino are described in a simple way.     

Simple probability models for project contracting

There are two aspects to the process of price setting using sealed bids. The project owner is interested in deploying the contract mechanism that will secure reliable service at the cheapest cost. On the other hand, each contractor bidding for the project is interested in winning the contract but at a price that assures him a reasonable profit margin. We use a parsimonious stochastic model to compare and contrast some commonly used contracts from the point of view of the project owner. We show that if the bidders are risk neutral, a Fixed Price contract results in the smallest expected procurement cost for the project owner. We introduce and analyze Menu contracts and show that the expected price of a Menu contract lies in between the prices of the Fixed Price and Cost Plus contracts for the same project. We analyze how risk aversion and collusion, which we model using concepts of stochastic dependence, impacts the average winning bid price.     

On the generalization of probabilistic transformation method

The probabilistic transformation method with the finite element analysis is a new technique to solve random differential equation. The advantage of this technique is finding the “exact” expression of the probability density function of the solution when the probability density function of the input is known. However the disadvantage is due to the characteristics (geometrics and materials) of the analyzed structure included in the random differential equation.

In this paper, a developed formula is used to generalize this technique by obtaining the “exact” joint probability density function of the solution in any situations, as well as the proposed technique for the non-linear case.     

On inferring the probability of misclassification by the linear discriminant function

Summary Let the two alternative populationsP ₁ andP ₂ from which the individual with measurements χ may have come beN(μ(1), Σ) andN(μ(2), Σ). Then the classification rule with minimum risk is to assign the individual toP ₁ orP ₂ according as (μ(2)-μ(1))′Σ^-1 x≶(1/2)(μ(2)-μ(1))′Σ^-1(μ(1)+μ(2))+c wherec is a constant depending on the prior probabilities ofP ₁ andP ₂ and the costs of the two kinds of misclassification. The probability of misclassifying an individual fromP ₂ by this rule is π₂₁=Φ(-δ/2+cδ^-1), where Φ(.) is the distribution function of anN(0, 1) and . (Since we are free to choose which population we shall callP ₂, it is not necessary to consider separately the probability of misclassifying an individual fromP ₁.) LetP ₂₁ denote the probability of misclassification of an individual fromP ₂ by the rule derived from the one mentioned by fixing μ(1), μ(2) and Σ at estimates andV and letP ₂₁ ^* be the probability of misclassification of an individual fromP ₂ when the classification rule is the one with minimum risk among those based on . The fiducial distributions of π₂₁,P ₂₁ andP ₂₁ ^* are determined. Point estimates and confidence intervals for π₂₁,P ₂₁ andP ₂₁ ^* are derived. Only easily available tables are needed to make fiducial inferences. An incidental result of some interest elsewhere as well is the distribution of a linear combination of a chi and an independent normal variable.     

On the relative belief transform

In this paper we discuss the semantics and properties of the relative belief transform, a probability transformation of belief functions closely related to the classical plausibility transform. We discuss its rationale in both the probability-bound and Shafer’s interpretations of belief functions. Even though the resulting probability (as it is the case for the plausibility transform) is not consistent with the original belief function, an interesting rationale in terms of optimal strategies in a non-cooperative game can be given in the probability-bound interpretation to both relative belief and plausibility of singletons. On the other hand, we prove that relative belief commutes with Dempster’s orthogonal sum, meets a number of properties which are the duals of those met by the relative plausibility of singletons, and commutes with convex closure in a similar way to Dempster’s rule. This supports the argument that relative plausibility and belief transform are indeed naturally associated with the D-S framework, and highlights a classification of probability transformations into two families, according to the operator they relate to. Finally, we point out that relative belief is only a member of a class of “relative mass” mappings, which can be interpreted as low-cost proxies for both plausibility and pignistic transforms.     

Scaling transformation and probability distributions for financial time series

The price of financial assets are, since [Bachelier L. Annales de l'Ecole Normale Supérieure 1900;3:XVII:21–86], considered to be described by a (discrete or continuous) time sequence of random variables, i.e., a stochastic process. Sharp scaling exponents or unifractal behavior of such processes has been reported in several works [Mandelbrot BB. J Business 1963;36:394–419; Peters EE. Chaos and order in the capital markets. New York: Wiley, 1991; Mantegna RN, Stanley HE. Nature 1995;376:46–49; Evertsz CJG. Fractals. 1995;3:609–616; Bouchaud JP, Potters M. Théorie des risques financiers. Aléa Saclay, 1997]. In this paper we investigate the question of scaling transformation of price processes by establishing a new connection between non-linear group theoretical methods and multifractal methods developed in mathematical physics. Using two sets of financial chronological time series, we show that the scaling transformation is a non-linear group action on the moments of the price increments. Its linear part has a spectral decomposition that puts in evidence a multifractal behavior of the price increments.     

Stationary probability distributions for multiresponse linear learning models

A necessary and sufficient condition is given for the existence of stationary probability distributions of a non-Markovian model with linear transition rule. Similar to the Markovian case, stationary probability distributions are characterized as eigenvectors of nonnegative matrices. The model studied includes as special cases the Markovian model as well as the linear learning model and has applications in psychological and biological research, in control theory, and in adaption theory.     

Ensemble Gaussian mixture models for probability density estimation

Estimation of probability density functions (PDF) is a fundamental concept in statistics. This paper proposes an ensemble learning approach for density estimation using Gaussian mixture models (GMM). Ensemble learning is closely related to model averaging: While the standard model selection method determines the most suitable single GMM, the ensemble approach uses a subset of GMM which are combined in order to improve precision and stability of the estimated probability density function. The ensemble GMM is theoretically investigated and also numerical experiments were conducted to demonstrate benefits from the model. The results of these evaluations show promising results for classifications and the approximation of non-Gaussian PDF.     

The probability density function to the random linear transport equation

We present a formula to calculate the probability density function of the solution of the random linear transport equation in terms of the density functions of the velocity and the initial condition. We also present an expression for the joint probability density function of the solution in two different points. Our results have shown good agreement with Monte Carlo simulations.     

On the index integral transformation with Nicholson's function as the kernel

The integral transformation, which is associated with the Nicholson function as the kernel, is introduced and investigated in the paper. This transformation is an integral, where integration is with respect to an index of the sum of squares of Bessel functions of the first and second kind. Composition representations and relationships with the Meijer K-transform, the Kontorovich-Lebedev transform, the Mellin transform, and the sine Fourier transform are given. We also present boundedness properties, a Parseval type equality, and an inversion formula.     

A method for inferring inequalities for probability values applied to complex networks

In this article, we infer inequalities representing (upper) bounds for real quantities. The obtained bound is parametric and, hence, it might be applicable to special problems. Here, we apply it to probabilities assigned to complex networks. © 2015 Wiley Periodicals, Inc. Complexity 21: 113–115, 2016