Optimizing fuzzy portfolio selection problems by parametric quadratic programming

This paper develops a robust method to describe fuzzy returns by employing parametric possibility distributions. The parametric possibility distributions are obtained by equivalent value (EV) reduction methods. For common type-2 triangular and trapezoidal fuzzy variables, their reduced fuzzy variables are studied in the current development. The parametric possibility distributions of reduced fuzzy variables are first derived, then the second moment formulas for the reduced fuzzy variables are established. Taking the second moment as a new risk measure, the reward-risk and risk-reward models are developed to optimize fuzzy portfolio selection problems. The mathematical properties of the proposed optimization models are analyzed, including the analytical representations for the second moments of linear combinations of reduced fuzzy variables as well as the convexity of second moments with respect to decision vectors. On the basis of the analytical representations for the second moments, the reward-risk and risk-reward models can be turned into their equivalent parametric quadratic convex programming problems, which can be solved by conventional solution methods or general-purpose software. Finally, some numerical experiments are performed to demonstrate the new modeling ideas and the efficiency of solution method.     

Optimizing fuzzy p-hub center problem with generalized value-at-risk criterion

In this study, we reduce the uncertainty embedded in secondary possibility distribution of a type-2 fuzzy variable by fuzzy integral, and apply the proposed reduction method to p-hub center problem, which is a nonlinear optimization problem due to the existence of integer decision variables. In order to optimize p-hub center problem, this paper develops a robust optimization method to describe travel times by employing parametric possibility distributions. We first derive the parametric possibility distributions of reduced fuzzy variables. After that, we apply the reduction methods to p-hub center problem and develop a new generalized value-at-risk (VaR) p-hub center problem, in which the travel times are characterized by parametric possibility distributions. Under mild assumptions, we turn the original fuzzy p-hub center problem into its equivalent parametric mixed-integer programming problems. So, we can solve the equivalent parametric mixed-integer programming problems by general-purpose optimization software. Finally, some numerical experiments are performed to demonstrate the new modeling idea and the efficiency of the proposed solution methods.     

Limit theorems for a supercritical branching process with immigration in a random environment

Let (Z_n) be a supercritical branching process with immigration in a random environment. Firstly, we prove that under a simple log moment condition on the offspring and immigration distributions, the naturally normalized population size W_n converges almost surely to a finite random variable W. Secondly, we show criterions for the non-degeneracy and for the existence of moments of the limit random variable W. Finally, we establish a central limit theorem, a large deviation principle and a moderate deviation principle about log Z_n.     

On the rate of decay of concentration functions of <Emphasis Type="Italic">n</Emphasis>-fold convolutions of probability distributions

Concentration functions of n-fold convolutions of probability distributions is shown to exhibit the following behavior. Let φ(n) be an arbitrary sequence tending to infinity as n tends to infinity, and ψ(x) be an arbitrary function tending to infinity as x tends to infinity. Then there exists a probability distribution F of a random variable X such that the mathematical expectation E _ψ(|X|) is infinite and, moreover, the upper limit of the sequence \(\sqrt n \phi \left( n \right)Q_n\) is equal to infinity, where Q _n is the maximal atom of the n-fold convolution of distribution F. Thus, no infinity conditions imposed on the moments can force the concentration functions of n-fold convolutions decay essentially faster than o(n ^?1/2).     

Sums of products of Ramanujan sums

The Ramanujan sum c _n (k) is defined as the sum of k-th powers of the primitive n-th roots of unity. We investigate arithmetic functions of r variables defined as certain sums of the products \({c_{m_1}(g_1(k))\cdots c_{m_r}(g_r(k))}\), where g ₁, . . . , g _r are polynomials with integer coefficients. A modified orthogonality relation of the Ramanujan sums is also derived.     

Sharp large deviations for sums of bounded from above random variables

We show large deviation expansions for sums of independent and bounded from above random variables. Our moderate deviation expansions are similar to those of Cram′er(1938), Bahadur and Ranga Rao(1960), and Sakhanenko(1991). In particular, our results extend Talagrand's inequality from bounded random variables to random variables having finite(2 + δ)-th moments, where δ∈(0, 1]. As a consequence,we obtain an improvement of Hoeffding's inequality. Applications to linear regression, self-normalized large deviations and t-statistic are also discussed.     

Dynamic bivariate normal copula

Normal copula with a correlation coefficient between-1 and 1 is tail independent and so it severely underestimates extreme probabilities. By letting the correlation coefficient in a normal copula depend on the sample size, H¨usler and Reiss(1989) showed that the tail can become asymptotically dependent. We extend this result by deriving the limit of the normalized maximum of n independent observations, where the i-th observation follows from a normal copula with its correlation coefficient being either a parametric or a nonparametric function of i/n. Furthermore, both parametric and nonparametric inference for this unknown function are studied, which can be employed to test the condition by H¨usler and Reiss(1989). A simulation study and real data analysis are presented too.     

Asymptotics for the partial sum and its maximum of dependent random variables<Superscript>*</Superscript>

Let X ₁,…,X _n be pairwise asymptotically independent or pairwise upper extended negatively dependent real-valued random variables. Under the condition that the distribution of the maximum of X ₁,…,X _n belongs to some subclass of heavy-tailed distributions, we investigate the asymptotic behavior of the partial sum and its maximum generated by dependent X ₁,…,X _n . As an application, we consider a discrete-time risk model with insurance and financial risks and derive the asymptotics for the finite-time ruin probability.     

Correlation Inequalities and Applications to Vector-Valued Gaussian Random Variables and Fractional Brownian Motion

In this paper we extend certain correlation inequalities for vector-valued Gaussian random variables due to Kolmogorov and Rozanov. The inequalities are applied to sequences of Gaussian random variables and Gaussian processes. For sequences of Gaussian random variables satisfying a correlation assumption, we prove a Borel-Cantelli lemma, maximal inequalities and several laws of large numbers. This extends results of Be?ka and Ciesielski and of Hytönen and the author. In the second part of the paper we consider a certain class of vector-valued Gaussian processes which are α-Hölder continuous in p-th moment. For these processes we obtain Besov regularity of the paths of order α. We also obtain estimates for the moments in the Besov norm. In particular, the results are applied to vector-valued fractional Brownian motions. These results extend earlier work of Ciesielski, Kerkyacharian and Roynette and of Hytönen and the author.     

A New Method for Bounding the Distance Between Sums of Independent Integer-Valued Random Variables

Let X ₁, X ₂,..., X _n and Y ₁, Y ₂,..., Y _n be two sequences of independent random variables which take values in ? and have finite second moments. Using a new probabilistic method, upper bounds for the Kolmogorov and total variation distances between the distributions of the sums \(\sum_{i=1}^{n}X_{i}\) and \(\sum_{i=1}^{n}Y_{i}\) are proposed. These bounds adopt a simple closed form when the distributions of the coordinates are compared with respect to the convex order. Moreover, they include a factor which depends on the smoothness of the distribution of the sum of the X _i ’s or Y _i ’s, in that way leading to sharp approximation error estimates, under appropriate conditions for the distribution parameters. Finally, specific examples, concerning approximation bounds for various discrete distributions, are presented for illustration.     

Semideviations of reduced fuzzy variables: a possibility approach

This paper proposes new methods to reduce the uncertain information embedded in the secondary possibility distribution of a type-2 fuzzy variable. Based on possibility measure, we define the lower value-at-risk (VaR) and upper VaR of a regular fuzzy variable, and develop the VaR-based reduction methods for type-2 fuzzy variables. The proposed VaR-based reduction methods generalize some existing reduction methods by introducing possibility level parameter in distribution functions. For VaR reduced fuzzy variables, we employ Lebesgue–Stieltjes (L–S) integral to define three $n$ th semideviations to gauge the risk resulted from asymmetric fuzzy uncertainty. Furthermore, we compute the mean values and semideviations of the VaR reduced fuzzy variables, and derive some useful analytical expressions. The theoretical results obtained in this paper have potential applications in practical risk management and engineering optimization problems.     

Exponential distance-based fuzzy clustering for interval-valued data

In several real life and research situations data are collected in the form of intervals, the so called interval-valued data. In this paper a fuzzy clustering method to analyse interval-valued data is presented. In particular, we address the problem of interval-valued data corrupted by outliers and noise. In order to cope with the presence of outliers we propose to employ a robust metric based on the exponential distance in the framework of the Fuzzy C-medoids clustering mode, the Fuzzy C-medoids clustering model for interval-valued data with exponential distance. The exponential distance assigns small weights to outliers and larger weights to those points that are more compact in the data set, thus neutralizing the effect of the presence of anomalous interval-valued data. Simulation results pertaining to the behaviour of the proposed approach as well as two empirical applications are provided in order to illustrate the practical usefulness of the proposed method.     

On the <Emphasis Type="Italic">n</Emphasis>-th row of the graded Betti table of an <Emphasis Type="Italic">n</Emphasis>-dimensional toric variety

We prove an explicit formula for the first nonzero entry in the n-th row of the graded Betti table of an n-dimensional projective toric variety associated with a normal polytope with at least one interior lattice point. This applies to Veronese embeddings of \(\mathbb {P}^n\). We also prove an explicit formula for the entire n-th row when the interior of the polytope is one-dimensional. All results are valid over an arbitrary field k.     

On the existence of invariant tori in nearly-integrable Hamiltonian systems with finitely differentiable perturbations

We prove the existence of invariant tori in Hamiltonian systems, which are analytic and integrable except a 2n-times continuously differentiable perturbation (n denotes the number of the degrees of freedom), provided that the moduli of continuity of the 2n-th partial derivatives of the perturbation satisfy a condition of finiteness (condition on an integral), which is more general than a Hölder condition. So far the existence of invariant tori could be proven only under the condition that the 2n-th partial derivatives of the perturbation are Hölder continuous.     

On teaching sets for 2-threshold functions of two variables

We consider k-threshold functions of n variables, i.e. the functions representable as the conjunction of k threshold functions. For n = 2, k = 2, we give upper bounds for the cardinality of the minimal teaching set depending on the various properties of the function.     

Order statistics of uncertain random variables with application to <Emphasis Type="Italic">k</Emphasis>-out-of-<Emphasis Type="Italic">n</Emphasis> system

Uncertain random variables are tools to deal with a mixture of uncertainty and randomness. A new concept of order statistics associated with uncertain random variables is proposed, and is applied to analyze k-out-of-n systems with uncertain random lifetimes. The chance distributions of order statistics of uncertain random variables are derived from the operational law of uncertain random variables. Finally, the reliability of k-out-of-n systems with uncertain random lifetimes is discussed.     

On basicity of perturbed system of exponents in Lebesgue space with variable summability factor

In this paper, the system of exponents \(\{ e^{i\lambda _n t} \} _{n \in Z} \) is considered, where {λ _n } ? R has the following asymptotic form: λ _n = n ? α sign n + O(|n|^?β). Basis properties of this system in Lebesgue space with variable summability factor are investigated.     

A new proof of a theorem of Petersen

Let M be an n-dimensional complete Riemannian manifold with Ricci curvature n- 1. By developing some new techniques, Colding(1996) proved that the following three conditions are equivalent: 1)dGH(M, S~n) → 0; 2) the volume of M Vol(M) → Vol(S~n); 3) the radius of M rad(M) →π. By developing a different technique, Petersen(1999) gave the 4th equivalent condition, namely he proved that the n + 1-th eigenvalue of M, λ_(n+1)(M) → n, is also equivalent to the radius of M, rad(M) →π, and hence the other two.In this paper, we use Colding's techniques to give a new proof of Petersen's theorem. We expect our estimates will have further applications.     

Bulk behaviour of some patterned block matrices

We investigate the bulk behaviour of singular values and/or eigenvalues of two types of block random matrices. In the first one, we allow unrestricted structure of order m × p with n × n blocks and in the second one we allow m × m Wigner structure with symmetric n × n blocks. Different rows of blocks are assumed to be independent while the blocks within any row satisfy a weak dependence assumption that allows for some repetition of random variables among nearby blocks. In general, n can be finite or can grow to infinity. Suppose the input random variables are i.i.d. with mean 0 and variance 1 with finite moments of all orders. We prove that under certain conditions, the Mar?enko-Pastur result holds in the first model when m → ∞ and \(\tfrac{m}{p} \to c \in (0,\infty )\), and the semicircular result holds in the second model when m → ∞. These in particular generalize the bulk behaviour results of Loubaton [10].     

Two-sample homogeneity tests based on divergence measures

The concept of f-divergences introduced by Ali and Silvey (J R Stat Soc (B) 28:131–142, 1996) provides a rich set of distance like measures between pairs of distributions. Divergences do not focus on certain moments of random variables, but rather consider discrepancies between the corresponding probability density functions. Thus, two-sample tests based on these measures can detect arbitrary alternatives when testing the equality of the distributions. We treat the problem of divergence estimation as well as the subsequent testing for the homogeneity of two-samples. In particular, we propose a nonparametric estimator for f-divergences in the case of continuous distributions, which is based on kernel density estimation and spline smoothing. As we show in extensive simulations, the new method performs stable and quite well in comparison to several existing non- and semiparametric divergence estimators. Furthermore, we tackle the two-sample homogeneity problem using permutation tests based on various divergence estimators. The methods are compared to an asymptotic divergence test as well as to several traditional parametric and nonparametric procedures under different distributional assumptions and alternatives in simulations. It turns out that divergence based methods detect discrepancies between distributions more often than traditional methods if the distributions do not differ in location only. The findings are illustrated on ion mobility spectrometry data.