Parametric Family Research Articles

Parametric dependence between random vectors via copula-based divergence measures

This article proposes copula-based dependence quantification between multiple groups of random variables of possibly different sizes via the family of Φ-divergences. An axiomatic framework for this purpose is provided, after which we focus on the absolutely continuous setting assuming copula densities exist. We consider parametric and semi-parametric frameworks, discuss estimation procedures, and report on asymptotic properties of the proposed estimators. In particular, we first concentrate on a Gaussian copula approach yielding explicit and attractive dependence coefficients for specific choices of Φ, which are more amenable for estimation. Next, general parametric copula families are considered, with special attention to nested Archimedean copulas, being a natural choice for dependence modelling of random vectors. The results are illustrated by means of examples. Simulations and a real-world application on financial data are provided as well.

Inverse matrix estimations by iterative methods with weight functions and their stability analysis

In this paper, we construct a parametric family of iterative methods to compute the inverse of a nonsingular matrix. This class is free of inverse operators. We prove the third-order of convergence under some conditions involving the parameter of the family. Moreover, a dynamical analysis is made for the first time to a matrix iterative method, finding intervals of stability, that include but are wider than those found in the convergence analysis. Numerical tests on large random matrices confirm the results found.

Integrable deformations of superintegrable quantum circuits

Superintegrable models are very special dynamical systems: they possess more conservation laws than what is necessary for integrability. This severely constrains their dynamical processes, and it often leads to their exact solvability, even in non-equilibrium situations. In this paper we consider special Hamiltonian deformations of superintegrable quantum circuits. The deformations break superintegrability, but they preserve integrability. We focus on a selection of concrete models and show that for each model there is an (at least) one parameter family of integrable deformations. Our most interesting example is the so-called Rule54 model. We show that the model is compatible with a one parameter family of Yang-Baxter integrable spin chains with six-site interaction. Therefore, the Rule54 model does not have a unique integrability structure, instead it lies at the intersection of a family of quantum integrable models.

Automatic deforestation detectors based on frequentist statistics and their extensions for other spatial objects

AbstractThis article is devoted to the problem of detection of forest and nonforest areas on Earth images. We propose two statistical methods to tackle this problem: one based on multiple hypothesis testing with parametric distribution families, another one—on nonparametric tests. The parametric approach is novel in the literature and relevant to a larger class of problems—detection of natural objects, as well as anomaly detection. We develop mathematical background for each of the two methods, build self‐sufficient detection algorithms using them and discuss practical aspects of their implementation. We also compare our algorithms with each other and with those from standard machine learning using satellite data.

Transversal family of non-autonomous conformal iterated function systems

We study Non-autonomous Iterated Function Systems (NIFSs) with overlaps. A NIFS on a compact subset X\subset\mathbb{R}^{m} is a sequence \Phi=(\{\phi^{(j)}_{i}\}_{i\in I^{(j)}})_{j=1}^{\infty} of collections of uniformly contracting maps \phi^{(j)}_{i}\colon X\rightarrow X , where I^{(j)} is a finite set. In comparison to usual iterated function systems, we allow the contractions \phi^{(j)}_{i} applied at each step j to depend on j . In this paper, we focus on a family of parameterized NIFSs on \mathbb{R}^{m} . Here, we do not assume the open set condition. We show that if a d -parameter family of such systems satisfies the transversality condition, then for almost every parameter value the Hausdorff dimension of the limit set is the minimum of m and the Bowen dimension. Moreover, we give an example of a family \{\Phi_{t}\}_{t\in U} of parameterized NIFSs such that \{\Phi_{t}\}_{t\in U} satisfies the transversality condition but \Phi_{t} does not satisfy the open set condition for any t\in U .

Nonparametric Estimation of Conditional Copula Using Smoothed Checkerboard Bernstein Sieves

Conditional copulas are useful tools for modeling the dependence between multiple response variables that may vary with a given set of predictor variables. Conditional dependence measures such as conditional Kendall’s tau and Spearman’s rho that can be expressed as functionals of the conditional copula are often used to evaluate the strength of dependence conditioning on the covariates. In general, semiparametric estimation methods of conditional copulas rely on an assumed parametric copula family where the copula parameter is assumed to be a function of the covariates. The functional relationship can be estimated nonparametrically using different techniques, but it is required to choose an appropriate copula model from various candidate families. In this paper, by employing the empirical checkerboard Bernstein copula (ECBC) estimator, we propose a fully nonparametric approach for estimating conditional copulas, which does not require any selection of parametric copula models. Closed-form estimates of the conditional dependence measures are derived directly from the proposed ECBC-based conditional copula estimator. We provide the large-sample consistency of the proposed estimator as well as the estimates of conditional dependence measures. The finite-sample performance of the proposed estimator and comparison with semiparametric methods are investigated through simulation studies. An application to real case studies is also provided.

Flexible-dimensional L-statistic for mean estimation of symmetric distributions

Estimating the mean of a population is a recurrent topic in statistics because of its multiple applications. If previous data is available, or the distribution of the deviation between the measurements and the mean is known, it is possible to perform such estimation by using L-statistics, whose optimal linear coefficients, typically referred to as weights, are derived from a minimization of the mean squared error. However, such optimal weights can only manage sample sizes equal to the one used to derive them, while in real-world scenarios this size might slightly change. Therefore, this paper proposes a method to overcome such a limitation and derive approximations of flexible-dimensional optimal weights. To do so, a parametric family of functions based on extreme value reductions and amplifications is proposed to be adjusted to the cumulative optimal weights of a given sample from a symmetric distribution. Then, the application of Yager’s method to derive weights for ordered weighted average (OWA) operators allows computing the approximate optimal weights for sample sizes close to the original one. This method is justified from the theoretical point of view by proving a convergence result regarding the cumulative weights obtained for different sample sizes. Finally, the practical performance of the theoretical results is shown for several classical symmetric distributions.

사회복지사 보상체계가 삶의 만족도에 미치는 영향

This study examined the parallel multiple mediating effect of work-life balance and family support on the effect of the social worker compensation system on life satisfaction. The life satisfaction of social workers is the most fundamental factor that prevents impaired quality of life, turnover, and burnout among social workers. Moreover, to improve the life satisfaction of social workers, it is important to consider work-life balance and family support together. Therefore, this study examined the relationship between the compensation system, life satisfaction, work-life balance, and family support among 475 social workers, and the results were as follows: First, it was found that the social worker compensation system had a positive effect on the parameter work-life balance and family support. Second, it was found that the parameters of work-life balance and family support both had positive effects on life satisfaction. Finally, it was found that the compensation system of social workers had an indirect effect on life satisfaction through certain parameters. Both work-life balance and family support, which are parameters, showed indirect effects on the path from the compensation system to life satisfaction. With these results, this study provided basic data for service and policy establishment to improve the compensation system for social workers, and it lastly suggested directions for follow-up studies.

Search algorithm on strongly regular graph by lackadaisical quantum walk

Quantum walk is a widely used method in designing quantum algorithms. In this article, we consider the lackadaisical discrete-time quantum walk (DTQW) on strongly regular graphs (SRG). When there is a single marked vertex in a SRG, we prove that lackadaisical DTQW can find the marked vertex with asymptotic success probability 1, with a quadratic speedup compared to classical random walk. This paper deals with any parameter family of SRG and argues that, by adding self-loops with proper weights, the asymptotic success probability can reach 1. The running time and asymptotic success probability matches the result of continuous-time quantum walk, and improves the result of DTQW.

Stochastic monotone wing property: A new dependence structure for copulas

In this study, we introduce a novel concept of a copula dependence structure termed the stochastic monotone wing property and investigate its associated properties. This concept is closely related to the monotone conditional variance property, which describes the behavior of the conditional variance of V given U=u as either an increasing or a decreasing function of u for a given random vector (U,V) with a certain distribution function. The need for such a dependence structure arises in various fields, including insurance, where a copula family with the monotone conditional variance property is needed for the joint distribution of frequency and average severity in the collective risk model. Beyond insurance, its applications extend to economics and climatology. Furthermore, this study proposes a new parametric copula family that exhibits the stochastic monotone wing property and provides a method for constructing copulas with this characteristic.

On the connectedness of multistationarity regions of small reaction networks

A multistationarity region is the part of a reaction network's parameter space that gives rise to multiple steady states. Mathematically, this region consists of the positive parameters for which a parametrized family of polynomial equations admits two or more positive roots. Much recent work has focused on analyzing multistationarity regions of biologically significant reaction networks and determining whether such regions are connected; indeed, a better understanding of the topology and geometry of such regions may help elucidate how robust multistationarity is to perturbations. Here we focus on the multistationarity regions of small networks, those with few species and few reactions. For two families of such networks – those with one species and up to three reactions, and those with two species and up to two reactions – we prove that the resulting multistationarity regions are connected. We also give an example of a network with one species and six reactions for which the multistationarity region is disconnected. Our proofs rely on the formula for the discriminant of a trinomial, a classification of small multistationary networks, and a recent result of Feliu and Telek that partially generalizes Descartes' rule of signs.

On approximating dependence function and its derivatives

Bivariate extreme value distributions can be used to model dependence of observations from random variables in extreme levels. There is no finite dimensional parametric family for these distributions, but they can be characterized by a certain one-dimensional function which is known as Pickands dependence function. In many applications the general approach is to estimate the dependence function with a non-parametric method and then conduct further analysis based on the estimate. Although this approach is flexible in the sense that it does not impose any special structure on the dependence function, its main drawback is that the estimate is not available in a closed form. This paper provides some theoretical results which can be used to find a closed form approximation for an exact or an estimate of a twice differentiable dependence function and its derivatives. We demonstrate the methodology by calculating approximations for symmetric and asymmetric logistic dependence functions and their second derivatives. We show that the theory can be even applied to approximating a non-parametric estimate of dependence function using a convex optimization algorithm. Other discussed applications include a procedure for testing whether an estimate of dependence function can be assumed to be symmetric and estimation of the concordance measures of a bivariate extreme value distribution. Finally, an Australian annual maximum temperature dataset is used to illustrate how the theory can be used to build semi-infinite and compact predictions regions.

A modified parameterization method for invariant Lagrangian tori for partially integrable Hamiltonian systems

In this paper we present an a-posteriori KAM theorem for the existence of an (n−d)-parameter family of d-dimensional isotropic invariant tori with Diophantine frequency vector ω∈Rd, of type (γ,τ), for n degrees of freedom Hamiltonian systems with (n−d) independent first integrals in involution. If the first integrals induce a Hamiltonian action of the (n−d)-dimensional torus, then we can produce n-dimensional Lagrangian tori with frequency vector of the form (ω,ωp), with ωp∈Rn−d. In the light of the parameterization method, we design a (modified) quasi-Newton method for the invariance equation of the parameterization of the torus, whose proof of convergence from an initial approximation, and under appropriate non-degeneracy conditions, is the object of this paper. We present the results in the analytic category, so the initial torus is real-analytic in a certain complex strip of width ρ, and the corresponding error in the functional equation is ɛ. We heavily use geometric properties and the so called automatic reducibility to deal directly with the functional equation and get convergence if γ−2ρ−2τ−1ɛ is small enough, in contrast with most of KAM results based on the parameterization method, that get convergence if γ−4ρ−4τɛ is small enough. The approach is suitable to perform computer assisted proofs.

Equilibria and bifurcations in contact dynamics

We provide a systematic study of equilibria of contact vector fields and the bifurcations that occur generically in 1-parameter families, and express the conclusions in terms of the Hamiltonian functions that generate the vector fields. Equilibria occur at points where the zero-level set of the Hamiltonian function is either singular or is tangent to the contact structure. The eigenvalues at an equilibrium have an interesting structure: there is always one particular real eigenvalue of any equilibrium, related to the contact structure, that we call the principal coefficient, while the other eigenvalues arise in quadruplets, similar to the symplectic case except they are translated by a real number equal to half the principal coefficient. There are two types of codimension 1 equilibria, named Type I, arising where the zero-set of the Hamiltonian is singular, and Type II where it is not, but there is a degeneracy related again to the principal coefficient and the contact of the zero-level set of the Hamiltonian with the contact structure. Both give rise generically to saddle-node bifurcations. Some special features include: (i) for Type II singularities, Hopf bifurcations cannot occur in dimension 3, but they may in dimension 5 or more; (ii) for Type I singularities, a fold-Hopf bifurcation can occur with codimension 1 in any dimension, and (iii) again for Type I, and in dimension at least 5, a fold-multi-Hopf bifurcation (where several pairs of eigenvalues pass through the imaginary axis simultaneously together with one through the origin) may also occur with codimension 1.

MDS, Hermitian almost MDS, and Gilbert–Varshamov quantum codes from generalized monomial-Cartesian codes

We construct new stabilizer quantum error-correcting codes from generalized monomial-Cartesian codes. Our construction uses an explicitly defined twist vector, and we present formulas for the minimum distance and dimension. Generalized monomial-Cartesian codes arise from polynomials in m variables. When m=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m=1$$\\end{document} our codes are MDS, and when m=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m=2$$\\end{document} and our lower bound for the minimum distance is 3, the codes are at least Hermitian almost MDS. For an infinite family of parameters, when m=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m=2$$\\end{document} we prove that our codes beat the Gilbert–Varshamov bound. We also present many examples of our codes that are better than any known code in the literature.

Orbital stability of periodic traveling waves in the [formula omitted]-Camassa–Holm equation

In this paper, we identify criteria that guarantees the nonlinear orbital stability of a given periodic traveling wave solution within the b-family Camassa–Holm equation. These periodic waves exist as 3-parameter families (up to spatial translations) of smooth traveling wave solutions, and their stability criteria are expressed in terms of Jacobians of the conserved quantities with respect to these parameters. The stability criteria utilizes a general Hamiltonian structure which exists for every b>1, and hence applies outside of the completely integrable cases (b=2 and b=3).

Estimation of the incubation time distribution in the singly and doubly interval censored model

We analyze nonparametric estimators for the distribution function of the incubation time in the singly and doubly interval censoring model. The classical approach is to use parametric families like Weibull, log‐normal or gamma distributions in the estimation procedure. We propose nonparametric estimates for functions of the observations, which stay closer to the data than the classical parametric methods. We also give explicit limit distributions for discrete versions of the models and apply this to compute confidence intervals. The methods complement the analysis of the continuous model in Groeneboom (2021, 2023). R scripts for computation of the estimates are provided in Groeneboom (2020).

Rationally extended harmonic oscillator potential, isospectral family and the uncertainty relations

We consider the rationally extended harmonic oscillator potential which is isospectral to the conventional one and whose solutions are associated with the exceptional, Xm- Hermite polynomials and discuss its various important properties for different even codimension of m. The uncertainty relations are obtained for different m and it is shown that for the ground state, the uncertainty increases as m increases. A one parameter (λ) family of exactly solvable isospectral potential corresponding to this extended harmonic oscillator potential is obtained. Special cases corresponding to the λ=0 and λ=−1, which give the Pursey and the Abraham-Moses potentials respectively, are discussed. The uncertainty relations for the entire isospectral family of potentials for different m and λ are also calculated.

Efficient parametric family of fourth-order Jacobian-free iterative vectorial schemes

AbstractIn this work, a multiparametric family of iterative vectorial fourth-order methods free of Jacobian matrices is proposed. A convergence analysis of this family is carried out as well as a study of its efficiency. Several numerical experiments are made in order to compare the behaviour of the proposed family with other competitive methods of the literature.

Sharp bounds on the survival function of exchangeable min-stable multivariate exponential sequences

Abstract We derive lower and upper bounds for the survival function of an exchangeable sequence of random variables, for which the scaled minimum of each finite subgroup has a univariate exponential distribution. These bounds are sharp in the sense that both bounds themselves are attained by exchangeable sequences of the same kind, for which the (non-scaled) minimum of each subgroup has the same univariate exponential distribution as the original sequence. This result is equivalent to inequalities between infinite-dimensional stable tail dependence functions, which leads to inequalities between multivariate extreme-value copulas. In addition, it is explained how an infinite-dimensional symmetric stable tail dependence function can be obtained from its upper bound by censoring certain distributional information. This technique is applied to derive new parametric families.