Weak convergence of the sequential empirical processes of residuals in TAR models

This paper studies the weak convergence of the sequential empirical process K n of the residuals in the threshold autoregressive(TAR)model of order p.Under some mild conditions,it is shown that K n converges weakly to a Kiefer process plus a random variable which converges to a multivariate normal.This differs from that given by Bai(1994)for a stationary autoregressive and moving average(ARMA)model.     

Clustering and classification via cluster-weighted factor analyzers

In model-based clustering and classification, the cluster-weighted model is a convenient approach when the random vector of interest is constituted by a response variable $Y$ and by a vector ${\varvec{X}}$ of $p$ covariates. However, its applicability may be limited when $p$ is high. To overcome this problem, this paper assumes a latent factor structure for ${\varvec{X}}$ in each mixture component, under Gaussian assumptions. This leads to the cluster-weighted factor analyzers (CWFA) model. By imposing constraints on the variance of $Y$ and the covariance matrix of ${\varvec{X}}$ , a novel family of sixteen CWFA models is introduced for model-based clustering and classification. The alternating expectation-conditional maximization algorithm, for maximum likelihood estimation of the parameters of all models in the family, is described; to initialize the algorithm, a 5-step hierarchical procedure is proposed, which uses the nested structures of the models within the family and thus guarantees the natural ranking among the sixteen likelihoods. Artificial and real data show that these models have very good clustering and classification performance and that the algorithm is able to recover the parameters very well.     

Efficient and robust tests for semiparametric models

In this paper, we investigate a hypothesis testing problem in regular semiparametric models using the Hellinger distance approach. Specifically, given a sample from a semiparametric family of \(\nu \)-densities of the form \(\{f_{\theta ,\eta }:\theta \in \Theta ,\eta \in \Gamma \},\) we consider the problem of testing a null hypothesis \(H_{0}:\theta \in \Theta _{0}\) against an alternative hypothesis \(H_{1}:\theta \in \Theta _{1},\) where \(\eta \) is a nuisance parameter (possibly of infinite dimensional), \(\nu \) is a \(\sigma \)-finite measure, \(\Theta \) is a bounded open subset of \(\mathbb {R}^{p}\), and \(\Gamma \) is a subset of some Banach or Hilbert space. We employ the Hellinger distance to construct a test statistic. The proposed method results in an explicit form of the test statistic. We show that the proposed test is asymptotically optimal (i.e., locally uniformly most powerful) and has some desirable robustness properties, such as resistance to deviations from the postulated model and in the presence of outliers.     

Statistical inference in constrained latent class models for multinomial data based on $$\phi $$-divergence measures

In this paper we explore the possibilities of applying \(\phi \)-divergence measures in inferential problems in the field of latent class models (LCMs) for multinomial data. We first treat the problem of estimating the model parameters. As explained below, minimum \(\phi \)-divergence estimators (M\(\phi \)Es) considered in this paper are a natural extension of the maximum likelihood estimator (MLE), the usual estimator for this problem; we study the asymptotic properties of M\(\phi \)Es, showing that they share the same asymptotic distribution as the MLE. To compare the efficiency of the M\(\phi \)Es when the sample size is not big enough to apply the asymptotic results, we have carried out an extensive simulation study; from this study, we conclude that there are estimators in this family that are competitive with the MLE. Next, we deal with the problem of testing whether a LCM for multinomial data fits a data set; again, \(\phi \)-divergence measures can be used to generate a family of test statistics generalizing both the classical likelihood ratio test and the chi-squared test statistics. Finally, we treat the problem of choosing the best model out of a sequence of nested LCMs; as before, \(\phi \)-divergence measures can handle the problem and we derive a family of \(\phi \)-divergence test statistics based on them; we study the asymptotic behavior of these test statistics, showing that it is the same as the classical test statistics. A simulation study for small and moderate sample sizes shows that there are some test statistics in the family that can compete with the classical likelihood ratio and the chi-squared test statistics.     

Integrated likelihood computation methods

Suppose a model has parameter \(\theta =(\psi , \lambda )\), where \(\psi \) is the parameter of interest and \(\lambda \) is a nuisance parameter. The integrated likelihood method eliminates \(\lambda \) from the likelihood function \(L(\psi , \lambda )\) by integrating with respect to a weight function \(\pi (\lambda | \psi )\). The resulting integrated likelihood function \(\bar{L}(\psi )\) can be used for inference for \(\psi \). However, the analytical form for the integrated likelihood is not always available. This paper discusses 12 different approaches to computing the integrated likelihood. Some methods were originally developed for other computation purposes and they are modified to fit in the integrated likelihood framework. Methods considered include direct numerical integration methods such as Monte Carlo integration method, importance sampling, Laplace method; marginal likelihood computation methods; and methods for computing the marginal posterior density. Simulation studies and real data example are presented to evaluate and compare these methods empirically.     

Hanf number for Scott sentences of computable structures

The Hanf number for a set S of sentences in \(\mathcal {L}_{\omega _1,\omega }\) (or some other logic) is the least infinite cardinal \(\kappa \) such that for all \(\varphi \in S\), if \(\varphi \) has models in all infinite cardinalities less than \(\kappa \), then it has models of all infinite cardinalities. Friedman asked what is the Hanf number for Scott sentences of computable structures. We show that the value is \(\beth _{\omega _1^{CK}}\). The same argument proves that \(\beth _{\omega _1^{CK}}\) is the Hanf number for Scott sentences of hyperarithmetical structures.     

Empirical likelihood bivariate nonparametric maximum likelihood estimator with right censored data

This article considers the estimation for bivariate distribution function (d.f.) \(F_0(t, z)\) of survival time \(T\) and covariate variable \(Z\) based on bivariate data where \(T\) is subject to right censoring. We derive the empirical likelihood-based bivariate nonparametric maximum likelihood estimator \(\hat{F}_n(t,z)\) for \(F_0(t,z)\) , which has an explicit expression and is unique in the sense of empirical likelihood. Other nice features of \(\hat{F}_n(t,z)\) include that it has only nonnegative probability masses, thus it is monotone in bivariate sense. We show that under \(\hat{F}_n(t,z)\) , the conditional d.f. of \(T\) given \(Z\) is of the same form as the Kaplan–Meier estimator for the univariate case, and that the marginal d.f. \(\hat{F}_n(\infty ,z)\) coincides with the empirical d.f. of the covariate sample. We also show that when there is no censoring, \(\hat{F}_n(t,z)\) coincides with the bivariate empirical d.f. For discrete covariate \(Z\) , the strong consistency and weak convergence of \(\hat{F}_n(t,z)\) are established. Some simulation results are presented.     

Robust Approximations for Pricing Asian Options and Volatility Swaps Under Stochastic Volatility

Abstract

We show that if the discounted Stock price process is a continuous martingale, then there is a simple relationship linking the variance of the terminal Stock price and the variance of its arithmetic average. We use this to establish a model-independent upper bound for the price of a continuously sampled fixed-strike arithmetic Asian call option, in the presence of non-zero time-dependent interest rates (Theorem 1.2). We also propose a model-independent lognormal moment-matching procedure for approximating the price of an Asian call, and we show how to apply these approximations under the Black–Scholes and Heston models (subsection 1.3). We then apply a similar analysis to a time-dependent Heston stochastic volatility model, and we show how to construct a time-dependent mean reversion and volatility-of-variance function, so as to be consistent with the observed variance swap curve and a pre-specified term structure for the variance of the integrated variance (Theorem 2.1). We characterize the small-time asymptotics of the first and second moments of the integrated variance (Proposition 2.2) and derive an approximation for the price of a volatility swap under the time-dependent Heston model ( Equation (52)), using the Brockhaus–Long approximation (Brockhaus, and Long, 2000 Brockhaus, O. and Long, D. 2000. Volatility Swaps made simple. Risk, 13(1) January: 92–96.  [Google Scholar]). We also outline a bootstrapping procedure for calibrating a piecewise-linear mean reversion level and volatility-of-volatility function (Subsection 2.3.2).     

Highly Incident Configurations with Chiral Symmetry

A geometric $k$ -configuration is a collection of points and straight lines in the plane so that $k$ points lie on each line and $k$ lines pass through this point. We introduce a new construction method for constructing $k$ -configurations with non-trivial dihedral or chiral (i.e., purely rotational) symmetry, for any $k \ge 3$ ; the configurations produced have $2^{k-2}$ symmetry classes of points and lines. The construction method produces the only known infinite class of symmetric geometric 7-configurations, the second known infinite class of symmetric geometric 6-configurations, and the only known 6-configurations with chiral symmetry.     

Testing with Exponentially Tilted Empirical Likelihood

Imposing restrictions without assuming underlying distributions to modelize complex realities is a valuable methodological tool. However, if a subset of restrictions were not correctly specified, the usual test-statistics for correctly specified models tend to reject erronously a simple null hypothesis. In this setting, we may say that the model suffers from misspecification. We study the behavior of empirical phi-divergence test-statistics, introduced in Balakrishnan et al. Statistics 49:951–977 (2015), by using the exponential tilted empirical likelihood estimators of Schennach Ann Stat 35:634–672 (2007), as a good compromise between the efficiency of the significance level for small sample sizes and the robustness under misspecification.     

Precise large deviations for dependent regularly varying sequences

We study a precise large deviation principle for a stationary regularly varying sequence of random variables. This principle extends the classical results of Nagaev (Theory Probab Appl 14:51–64, 193–208, 1969) and Nagaev (Ann Probab 7:745–789, 1979) for iid regularly varying sequences. The proof uses an idea of Jakubowski (Stoch Proc Appl 44:291–327, 1993; 68:1–20, 1997) in the context of central limit theorems with infinite variance stable limits. We illustrate the principle for stochastic volatility models, real valued functions of a Markov chain satisfying a polynomial drift condition and solutions of linear and non-linear stochastic recurrence equations.     

A data driven equivariant approach to constrained Gaussian mixture modeling

Maximum likelihood estimation of Gaussian mixture models with different class-specific covariance matrices is known to be problematic. This is due to the unboundedness of the likelihood, together with the presence of spurious maximizers. Existing methods to bypass this obstacle are based on the fact that unboundedness is avoided if the eigenvalues of the covariance matrices are bounded away from zero. This can be done imposing some constraints on the covariance matrices, i.e. by incorporating a priori information on the covariance structure of the mixture components. The present work introduces a constrained approach, where the class conditional covariance matrices are shrunk towards a pre-specified target matrix \(\varvec{\varPsi }.\) Data-driven choices of the matrix \(\varvec{\varPsi },\) when a priori information is not available, and the optimal amount of shrinkage are investigated. Then, constraints based on a data-driven \(\varvec{\varPsi }\) are shown to be equivariant with respect to linear affine transformations, provided that the method used to select the target matrix be also equivariant. The effectiveness of the proposal is evaluated on the basis of a simulation study and an empirical example.     

Empirical likelihood inference for estimating equation with missing data

In this article, empirical likelihood inference for estimating equation with missing data is considered. Based on the weighted-corrected estimating function, an empirical log-likelihood ratio is proved to be a standard chi-square distribution asymptotically under some suitable conditions. This result is different from those derived before. So it is convenient to construct confidence regions for the parameters of interest. We also prove that our proposed maximum empirical likelihood estimator θ is asymptotically normal and attains the semiparametric efficiency bound of missing data. Some simulations indicate that the proposed method performs the best.     

Regular prism tilings in {{\widetilde{\bf SL_{2}R}}} space

\({{\widetilde{\bf SL_{2}R}}}\) geometry is one of the eight 3-dimensional Thurston geometries, it can be derived from the 3-dimensional Lie group of all 2 × 2 real matrices with determinant one. Our aim is to describe and visualize the regular infinite or bounded p-gonal prism tilings in \({{\widetilde{\bf SL_{2}R}}}\) . For this purpose we introduce the notion of infinite and bounded prisms, prove that there exist infinitely many regular infinite p-gonal face-to-face prism tilings \({\mathcal{T}^i_p(q)}\) and infinitely many regular bounded p-gonal non-face-to-face \({{\widetilde{\bf SL_{2}R}}}\) prism tilings \({\mathcal{T}_p(q)}\) for integer parameters \({p,q; 3 \leq p, \frac{2p}{p-2} < q}\) . Moreover, we describe the symmetry group of \({\mathcal{T}_p(q)}\) via its index 2 rotational subgroup, denoted by pq2 ₁ . Surprisingly this group already occurred in our former work (Molnár et al., J Geometry, 95:91–133, 2009) in another context. We also develop a method to determine the data of the space filling regular infinite and bounded prism tilings. We apply the above procedure to \({\mathcal{T}^i_3(q)}\) and \({\mathcal{T}_3(q)}\) where 6 < q and visualize them and the corresponding tilings. E. Molnár showed, that homogeneous 3-spaces have a unified interpretation in the projective 3-sphere \({\mathcal{PS}^3}\) and 3-space \({\mathcal{P}^3({\bf V}^4,{\bf V}_4, {\bf R})}\) . In our work we will use this projective model of \({{\widetilde{\bf SL_{2}R}}}\) and in this manner the prisms and prism tilings can be visualized on the Euclidean screen of a computer.     

A Few Properties of Just Infinite Algebras

In this article we prove a few interesting properties of just infinite algebras. Bartholdi (2006 Bartholdi , L. ( 2006 ). Branch rings, thinned rings, tree enveloping rings . Israel J. Math. 154 : 93 – 139 .[Crossref], [Web of Science ®] , [Google Scholar]), defines a particular class of just infinite algebras and demonstrates various properties of these examples. One such property, which is tedious to prove for his specific examples, is primality. We prove here that, in fact, all just infinite algebras are prime. We then consider two corollaries of this theorem; one suggests a weaker definition of just infinite for finitely generated algebras and the other examines the specific case of just infinite algebras which also satisfy a polynomial identity.     

On a Regularized Family of Models for Homogeneous Incompressible Two-Phase Flows

We consider a general family of regularized models for incompressible two-phase flows based on the Allen–Cahn formulation in \(n\) -dimensional compact Riemannian manifolds for \(n=2,3\) . The system we consider consists of a regularized family of Navier–Stokes equations (including the Navier–Stokes- \(\alpha \) -like model, the Leray- \(\alpha \) model, the modified Leray- \(\alpha \) model, the simplified Bardina model, the Navier–Stokes–Voight model, and the Navier–Stokes model) for the fluid velocity \(u\) suitably coupled with a convective Allen–Cahn equation for the order (phase) parameter \(\phi \) . We give a unified analysis of the entire three-parameter family of two-phase models using only abstract mapping properties of the principal dissipation and smoothing operators and then use assumptions about the specific form of the parameterizations, leading to specific models, only when necessary to obtain the sharpest results. We establish existence, stability, and regularity results and some results for singular perturbations, which as special cases include the inviscid limit of viscous models and the \(\alpha \rightarrow 0\) limit in \(\alpha \) models. Then we show the existence of a global attractor and exponential attractor for our general model and establish precise conditions under which each trajectory \(\left( u,\phi \right) \) converges to a single equilibrium by means of a Lojasiewicz–Simon inequality. We also derive new results on the existence of global and exponential attractors for the regularized family of Navier–Stokes equations and magnetohydrodynamics models that improve and complement the results of Holst et al. (J Nonlinear Sci 20(5):523–567, 2010). Finally, our analysis is applied to certain regularized Ericksen–Leslie models for the hydrodynamics of liquid crystals in \(n\) -dimensional compact Riemannian manifolds.     

Heat Kernel Empirical Laws on $${\mathbb {U}}_N$$ and $${\mathbb {GL}}_N$$GLN

This paper studies the empirical laws of eigenvalues and singular values for random matrices drawn from the heat kernel measures on the unitary groups \({\mathbb {U}}_N\) and the general linear groups \({\mathbb {GL}}_N\), for \(N\in {\mathbb {N}}\). It establishes the strongest known convergence results for the empirical eigenvalues in the \({\mathbb {U}}_N\) case, and the first known almost sure convergence results for the eigenvalues and singular values in the \({\mathbb {GL}}_N\) case. The limit noncommutative distribution associated with the heat kernel measure on \({\mathbb {GL}}_N\) is identified as the projection of a flow on an infinite-dimensional polynomial space. These results are then strengthened from variance estimates to \(L^p\) estimates for even integers p.     

Blockwise bootstrap of the estimated empirical process based on $$\psi $$-weakly dependent observations

The distributional convergence of the bootstrapped estimated empirical process is shown and bootstrap consistency in the \(\sup \)-norm for test statistics based on that process. Bootstrapping the estimated empirical process has up to now been considered by assuming independence of the observations, where we give up this assumption now and allow the observations to be \(\psi \)-weakly dependent in the sense of Doukhan and Louhichi (Stoch Proc Appl 84:313–342, 1999). Due to the fact that no model assumptions on the original process are made, a nonparametric blockwise bootstrap procedure is used, which has previously been used in empirical process theory based on mixing observations. Here, we succeeded in proving that assuming \(l=o(n)\) and \(l\rightarrow \infty \) as only conditions for the blocklength is sufficient to show convergence of the bootstrap process to the same limit as for the original process under \({\mathcal {H}}_0\), which is the weakest condition that has been imposed in that context up to now.     

Jackknife empirical likelihood for the difference of two volumes under ROC surfaces

The volume under a surface (VUS) is an effective measure for evaluating the discriminating power of a diagnostic test with three ordinal diagnostic groups. In this paper, we investigate the difference of two correlated VUS’s to compare two treatments for discrimination of three-class classification data. A jackknife empirical likelihood (JEL) procedure is employed to avoid the variance estimation in the existing methods. We prove that the limiting distribution of the empirical log-likelihood ratio statistic follows a \(\chi ^2\) distribution. Extensive numerical studies show that the JEL confidence intervals outperform those based on the normal approximation method. The proposed method is also applied to the Alzheimer’s disease data.     

On Infinite–finite Duality Pairs of Directed Graphs

The $({\mathcal{A}},{\mathcal{D}})$ duality pairs play a crucial role in the theory of general relational structures and in Constraint Satisfaction Problems. The case where both sides are finite is fully characterized. The case where both sides are infinite seems to be very complex. It is also known that no finite–infinite duality pair is possible if we make the additional restriction that both classes are antichains. In this paper (which is the first one of a series) we start the detailed study of the infinite–finite case. Here we concentrate on directed graphs. We prove some elementary properties of the infinite–finite duality pairs, including lower and upper bounds on the size of ${\mathcal{D}}$ , and show that the elements of ${\mathcal{A}}$ must be equivalent to forests if ${\mathcal{A}}$ is an antichain. Then we construct instructive examples, where the elements of ${\mathcal{A}}$ are paths or trees. Note that the existence of infinite–finite antichain dualities was not previously known.