Local asymptotic normality of multivariate ARMA processes with a linear trend

The local asymptotic normality (LAN) property is established for multivariate ARMA models with a linear trend or, equivalently, for multivariate general linear models with ARMA error term. In contrast with earlier univariate results, the central sequence here is correlogram-based, i.e. expressed in terms of a generalized concept of residual cross-covariance function.     

On asymptotic theory for multivariate GARCH models

The paper investigates the asymptotic theory for a multivariate GARCH model in its general vector specification proposed by Bollerslev, Engle and Wooldridge (1988) [4], known as the VEC model. This model includes as important special cases the so-called BEKK model and many versions of factor GARCH models, which are often used in practice. We provide sufficient conditions for strict stationarity and geometric ergodicity. The strong consistency of the quasi-maximum likelihood estimator (QMLE) is proved under mild regularity conditions which allow the process to be integrated. In order to obtain asymptotic normality, the existence of sixth-order moments of the process is assumed.     

Local asymptotic mixed normality of transformed Gaussian models for random fields

Local asymptotic mixed normality (LAMN) of a class of transformed Gaussian models for discretely observed random fields is proved. The original Gaussian random field is assumed to be the product of a deterministic process and a process with independent increments. The transformed process is observed only on discrete lattice points in the unit cube and fixed domain asymptotics is investigated. This model is useful for modeling random fields with non-Gaussian marginal distributions.     

Local asymptotic normality of a sequential model for marked point processes and its applications

This paper deals with statistical inference problems for a special type of marked point processes based on the realization in random time intervals [0,_u]. Sufficient conditions to establish the local asymptotic normality (LAN) of the model are presented, and then, certain class of stopping times _u satisfying them is proposed. Using these stopping rules, one can treat the processes within the framework of LAN, which yields asymptotic optimalities of various inference procedures. Applications for compound Poisson processes and continuous time Markov branching processes (CMBP) are discussed. Especially, asymptotically uniformly most powerful tests for criticality of CMBP can be obtained. Such tests do not exist in the case of the non-sequential approach. Also, asymptotic normality of the sequential maximum likelihood estimators (MLE) of the Malthusian parameter of CMBP can be derived, although the non-sequential MLE is not asymptotically normal in the supercritical case.     

On asymptotic normality of pseudo likelihood estimates for pairwise interaction processes

We consider point processes defined through a pairwise interaction potential and admitting a two-dimensional sufficient statistic. It is shown that the pseudo maximum likelihood estimate can be stochastically normed so that the limiting distribution is a standard normal distribution. This result is true irrespectively of the possible existence of phase transitions. The work here is an extension of the work Guyon and Künsch (1992,Lecture Notes in Statist.,74, Springer, New York) and is based on viewing a point process interchangeably as a lattice field.     

PP multivariate random weighting method

According to the Projection Pursuit (PP) method and the random weighting method, we propose a PP random weighting method, and set up the asymptotic distribution theory and strong limit theorem of PP random weighting empirical process. Applying this method, we obtain two kinds of goodness-of-fit test for a multivariate distribution function, i.e., we get the random weighting approximations of PP Kolmogorov Smirnov statistics (PPKS) and PP Smirnov Cramér Von Mises statistics (PPSC), we prove that the asymptotic distribution of PPKS and PPSC are the same as those of their respective random weighting approximations.Supported by the National Natural Science Foundation of China.     

Local structure of bifurcation curves for nonlinear Sturm-Liouville problems

We consider the nonlinear bifurcation problem arising in population dynamics and nonlinear Schrödinger equation:

     

Consistency and asymptotic normality of the maximum quasi-likelihood estimator in quasi-likelihood nonlinear models with random regressors

This paper proposes some regularity conditions, which result in the existence, strong consistency and asymptotic normality of maximum quasi-likelihood estimator （MQLE） in quasi-likelihood nonlinear models （QLNM） with random regressors. The asymptotic results of generalized linear models （GLM） with random regressors are generalized to QLNM with random regressors.     

Local asymptotic normality in a stationary model for spatial extremes

De Haan and Pereira (2006) [6] provided models for spatial extremes in the case of stationarity, which depend on just one parameter β>0 measuring tail dependence, and they proposed different estimators for this parameter. We supplement this framework by establishing local asymptotic normality (LAN) of a corresponding point process of exceedances above a high multivariate threshold. Standard arguments from LAN theory then provide the asymptotic minimum variance within the class of regular estimators of β. It turns out that the relative frequency of exceedances is a regular estimator sequence with asymptotic minimum variance, if the underlying observations follow a multivariate extreme value distribution or a multivariate generalized Pareto distribution.     

Local asymptotic normality for progressively censored likelihood ratio statistics and applications

Let X_n,1 ≤ X_n,2 ≤ … ≤ X_n,n be the ordered variables corresponding to a random sample of size n with respect to a family of probability measures {P_θ:θ ∈ Θ} where Θ is an open subset of the real line. In many practical situations the X_n,i are the observables and experimentation must be curtailed prior to X_n,n. If τ_n is a stopping variable adapted to the σ-fields {σ(X_n,1,…,X_n,k): 1 ≤ k ≤ n} and P_n,θ the projection of P_θ onto σ(X_n,1,…,X_n,τn), the local asymptotic normality of the stopped progressively censored likelihood ratio statistics is established with θ, $\text{θ}{}_{\mathrm{n}}\text{= θ + un}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}\text{∈ Θ}$ and θ, u held fixed, under certain conditions on the underlying distribution and on τ_n. Conditions are also given to ensure the local asymptotic normality of likelihood ratio statistics where the underlying observations are given in a series scheme.     

Second order sensitivity analysis and asymptotic theory of parametrized nonlinear programs

In this paper we study second-order differential properties of an optimal-value function(x). It is shown that under certain conditions(x) possesses second-order directional derivatives, which can be calculated by solving corresponding quadratic programs. Also upper and lower bounds on these derivatives are introduced under weaker assumptions. In particular we show that the second-order directional derivative is infinite if the corresponding quadratic program is unbounded. Finally sensitivity results are applied to investigate asymptotics of estimators in parametrized nonlinear programs.     

Oscillation and asymptotic behavior for second-order nonlinear perturbed differential equations

In this paper, for all regular solutions of a class of second-order nonlinear perturbed differential equations, new oscillation criteria are established. Asymptotic behavior for forced equations is also discussed.     

Asymptotic normality for discretely observed Markov jump processes with an absorbing state

For a continuous-time Markov process, occasionally, only discrete-time observations are available. For a simple sample of homogeneous Markov jump processes with an absorbing state, observed each on a stochastic grid of time points, we establish asymptotic normality of the maximum likelihood estimator and close the gap in Kremer and Weißbach (2013). By showing that the solution of the Kolmogorov backward equation system is continuous differentiable, we can apply results for M-estimators.     

Local asymptotic attractivity for nonlinear quadratic functional integral equations

In this paper, an existence result for local asymptotic attractivity of the solutions is proved for a nonlinear quadratic functional integral equation under certain growth conditions which in turn gives the existence as well as asymptotic stability of solutions. A couple of examples are provided for indicating the natural realizations of abstract theory presented in the paper.     

Comparison of the asymptotic stability properties for two multirate strategies

This paper contains a comparison of the asymptotic stability properties for two multirate strategies. For each strategy, the asymptotic stability regions are presented for a 2×2 test problem and the differences between the results are discussed. The considered multirate schemes use Rosenbrock type methods as the main time integration method and have one level of temporal local refinement. Some remarks on the relevance of the results for 2×2 test problems are presented.     

On the construction of Wold decomposition for multivariate stationary processes

A method to construct the Wold decomposition for multivariate stationary stochastic processes x_k, k Z, is presented. The method is based on orthogonal decompositions for x_k, k Z, obtained by forming orthogonal projections of x_k, k Z, onto its component processes , k Z, j = 1, …, q. The method does not give a complete solution to the Wold decomposition problem.     

Simulated annealing with asymptotic convergence for nonlinear constrained optimization

In this paper, we present constrained simulated annealing (CSA), an algorithm that extends conventional simulated annealing to look for constrained local minima of nonlinear constrained optimization problems. The algorithm is based on the theory of extended saddle points (ESPs) that shows the one-to-one correspondence between a constrained local minimum and an ESP of the corresponding penalty function. CSA finds ESPs by systematically controlling probabilistic descents in the problem-variable subspace of the penalty function and probabilistic ascents in the penalty subspace. Based on the decomposition of the necessary and sufficient ESP condition into multiple necessary conditions, we present constraint-partitioned simulated annealing (CPSA) that exploits the locality of constraints in nonlinear optimization problems. CPSA leads to much lower complexity as compared to that of CSA by partitioning the constraints of a problem into significantly simpler subproblems, solving each independently, and resolving those violated global constraints across the subproblems. We prove that both CSA and CPSA asymptotically converge to a constrained global minimum with probability one in discrete optimization problems. The result extends conventional simulated annealing (SA), which guarantees asymptotic convergence in discrete unconstrained optimization, to that in discrete constrained optimization. Moreover, it establishes the condition under which optimal solutions can be found in constraint-partitioned nonlinear optimization problems. Finally, we evaluate CSA and CPSA by applying them to solve some continuous constrained optimization benchmarks and compare their performance to that of other penalty methods.     

Asymptotic optimal inference for multivariate branching-Markov processes via martingale estimating functions and mixed normality

Multivariate tree-indexed Markov processes are discussed with applications. A Galton-Watson super-critical branching process is used to model the random tree-indexed process. Martingale estimating functions are used as a basic framework to discuss asymptotic properties and optimality of estimators and tests. The limit distributions of the estimators turn out to be mixtures of normals rather than normal. Also, the non-null limit distributions of standard test statistics such as Wald, Rao’s score, and likelihood ratio statistics are shown to have mixtures of non-central chi-square distributions. The models discussed in this paper belong to the local asymptotic mixed normal family. Consequently, non-standard limit results are obtained.