Asymptotic normality of in- and out-degree counts in a preferential attachment model

Preferential attachment in a directed scale-free graph is an often used paradigm for modeling the evolution of social networks. Social network data is usually given in a format allowing recovery of the number of nodes with in-degree i and out-degree j. Assuming a model with preferential attachment, formal statistical procedures for estimation can be based on such data summaries. Anticipating the statistical need for such node-based methods, we prove asymptotic normality of the node counts. Our approach is based on a martingale construction and a martingale central limit theorem.     

Convergence results for strict C-sequences

We examine properties of strict C-sequences. These sequences are a generalization both of eventual martingales and of quasimartingales. Several global and local convergence results are proved. The transform and the quadratic variation of a strict C-sequence are also studied. A comparison with various martingale generalizations is established.     

Linear Models with Doubly Exchangeable Distributed Errors

We study the general linear model (GLM) with doubly exchangeable distributed error for m observed random variables. The doubly exchangeable general linear model (DEGLM) arises when the m-dimensional error vectors are “doubly exchangeable,” jointly normally distributed, which is a much weaker assumption than the independent and identically distributed error vectors as in the case of GLM or classical GLM (CGLM). We estimate the parameters in the model and also find their distributions. We show that the tests of intercept and slope are possible in DEGLM as a particular case using parametric bootstrap as well as multivariate Satterthwaite approximation.     

A Modified Empirical Martingale Simulation for Financial Derivative Pricing

This article extends the empirical martingale simulation (EMS) method from using a risk-neutral measure to using a dynamic measure for financial derivative pricing. Although the EMS is shown to be capable of obtaining consistent estimate of financial derivative prices in a more efficient way than the standard Monte Carlo simulation procedure, it can proceed only under a risk-neutral framework. In practice, however, it is cumbersome to obtain the explicit expression of a risk-neutral model when dealing with a complex model. To alleviate this difficulty, we compute the financial derivative prices under the dynamic model and impose the martingale property on the simulated sample paths of both the change of measure process and the underlying asset prices under the dynamic P measure. Hence, we call this modification the empirical P-martingale simulation (EPMS). The strong consistency of the EPMS is established and its efficiency is performed by simulation in the GARCH framework. Simulation results shows that EPMS has the similar variance reduction as the EMS method in option pricing if the risk-neutral model can be obtained, and is more efficient than the standard Monte Carlo simulation in most cases.     

A martingale residual diagnostic for longitudinal and recurrent event data

One method of assessing the fit of an event history model is to plot the empirical standard deviation of standardised martingale residuals. We develop an alternative procedure which is valid also in the presence of measurement error and applicable to both longitudinal and recurrent event data. Since the covariance between martingale residuals at times t ₀ and t > t ₀ is independent of t, a plot of these covariances should, for fixed t ₀, have no time trend. A test statistic is developed from the increments in the estimated covariances, and we investigate its properties under various types of model misspecification. Applications of the approach are presented using two Brazilian studies measuring daily prevalence and incidence of infant diarrhoea and a longitudinal study into treatment of schizophrenia.     

An Approximation of Subfractional Brownian Motion

In this article, we obtain an approximation theorem for subfractional Brownian motion with H > 1/2, using martingale differences. The proof involves the tightness and identification of finite dimensional distributions.     

Inference for random coefficient INAR(k) with the occasional level shift random noise based on dual empirical likelihood

The INAR(k) model has been widely used in various kinds of fields. However, there are little discussions about the INAR(k) model with the occasional level shift random noise. In this paper, the maximum likelihood estimation of parameter based on martingale difference sequence is given, the log empirical likelihood ratio test statistic is obtained and the test statistic converges to chi-square distribution, we prove that the confidence region of the parameter is convex. Furthermore, the numerical simulation of the proposed INAR(k) model is given, which illustrates the effectiveness of the model. Then, the proofs of asymptotic results are given in the Appendix.     

Complete convergence for weighted sums of mixingale sequences and statistical applications

In this paper, the complete convergence of weighted sums of L^r-mixingale is established, from which the complete convergence of martingale differences is also derived. As statistical applications, non parametric regression model and simpler linear errors-in-variables model with mixingale errors are discussed.     

Number of Working Components in Consecutive k-out-of-n System While It Is Working

In this article we study the distribution and expected value of the number of working components at time t in a consecutive k-out-of-n system under the condition that it is working at time t. We provide the exact distribution of the corresponding conditional random variable and compute its expected value for the system consisting of exchangeable dependent components. The results are also extended to any coherent system by the aid of system signature. Finally, we present illustrative and computational results for some systems having Lomax components.     

Strong consistency of conditional least squares estimators in multiple regime threshold autoregressive models

In this paper we analyse the conditional least squares estimators of the parameters of a multiple regime threshold AR(1) model and prove that under certain conditions these are strongly consistent. We assume that the error process in each regime is amartingale difference sequence. Then we deal with strong consistency of the natural estimator of the error variance in each regime.     

Distributions of simple patterns in some kinds of exchangeable sequences

In this article, we consider the distributions of simple patterns in some types of sequences of infinite exchangeable multi-state trials. The distributions on exchangeable multi-state trials are considered in terms of an extension of de Finetti's theorem. As an application of partially exchangeable sequences, distributions on a Markov exchangeable sequence are studied. Furthermore, we propose a new type of partially exchangeable sequence and examine its properties. In addition, we discuss the distribution theory in the case of the finite exchangeable sequences. The results presented here provide a wide framework for developing the exact distribution theory of simple patterns. Finally, some examples are given in order to illustrate our theoretical results.     

Omnibus Tests of The Martingale Assumption in The Analysisof Recurrent Failure Time Data

The Andersen-Gill multiplicative intensity(MI) model is well-suited to the analysis of recurrent failuretime data. The fundamental assumption of the MI model is thatthe process M_i(t) for subjects i=1,,n,defined to be the difference between a subject's counting processand compensator, i.e., N_i(t) A_i(t); >0,is a martingale with respect to some filtration. We propose omnibusprocedures for testing this assumption. The methods are basedon transformations of the estimated martingale residual process ^M _i (t) a function of consistent estimatesof the log-intensity ratios and the baseline cumulative hazard.Under a correctly specified model, the expected value of ^M _i (t)is approximately equal to zero with approximately uncorrelatedincrements. These properties are exploited in the proposed testingprocedures. We examine the effects of censoring and covariateeffects on the operating characteristics of the proposed methodsvia simulation. The procedures are most sensitive to the omissionof a time-varying continuous covariate. We illustrate use ofthe methods in an analysis of data from a clinical trial involvingpatients with chronic granulatomous disease.     

Stochastic monotonicity and comparability of Markov chains with block-monotone transition matrices and their applications to queueing systems

ABSTRACT

Motivated by various applications in queueing theory, this article is devoted to the stochastic monotonicity and comparability of Markov chains with block-monotone transition matrices. First, we introduce the notion of block-increasing convex order for probability vectors, and characterize the block-monotone matrices in the sense of the block-increasing order and block-increasing convex order. Second, we characterize the Markov chain with general transition matrix by martingale and provide a stochastic comparison of two block-monotone Markov chains under the two block-monotone orders. Third, the stochastic comparison results for the Markov chains corresponding to the discrete-time GI/G/1 queue with different service distributions under the two block-monotone orders are given, and the lower bound and upper bound of the Markov chain corresponding to the discrete-time GI/G/1 queue in the sense of the block-increasing convex order are found.     

Asymptotic properties for estimates of nonparametric regression model with martingale difference errors

Consider the nonparametric regression model with martingale difference errors. Nonparametric estimator g _n (x) of regression function g(x) will be introduced, and its asymptotic properties are studied. In particular, the pointwise and uniform convergence of g _n (x) and its asymptotic normality will be investigated. This extends the earlier work on independent random errors.     

Regime-switching pure jump processes and applications in the valuation of mortality-linked products

In this paper, we study the pricing of longevity bonds and an insurance contract on multiple lives in a regime-switching market driven by an underlying continuous-time Markov chain. For modeling dependent mortality, we make use of a Markov chain and some shot noise processes with regime switching. By using a martingale method, we give semi-analytical expressions for the price of longevity bonds and the premium of an insurance contract on the kth person to die.     

On the Mean Remaining Strength of the k-Out-of-n:F System with Exchangeable Components

In this article, we consider the mean remaining strength of a k-out-of-n:F system in the stress–strength setup for the exchangeable components. We provide some results for parallel and series systems under this setup, where X₁, X₂, …, X_n are the strengths of the components designed under the common stress. An illustrative example is given for the k-out-of- n:F system using the multivariate FGM distribution.     

Inverse Probability of Censoring Weighted U‐statistics for Right‐Censored Data with an Application to Testing Hypotheses

Abstract. A right‐censored version of a U ‐statistic with a kernel of degree m 1 is introduced by the principle of a mean preserving reweighting scheme which is also applicable when the dependence between failure times and the censoring variable is explainable through observable covariates. Its asymptotic normality and an expression of its standard error are obtained through a martingale argument. We study the performances of our U ‐statistic by simulation and compare them with theoretical results. A doubly robust version of this reweighted U ‐statistic is also introduced to gain efficiency under correct models while preserving consistency in the face of model mis‐specifications. Using a Kendall's kernel, we obtain a test statistic for testing homogeneity of failure times for multiple failure causes in a multiple decrement model. The performance of the proposed test is studied through simulations. Its usefulness is also illustrated by applying it to a real data set on graft‐versus‐host‐disease.     

Exchange option pricing in jump-diffusion models based on esscher transform

In the real world, we introduce a dynamic model about the risky asset which is governed by Brownian motion, stationary compound Poisson process and its compensation process. By choosing Esscher transform parameters, we obtain a risk-neural measure Q under which the discounted value of the risky underlying asset is a martingale. Then, we give the pricing formulas of Exchange option by change of numeraire. At last, we analyze the option pricing formula and provide numerical illustrations by introducing BBY stock and SBUX stock.     

Nonlinear regression in a trivariate elliptical distribution via a linear combination of order statistics

In this paper, by assuming that (X, Y ₁, Y ₂)^T has a trivariate elliptical distribution, we derive the exact joint distribution of X and a linear combination of order statistics from (Y ₁, Y ₂)^T and show that it is a mixture of unified bivariate skew-elliptical distributions. We then derive the corresponding marginal and conditional distributions for the special case of t kernel. We also present these results for an exchangeable case with t kernel and illustrate the established results with an air-pollution data.     

Complete consistency for the estimator of nonparametric regression model based on martingale difference errors

Abstract

In this paper, we study the complete consistency for the estimator of nonparametric regression model based on martingale difference errors, and obtain the convergence rates of the complete consistency by using the inequalities for martingale difference sequence. Finally, some simulations are illustrated.