Optimal Designs for Approximating a Stochastic Process with Respect to a Minimax Criterion

We study the problem of approximating a stochastic process Y = {Y(t: t ∈ T} with known and continuous covariance function R on the basis of finitely many observations Y(t ₁,), …, Y(t _n ). Dependent on the knowledge about the mean function, we use different approximations ? and measure their performance by the corresponding maximum mean squared error sub _t∈T E(Y(t) ? ?(t))². For a compact T ? ? ^p we prove sufficient conditions for the existence of optimal designs. For the class of covariance functions on T ² = [0, 1]² which satisfy generalized Sacks/Ylvisaker regularity conditions of order zero or are of product type, we construct sequences of designs for which the proposed approximations perform asymptotically optimal.     

Sojourn time in an M/M/1 processor sharing queue with permanent customers

We explicitly compute the sojourn time distribution of an arbitrary customer in an M/M/1 processor sharing (PS) queue with permanent customers. We notably exhibit the orthogonal structure associated with this queuing system and we show how sieved Pollaczek polynomials and their associated orthogonality measure can be used to obtain an explicit representation for the complementary cumulative distribution function of the sojourn time of a customer. This explicit formula subsequently allows us to compute the two first moments of this random variable and to study the asymptotic behavior of its distribution. The most salient result is that the decay rate depends on the load of the system and the number K of permanent customers. When the load is above a certain threshold depending on K, the decay rate is identical to that of a regular M/M/1 PS queue.     

Estimation of covariance matrix via the sparse Cholesky factor with lasso

In this paper, we discuss a parsimonious approach to estimation of high-dimensional covariance matrices via the modified Cholesky decomposition with lasso. Two different methods are proposed. They are the equi-angular and equi-sparse methods. We use simulation to compare the performance of the proposed methods with others available in the literature, including the sample covariance matrix, the banding method, and the L₁-penalized normal loglikelihood method. We then apply the proposed methods to a portfolio selection problem using 80 series of daily stock returns. To facilitate the use of lasso in high-dimensional time series analysis, we develop the dynamic weighted lasso (DWL) algorithm that extends the LARS-lasso algorithm. In particular, the proposed algorithm can efficiently update the lasso solution as new data become available. It can also add or remove explanatory variables. The entire solution path of the L₁-penalized normal loglikelihood method is also constructed.     

Marker processes in survival analysis

In the development of many diseases there are often associated variables which continuously measure the progress of an individual towards the final expression of the disease (failure). Such variables are stochastic processes, here called marker processes, and, at a given point in time, they may provide information about the current hazard and subsequently on the remaining time to failure. Here we consider a simple additive model for the relationship between the hazard function at time t and the history of the marker process up until time t. We develop some basic calculations based on this model. Interest is focused on statistical applications for markers related to estimation of the survival distribution of time to failure, including (i) the use of markers as surrogate responses for failure with censored data, and (ii) the use of markers as predictors of the time elapsed since onset of a survival process in prevalent individuals. Particular attention is directed to potential gains in efficiency incurred by using marker process information.     

Estimated Generalized Least Squares for a Heteroscedastic Regression Model

This paper considers the general linear regression model yc = X₁β+u_t under the heteroscedastic structure E(u_t) = 0, E(u²) =σ²- (X_tβ)², E(u_t u_s) = 0, tæs, t, s= 1, T. It is shown that any estimated GLS estimator for β is asymptotically equivalent to the GLS estimator under some regularity conditions. A three-step GLS estimator, which calls upon the assumption E(u_t²) =s?²(X,β)² for the estimation of the disturbance covariance matrix, is considered.     

Order statistics from trivariate normal and -distributions in terms of generalized skew-normal and skew- distributions

We consider here a generalization of the skew-normal distribution, GSN(λ₁,λ₂,ρ), defined through a standard bivariate normal distribution with correlation ρ, which is a special case of the unified multivariate skew-normal distribution studied recently by Arellano-Valle and Azzalini [2006. On the unification of families of skew-normal distributions. Scand. J. Statist. 33, 561–574]. We then present some simple and useful properties of this distribution and also derive its moment generating function in an explicit form. Next, we show that distributions of order statistics from the trivariate normal distribution are mixtures of these generalized skew-normal distributions; thence, using the established properties of the generalized skew-normal distribution, we derive the moment generating functions of order statistics, and also present expressions for means and variances of these order statistics.Next, we introduce a generalized skew-t_ν distribution, which is a special case of the unified multivariate skew-elliptical distribution presented by Arellano-Valle and Azzalini [2006. On the unification of families of skew-normal distributions. Scand. J. Statist. 33, 561–574] and is in fact a three-parameter generalization of Azzalini and Capitanio's [2003. Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t distribution. J. Roy. Statist. Soc. Ser. B 65, 367–389] univariate skew-t_ν form. We then use the relationship between the generalized skew-normal and skew-t_ν distributions to discuss some properties of generalized skew-t_ν as well as distributions of order statistics from bivariate and trivariate t_ν distributions. We show that these distributions of order statistics are indeed mixtures of generalized skew-t_ν distributions, and then use this property to derive explicit expressions for means and variances of these order statistics.     

Commuting Matrices in the Queue Length and Sojourn Time Analysis of MAP/MAP/1 Queues

Queues with Markovian arrival and service processes, i.e., MAP/MAP/1 queues, have been useful in the analysis of computer and communication systems and different representations for their stationary sojourn time and queue length distribution have been derived. More specifically, the class of MAP/MAP/1 queues lies at the intersection of the class of QBD queues and the class of semi-Markovian queues. While QBD queues have a matrix exponential representation for their queue length and sojourn time distribution of order N and N², respectively, where N is the size of the background continuous time Markov chain, the reverse is true for a semi-Markovian queue. As the class of MAP/MAP/1 queues lies at the intersection, both the queue length and sojourn time distribution of a MAP/MAP/1 queue has an order N matrix exponential representation. The aim of this article is to understand why the order N² distributions of the sojourn time of a QBD queue and the queue length of a semi-Markovian queue can be reduced to an order N distribution in the specific case of a MAP/MAP/1 queue. We show that the key observation exists in establishing the commutativity of some fundamental matrices involved in the analysis of the MAP/MAP/1 queue.     

Models for Extremal Dependence Derived from Skew‐symmetric Families

Skew‐symmetric families of distributions such as the skew‐normal and skew‐t represent supersets of the normal and t distributions, and they exhibit richer classes of extremal behaviour. By defining a non‐stationary skew‐normal process, which allows the easy handling of positive definite, non‐stationary covariance functions, we derive a new family of max‐stable processes – the extremal skew‐t process. This process is a superset of non‐stationary processes that include the stationary extremal‐t processes. We provide the spectral representation and the resulting angular densities of the extremal skew‐t process and illustrate its practical implementation.     

Estimation of Parameters of a Clipped MA(1) Process

Let {X _t , t ∈ ?} be a sequence of iid random variables with an absolutely continuous distribution. Let a > 0 and c ∈ ? be some constants. We consider a sequence of 0-1 valued variables {ξ _t , t ∈ ?} obtained by clipping an MA(1) process X _t  ? aX _t?1 at the level c, i.e., ξ _t  = I[X _t  ? aX _t?1 < c] for all t ∈ ?. We deal with the estimation problem in this model. Properties of the estimators of the parameters a and c, the success probability p, and the 1-lag autocorrelation r ₁ are investigated. A numerical study is provided as an illustration of the theoretical results.     

COVARIANCES FOR FIXED INTERVAL SMOOTHED KALMAN FILTER PARAMETER ESTIMATES

Given time series data for fixed interval t= 1,2,…, M with non-autocorrelated innovations, the regression formulae for the best linear unbiased parameter estimates at each time t are given by the Kalman filter fixed interval smoothing equations. Formulae for the variance of such parameter estimates are well documented. However, formulae for covariance between these fixed interval best linear parameter estimates have previously been derived only for lag one. In this paper more general formulae for covariance between fixed interval best linear unbiased estimates at times t and t - l are derived for t= 1,2,…, M and l= 0,1,…, t - 1. Under Gaussian assumptions, these formulae are also those for the corresponding conditional covariances between the fixed interval best linear unbiased parameter estimates given the data to time M. They have application, for example, in determination via the expectation-maximisation (EM) algorithm of exact maximum likelihood parameter estimates for ARMA processes expressed in statespace form when multiple observations are available at each time point.     

Estimation in M/Er /1 queueing systems

In this paper, the maximum likelihood estimates of the parameters for the M/E_r /1 queueing model are derived when the queue size at each departure point is observed. A numerical example is generated by simulating a finite Markov chain to illustrate the methodology for estimating the parameters with variable Erlang service time distribution. The problem of hypothesis testing and simultaneous Confidence regions of the parameter is also investigated.0     

A Class of Convolution‐Based Models for Spatio‐Temporal Processes with Non‐Separable Covariance Structure

Abstract. In this article, we propose a new parametric family of models for real‐valued spatio‐temporal stochastic processes S ( x , t ) and show how low‐rank approximations can be used to overcome the computational problems that arise in fitting the proposed class of models to large datasets. Separable covariance models, in which the spatio‐temporal covariance function of S ( x , t ) factorizes into a product of purely spatial and purely temporal functions, are often used as a convenient working assumption but are too inflexible to cover the range of covariance structures encountered in applications. We define positive and negative non‐separability and show that in our proposed family we can capture positive, zero and negative non‐separability by varying the value of a single parameter.     

Selection of Variables in Multivariate Regression Models for Large Dimensions

The Akaike information criterion, AIC, and Mallows’ C _p statistic have been proposed for selecting a smaller number of regressors in the multivariate regression models with fully unknown covariance matrix. All of these criteria are, however, based on the implicit assumption that the sample size is substantially larger than the dimension of the covariance matrix. To obtain a stable estimator of the covariance matrix, it is required that the dimension of the covariance matrix is much smaller than the sample size. When the dimension is close to the sample size, it is necessary to use ridge-type estimators for the covariance matrix. In this article, we use a ridge-type estimators for the covariance matrix and obtain the modified AIC and modified C _p statistic under the asymptotic theory that both the sample size and the dimension go to infinity. It is numerically shown that these modified procedures perform very well in the sense of selecting the true model in large dimensional cases.     

Characterizations of the poisson process using,the variance

Let γ(t) be the residual life at time t of the renewal process {A(t), t > 0}, which has F as the common distribution function of the inter-arrival times. In this article we prove that if Var(γ(t)) is constant, then F will be exponentially or geometrically distributed under the assumption F is continuous or discrete respectively. An application and a related example also are given.     

APPROXIMATION OF EXCEEDANCE PROCESSES IN LARGE POPULATIONS

We consider a population of n individuals. Each of these individuals generates a discrete time branching stochastic process. We study the number of ancestors S(n,t) whose offspring at time t exceeds level θ(t), where θ(t) is some positive valued function. It is proved that S(n,t) may be approximated as t → ∞ and n → ∞ by some stochastic processes with independent increments.

     

An RG-Factorization Approach for a BMAP/M/1 Generalized Processor-Sharing Queue

ABSTRACT

In this paper, we study a BMAP/M/1 generalized processor-sharing queue. We propose an RG-factorization approach, which can be applied to a wider class of Markovian block-structured processor-sharing queues. We obtain the expressions for both the distribution of the stationary queue length and the Laplace transform of the sojourn time distribution. From these two expressions, we develop an algorithm to compute the mean and variance of the sojourn time approximately.     

Response Time Distribution in a D-MAP/PH/1 Queue with General Customer Impatience

ABSTRACT

This paper presents two methods to calculate the response time distribution of impatient customers in a discrete-time queue with Markovian arrivals and phase-type services, in which the customers’ patience is generally distributed (i.e., the D-MAP/PH/1 queue). The first approach uses a GI/M/1 type Markov chain and may be regarded as a generalization of the procedure presented in Van Houdt ^[14] Van Houdt , B. ; Lenin , R. B. ; Blondia , C. Delay distribution of (im)patient customers in a discrete time D-MAP/PH/1 queue with age dependent service times Queueing Systems and Applications 2003 , 45 1 , 59 – 73 . [CROSSREF]  [Google Scholar] for the D-MAP/PH/1 queue, where every customer has the same amount of patience. The key construction in order to obtain the response time distribution is to set up a Markov chain based on the age of the customer being served, together with the state of the D-MAP process immediately after the arrival of this customer. As a by-product, we can also easily obtain the queue length distribution from the steady state of this Markov chain.

We consider three different situations: (i) customers leave the system due to impatience regardless of whether they are being served or not, possibly wasting some service capacity, (ii) a customer is only allowed to enter the server if he is able to complete his service before reaching his critical age and (iii) customers become patient as soon as they are allowed to enter the server. In the second part of the paper, we reduce the GI/M/1 type Markov chain to a Quasi-Birth-Death (QBD) process. As a result, the time needed, in general, to calculate the response time distribution is reduced significantly, while only a relatively small amount of additional memory is needed in comparison with the GI/M/1 approach. We also include some numerical examples in which we apply the procedures being discussed.     

First-exit times for compound poisson processes for some types of positive and negative jumps

We consider the one-sided and the two-sided first-exit problem for a compound Poisson process with linear deterministic decrease between positive and negative jumps. This process (X(t)) _t≥0 occurs as the workload process of a single-server queueing system with random workload removal, which we denote by M/G _u /G _d /1, where G _u (G _d ) stands for the distribution of the upward (downward) jumps; other applications are to cash management, dams, and several related fields. Under various conditions on G _u and G _d (assuming e.g. that one of them is hyperexponential, Erlang or Coxian), we derive the joint distribution of τ _y =inf{t≥0|X(t)?(0,y)}, y>0, and X(τ _y ) as well as that of T=inf{t≥0|X(t)≤0} and X(T). We also determine the distribution of sup{X(t)|0≤t≤T}.     

A new robust Kalman filter for filtering the microstructure noise

We propose a robust Kalman filter (RKF) to estimate the true but hidden return when microstructure noise is present. Following Zhou's definition, we assume the observed return Y_t is the result of adding microstructure noise to the true but hidden return X_t. Microstructure noise is assumed to be independent and identically distributed (i.i.d.); it is also independent of X_t. When X_t is sampled from a geometric Brownian motion process to yield Y_t, the Kalman filter can produce optimal estimates of X_t from Y_t. However, the covariance matrix of microstructure noise and that of X_t must be known for this claim to hold. In practice, neither covariance matrix is known so they must be estimated. Our RKF, in contrast, does not need the covariance matrices as input. Simulation results show that the RKF gives essentially identical estimates to the Kalman filter, which has access to the covariance matrices. As applications, estimated X_t can be used to estimate the volatility of X_t.     

The covariance of the backward and forward recurrence times in a renewal process: the stationary case and asymptotics for the ordinary case

In a Poisson process, it is well-known that the forward and backward recurrence times at a given time point t are independent random variables. In a renewal process, although the joint distribution of these quantities is known (asymptotically), it seems that very few results regarding their covariance function exist. In the present paper, we study this covariance and, in particular, we state both necessary and sufficient conditions for it to be positive, zero or negative in terms of reliability classifications and the coefficient of variation of the underlying inter-renewal and the associated equilibrium distribution. Our results apply either for an ordinary renewal process in the steady state or for a stationary process.