autoregressive gaussian random vectors of first order

Autoregressive Gaussian Random Vectors of First Order

Gaussian Random Vectors

1. The multivariate normal distribution Let X := (X1 � � � � �X�) be a random vector. We say that X is a Gaussian random vector if we can write X = μ + AZ� where μ ∈ R, A is an � × � matrix and Z := (Z1 � � � � �Z�) is a �-vector of i.i.d. standard normal random variables. Proposition 1. Let X be a Gaussian random vector, as above. Then, EX = μ� Var(X) := Σ = AA� and MX(�) = e � μ+ 1 2 �A���2 =...

Circularly-Symmetric Gaussian random vectors

A number of basic properties about circularly-symmetric Gaussian random vectors are stated and proved here. These properties are each probably well known to most researchers who work with Gaussian noise, but I have not found them stated together with simple proofs in the literature. They are usually viewed as too advanced or too detailed for elementary texts but are used (correctly or incorrect...

A Modified Weighted Symmetric Estimator for a Gaussian First-Order Autoregressive Model with Additive Outliers

Guttman and Tiao [1], and Chang [2] showed that the effect of outliers may cause serious bias in estimating autocorrelations, partial correlations, and autoregressive moving average parameters (cited in Chang et al. [3]). This paper presents a modified weighted symmetric estimator for a Gaussian first-order autoregressive AR(1) model with additive outliers. We apply the recursive median adjustm...

Numerical solution of second-order stochastic differential equations with Gaussian random parameters

In this paper, we present the numerical solution of ordinary differential equations (or SDEs), from each order especially second-order with time-varying and Gaussian random coefficients. We indicate a complete analysis for second-order equations in special case of scalar linear second-order equations (damped harmonic oscillators with additive or multiplicative noises). Making stochastic differe...

Conditional Maximum Likelihood Estimation of the First-Order Spatial Integer-Valued Autoregressive (SINAR(1,1)) Model

‎Recently a first-order Spatial Integer-valued Autoregressive‎ ‎SINAR(1,1) model was introduced to model spatial data that comes‎ ‎in counts citep{ghodsi2012}‎. ‎Some properties of this model‎ ‎have been established and the Yule-Walker estimator has been‎ ‎proposed for this model‎. ‎In this paper‎, ‎we introduce the...