Parameter Estimation for the Tail Distribution of a Random Sequence

The tail Y_t = X_t – u of a random sequence {X_t ∈ , t ∈ } with identically distributed X_t is approximated by the generalized Pareto distribution according to the extreme value theory, wherein Y_t occurs in clusters because of the dependence in the random sequence. Nevertheless, the parameters of the generalized Pareto distribution are estimated by the same methods as in the case of independent and identically distributed Y_t, provided that there is independence between the clusters of Y_t. The estimation variances and confidence intervals can be estimated by the jackknife method. The approaches are theoretically discussed and verified by extensive numerical researches.     

Estimating moving average parameters in the presence of measurement error

Suppose that a moving average time series X_t is not observed, but instead Y_t = X_t + ?_t is observed, where ?_t, is measurement error. Estimation of the parameters of X_t has previously been considered under the assumption that X_t and ?_t are uncorrelated. The case where X_t and ?_t have known cross covariances is considered here, and a method is described for estimating the parameters of X_t. A simulation compares four estimators for a MA(1) series parameter in the presence of measurement error.     

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.     

Estimation of an autoregressive semiparametric model with exogenous variables

In many autoregressive relationships, there are observed external influences. This paper deals with the estimation of the multivariate model X_t+1= φ(X_t,…,X_t−r+1) + ψ(Y_t) + ε_t, where φ(·) is an unknown nonlinear function, ∫ the exogenous variable concerning ψ(·). Two cases are considered: ψ(·) is linear ψ(Y_t) = AY_t, where A is an unknown parameter, and ψ(·) the nonlinear function corresponding to a series expansion. In the latter situation, the method of estimation is ‘seminonparametric’. We first isolate and estimate parametrically the exogenous part, and then estimate nonparametrically the endogenous part ψ(·).     

A Note on the Predictors of Differenced Sequences

In this note we consider the problems of optimal linear prediction (o.l.p.) and the minimum mean squared error prediction (m.m.s.e.p.) of a sequence X_t, which fits to a stationary and invertible ARMA model through the filter (1 - B^s)^d X_t= Y_t. It is shown that these two predictors are not identical in general from the theoretical point of view. Permitting the degree of differencing d to take any real value, a set of conditions for these commonly applied prediction formulas to be identical is given.     

Bias of the lse estimator of the first order autoregressive model under tukey contamination

Results of an exhaustive study of the bias of the least square estimator (LSE) of an first order autoregression coefficient α in a contaminated Gaussian model are presented. The model describes the following situation. The process is defined as X_t = α X_t-1 + Y_t . Until a specified time T, Y_t are iid normal N(0, 1). At the moment T we start our observations and since then the distribution of Y_t, t≥T, is a Tukey mixture T(εσ) = (1 – ε)N(0,1) + εN(0, σ²). Bias of LSE as a function of α and ε, and σ² is considered. A rather unexpected fact is revealed: given α and ε, the bias does not change montonically with σ (“the magnitude of the contaminant”), and similarly, given α and σ, the bias is not growing with ε (“the amount of contaminants”).     

Prediction intervals with a Dirichlet-process prior distribution

Let X₁, …,X_n, and Y₁, … Y_n be consecutive samples from a distribution function F which itself is randomly chosen according to the Ferguson (1973) Dirichlet-process prior distribution on the space of distribution functions. Typically, prediction intervals employ the observations X₁,…, X_n in the first sample in order to predict a specified function of the future sample Y₁, …, Y_n. Here one- and two-sided prediction intervals for at least q of N future observations are developed for the situation in which, in addition to the previous sample, there is prior information available. The information is specified via the parameter α of the Dirichlet process prior distribution.     

Concomitants of order statistics from multivariate elliptical distributions

In this paper, by considering a 2n-dimensional elliptically contoured random vector (X^T,Y^T)^T=(X₁,…,X_n,Y₁,…,Y_n)^T, we derive the exact joint distribution of linear combinations of concomitants of order statistics arising from X. Specifically, we establish a mixture representation for the distribution of the rth concomitant order statistic, and also for the joint distribution of the rth order statistic and its concomitant. We show that these distributions are indeed mixtures of multivariate unified skew-elliptical distributions. The two most important special cases of multivariate normal and multivariate t distributions are then discussed in detail. Finally, an application of the established results in an inferential problem is outlined.     

Some competitors of the mood test for the two-sample scale problem

Let X_l,…,X_n (Y_l,…,Y_m) be a random sample from an absolutely continuous distribution with distribution function F(G).A class of distribution-free tests based on U-statistics is proposed for testing the equality of F and G against the alternative that X's are more dispersed then Y's. Let 2 ? C ? n and 2 ? d ? m be two fixed integers. Let ?_c,d(X_il,…,X_ic ; Y_jl,…,X_jd)=1(-1)when max as well as min of {X_il,…,X_ic ; Y_jl,…,Y_jd } are some X_i's (Y_j's)and zero oterwise. Let S_c,d be the U-statistic corresponding to ?_c,d.In case of equal sample sizes, S22 is equivalent to Mood's Statistic.Large values of S_c,d are significant and these tests are quite efficient     

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.     

Regression-based orderings and measures of stochastic depence

General axiomatic approach to the so-called global dependence of a random variable xon a random vector Y= Y _t,Y _n) is proposed. natural orderings and measures of global dependence are discussed and examplified by some real and function-valued measures of dependence. orderings and measures to be introduced are referred o as regression-based as they depend only on the distributions of EX|Y X.     

Seasonal Time Series and Transfer Function Modeling

A simple modification is suggested for the construction of transfer function models relating an output variable Y_t to an input variable X_t when the model for X_t contains operators that cancel out. In addition, the evaluation of transfer function models is discussed by comparing the forecasts with the actual observations.     

Characterization of the geometric distribution by the form of a predictor

The problem of prodicting max (X₁X₂) given min(X₁X₂) is considered when Y₁and Y₂ are i.i.d random variables making positive integral values. It is provea tnat tne oest predictor is a linear Function or min(X₁,X₂); with unit slope Iff X₁, and X₂ have geometric distributions. As an extension of this result, the geometric distribution is characterized by the constancy of regression of min(X₁?X₂|, c) on inin(X₁,X₂) where c is any positive integer.     

Bivariate extreme value distributions

Let (X_i, Y_i), i = 1, 2,…, n, be n independent observations from a bivariate population and let X_(n) = max X_i and Y_(n) = max Y_i. This article gives a necessary and sufficient condition for the weak convergence of the distribution function of (X_(n), Y_(n)) to a nondegenerate distribution.     

Non-parametric estimation of an acceleration parameter

Let X₁,… X_m be a random sample of m failure times under normal conditions with the underlying distribution F(x) and Y₁,…,Y_n a random sample of n failure times under accelerated condititons with underlying distribution G(x);G(x)=1?[1?F(x)]^θ with θ being the unknown parameter under study.Define:U_ij=1 otherwise.The joint distribution of _ijdoes not involve the distribution F and thus can be used to estimate the acceleration parameter θ.The second approach for estimating θ is to use the ranks of the Y-observations in the combined X- and Y-samples.In this paper we establish that the rank of the Y-observations in the pooled sample form a sufficient statistic for the information contained in the U_ii 's about the parameter θ and that there does not exist an unbiassed estimator for the parameter θ.We also construct several estimators and confidence interavals for the parameter θ.     

On Censored Bivariate Random Variables: Copula,Characterization, and Estimation

Let (X, Y) be a bivariate random vector with joint distribution function F_{X, Y}(x, y) = C(F(x), G(y)), where C is a copula and F and G are marginal distributions of X and Y, respectively. Suppose that (X_i, Y_i), i = 1, 2, …, n is a random sample from (X, Y) but we are able to observe only the data consisting of those pairs (X_i, Y_i) for which X_i ? Y_i. We denote such pairs as (X*_i, Y_i*), i = 1, 2, …, ν, where ν is a random variable. The main problem of interest is to express the distribution function F_{X, Y}(x, y) and marginal distributions F and G with the distribution function of observed random variables X* and Y*. It is shown that if X and Y are exchangeable with marginal distribution function F, then F can be uniquely determined by the distributions of X* and Y*. It is also shown that if X and Y are independent and absolutely continuous, then F and G can be expressed through the distribution functions of X* and Y* and the stress–strength reliability P{X ? Y}. This allows also to estimate P{X ? Y} with the truncated observations (X*_i, Y_i*). The copula of bivariate random vector (X*, Y*) is also derived.     

A Note on Stagewise Regression

Suppose one estimates the coefficient β₂ in E[Y] = β₀ + β₁ X ₁ + β₂ X ₂ by stagewise regression. That is, first the model E[Y] ≌ β₀ + β₁ X ₁ is fit using simple linear regression followed by a simple linear regression of the residuals from this model on X ₂ to yield the estimator β₂. The ratio of the squared t statistic for the estimate b ₂ from multiple regression to the squared t statistic for β₂ is greater than or equal to 1.0 and is shown to be a convenient function of correlation coefficients among Y, X ₁, and X ₂. Examination of stagewise regression can provide useful insights when introducing concepts of multiple regression.     

Bivariate Aging Properties under Archimedean Dependence Structures

Let X  = (X, Y) be a pair of lifetimes whose dependence structure is described by an Archimedean survival copula, and let X _t  = [(X ? t, Y ? t) | X > t, Y > t] denotes the corresponding pair of residual lifetimes after time t ≥ 0. Multivariate aging notions, defined by means of stochastic comparisons between X and X _t , with t ≥ 0, were studied in Pellerey (2008 Pellerey , F. ( 2008 ). On univariate and bivariate aging for dependent lifetimes with Archimedean survival copulas . Kybernetika 44 : 795 – 806 .[Web of Science ®] , [Google Scholar]), who considered pairs of lifetimes having the same marginal distribution. Here, we present the generalizations of his results, considering both stochastic comparisons between X _t and X _t+s for all t, s ≥ 0 and the case of dependent lifetimes having different distributions. Comparisons between two different pairs of residual lifetimes, at any time t ≥ 0, are discussed as well.     

Nonparametric two‐step regression estimation when regressors and error are dependent

This paper considers estimation of the function g in the model Y_t = g(X_t ) + ?_t when E(?_t|Xt) ≠ 0 with nonzero probability. We assume the existence of an instrumental variable Z_t that is independent of ?_t, and of an innovation ηt = X_t — E(X_t|Z_t). We use a nonparametric regression of X_t on Z_t to obtain residuals η_t, which in turn are used to obtain a consistent estimator of g. The estimator was first analyzed by Newey, Powell & Vella (1999) under the assumption that the observations are independent and identically distributed. Here we derive a sample mean‐squared‐error convergence result for independent identically distributed observations as well as a uniform‐convergence result under time‐series dependence.