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.     

Strict stationarity of ar(p) processes generated by nonlinear random functions with additive perturbations

Let {X_n} be a generalized autoregressive process of order ρ defined by X_n=Φ_n(X_n-ρ,…,X_n-1)-η_m, where {φ_n} is a sequence of i.i.d. random maps taking values on H, and {η_n} is a sequence of i.i.d. random variables. Let H be a collection of Borel measurable functions on R_P to R. By considering the associated Markov process, we obtain sufficient conditions for stationarity, (geometric) ergodicity of {X_n}.     

Partial and inverse autocorrelations in portmanteau-type tests for time series

We present a decomposition of the correlation coefficient between x_t and x_t?k into three terms that include the partial and inverse autocorrelations. The first term accounts for the portion of the autocorrelation that is explained by the inner variables {x_t?1 , x_t?2 , …, x _{t? k+1}}, the second one measures the portion explained by the outer variables {x _t+1, x _t+2, } ∪ {x _t?k?1, x _t?k?2,…} and the third term measures the correlation between x _t and x_t?k given all other variables. These terms, squared and summed, can form the basis of three portmanteau-type tests that are able to detect both deviation from white noise and lack of fit of an entertained model. Quantiles of their asymptotic sample distributions are complicated to derive at an adequate level of accuracy, so they are approximated using the Monte Carlo method. A simulation experiment is carried out to investigate significance levels and power of each test, and compare them to the portmanteau test.     

More bounds for moments of some rank statistics

Let X₁,…,X₇ be i.i.d. random variables with a common continuous distribution F, Two parameters, μ(F) = P(X₁ < X₅ and X₁+X₄ < X₂+X₃) and λ(F) = P(X₁+X₄ < X₂+X₃ and X₁+X₇ < X₅+X₆), which appear in the moments of some rank statistics have been studied by several authors. It is shown that the existing lower bound, 3/10 ≤ μ(F) can be improved to 3/10 < μ(F) and that no further improvement is possible. It is also shown that the existing upper bounds μ(F) ≤ (2^1/2+6)/24 ≈ 0.30893 and λ(F) ≤ 7/24 ≈ 0.29167 can be improved to [14+(2/3)^1/2]/48 ≈ 0.30868 and {7 ? [1 ? (2/3)^1/2]²/4}/24 ≈ 0.29132.     

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.     

On the quantile process based on the autoregressive residuals

Let {X_t} be the stationary AR(p) process satisfying the difference equation X_t=β₁X_t−1 + … + β_pX_t−p+ε_t, where {ε_t} is a sequence of iid random variables with mean zero and finite variance. Motivated by a goodness of fit test on the true errors {ε_t}, we are led to study the asymptotic behavior of the quantile process based on residuals (the residual quantile process). Particularly, we concentrate on the deviations between the residual quantile process and the empirical process based on the true errors. In this asymptotic study, it is shown that the deviations converge to zero in probability uniformly over certain intervals with specific order as sample size increases. Here, these intervals are allowed to vary with the sample size n and converge to the unit interval as n goes to infinity. Then, based on our result and the strong approximation result of Csörgö and Révész (1978), we propose a goodness of fit test statistic of which limiting distribution is the same as of a functional form of a standard Brownian bridge.     

Bayesian estimation of an AR(1) process with exponential white noise

The object of this paper is a Bayesian analysis of the autoregressive model X _t ?=?ρX _t?1?+?Y _t where 0?Y _t are independent random variables with an exponential distribution of parameter θ. Our study generalizes some results obtained by Turkmann (1990 Amaral Turkmann, M. A. (1990). Bayesian analysis of an autoregressive process with exponential white noise. Statistics, 4: 601–608.  [Google Scholar]). Our analysis is based on a more general non-informative prior which allows us to improve the estimators of ρ and θ.     

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(·).     

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.     

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.     

Precise large deviations for the difference of two sums of WUOD and non identically distributed random variables with dominatedly varying tails

In this article, we study large deviations for non random difference ∑^n₁(t)_{j = 1}X_1j ? ∑^n₂(t)_{j = 1}X_2j and random difference ∑^N₁(t)_{j = 1}X_1j ? ∑^N₂(t)_{j = 1}X_2j, where {X_1j, j ? 1} is a sequence of widely upper orthant dependent (WUOD) random variables with non identical distributions {F_1j(x), j ? 1}, {X_2j, j ? 1} is a sequence of independent identically distributed random variables, n₁(t) and n₂(t) are two positive integer-valued functions, and {N_i(t), t ? 0}²_{i = 1} with EN_i(t) = λ_i(t) are two counting processes independent of {X_ij, j ? 1}²_{i = 1}. Under several assumptions, some results of precise large deviations for non random difference and random difference are derived, and some corresponding results are extended.     

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.     

ON TRANSITION PROBABILITIES OF SKIP-FREE MARKOV CHAINS

Consider an ergodic Markov chain X(t) in continuous time with an infinitesimal matrix Q = (q_ij) defined on a finite state space {0, 1,…, N}. In this note, we prove that if X(t) is skip-free positive (negative, respectively), i.e., q_ij, = 0 for j > i+ 1 (i > j+ 1), then the transition probability p_ij(t) = Pr[X(t)=j | X(0) =i] can be represented as a linear combination of p_0N(t) (p^(m)_(N0)(t)), 0 ≤ m ≤N, where f^(m)(t) denotes the mth derivative of a function f(t) with f⁽⁰⁾(t) =f(t). If X(t) is a birth-death process, then p_ij(t) is represented as a linear combination of p_0N^(m)(t), 0 ≤m≤N - |i-j|.     

A New Test for New Better Than Used in Expectation Lifetimes

The mean residual life of a non negative random variable X with a finite mean is defined by M(t) = E[X ? t|X > t] for t ? 0. A popular nonparametric model of aging is new better than used in expectation (NBUE), when M(t) ? M(0) for all t ? 0. The exponential distribution lies at the boundary. There is a large literature on testing exponentiality against NBUE alternatives. However, comparisons of tests have been made only for alternatives much stronger than NBUE. We show that a new Kolmogorov-Smirnov type test is much more powerful than its competitors in most cases.     

A characterization of the inverse autocorrelation function

The inverse autocorrelation function of a weakly stationary stochastic process X_t at lag h, γⁱ _h, is shown to equal the negative of the partial correlation between random variables X_t and X_t+h after elimination of the influence of random variables X_k, k≠t5,t+h.     

Bayes Estimation of Shift Point in Geometric Sequence

A sequence of independent lifetimes X ₁, X ₂,…, X _m , X _m+1,… X _n were observed from geometric population with parameter q ₁ but later it was found that there was a change in the system at some point of time m and it is reflected in the sequence after X _m by change in parameter q ₂. The Bayes estimates of m, q ₁, q ₂, reliability R ₁ (t) and R ₂ (t) at time t are derived for symmetric and asymmetric loss functions under informative and non informative priors. A simulation study is carried out.     

Prediction of Variables Via Their Order Statistics in Bivariate Elliptical Distributions with Application in the Financial Markets

Assuming that (X₁, X₂) has a bivariate elliptical distribution, we obtain an exact expression for the joint probability density function (pdf) as well as the corresponding conditional pdfs of X₁ and X₍₂₎ ? max?{X₁, X₂}. The problem is motivated by an application in financial markets. Exchangeable random variables are discussed in more detail. Two special cases of the elliptical distributions that is the normal and the student’s t models are investigated. For illustrative purposes, a real data set on the total personal income in California and New York is analyzed using the results obtained. Finally, some concluding remarks and further works are discussed.     

Bayesian Estimation of Change Point in Inverse Weibull Sequence

A sequence of independent lifetimes X ₁,…, X _m , X _m+1,…, X _n were observed from inverse Weibull distribution with mean stress θ₁ and reliability R ₁(t ₀) at time t ₀ but later it was found that there was a change in the system at some point of time m and it is reflected in the sequence after X _m by change in mean stress θ₁ and in reliability R ₂(t ₀) at time t ₀. The Bayes estimators of m, R ₁(t ₀) and R ₂(t ₀) are derived when a poor and a more detailed prior information is introduced into the inferential procedure. The effects of correct and wrong prior information on the Bayes estimators are studied.     

FIRST-EXIT TIMES FOR POISSON SHOT NOISE

For a shot-noise process X(t) with Poisson arrival times and exponentially diminishing shocks of i.i.d. sizes, we consider the first time T _b at which a given level b > 0 is exceeded. An integral equation for the joint density of T _b and X(T _b ) is derived and, for the case of exponential jumps, solved explicitly in terms of Laplace transforms (LTs). In the general case we determine the ordinary LT of the function ? P(T _b > t) in terms of certain LTs derived from the distribution function H(x; t) = P(X(t) ≤ x), considered as a function of both variables x and t. Moreover, for G(t, u) = P(T _b > t, X(t) < u), that is the joint distribution function of sup_{0 ≤ s ≤ t} X(s) and X(t), an integro-differential equation is presented, whose unique solution is G(t, u).     

Precise large deviations for the difference of two sums of END random variables with heavy tails

Assume that there are two types of insurance contracts in an insurance company, and the ith related claims are denoted by {X_ij, j ? 1}, i = 1, 2. In this article, the asymptotic behaviors of precise large deviations for non random difference ∑^n₁(t)_{j = 1}X_1j ? ∑^n₂(t)_{j = 1}X_2j and random difference ∑^N₁(t)_{j = 1}X_1j ? ∑^N₂(t)_{j = 1}X_2j are investigated, and under several assumptions, some corresponding asymptotic formulas are obtained.