A regression problem concerning stationary processes

Heble, M. P.

doi:10.1090/s0002-9947-1961-0124144-7

Cited by 5 publications

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“…For further references see Kholevo (1969) and Hannan (1970). The first attempt at a general solution of the continuous time problem is found in Heble (1961). However, Theorem 2 is incorrect, as is pointed out in Kholevo (1969).…”

Section: Introductionmentioning

confidence: 99%

Linear regression in continuous time

Hannan

1975

J. Aust. Math. Soc.

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We consider a regression relation of the from wherein y(t) and x(t) are real (column) vectors of q and p components and e(t) is real and is generated by a stationary generalised vector process of q components with zero mean and covariance function (a q rowed matrix) Γ(t–s) = E{x(s)x(t)′}. (See Hannan (1970; pages 23–26, 91–94) and references therein for definitions of terms used.) We assume e(t) to be independent of x(s) for all s, t. Thus we may regard x(t) as a fixed time function and not stochastic and we shall henceforth do that. We take Γ(t) to be continuous and to correspond to an absolutely continuous spectral function with spectral density which is uniformly bounded and continuous. Then we have We do not exclude the possibility that for theyth diagonal element, fjj, of fwe have

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Section: Introductionmentioning

confidence: 99%

Linear regression in continuous time

Hannan

1975

J. Aust. Math. Soc.

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On the estimation of the regression coefficients of a continuous parameter process with stationary residual

Yan

Liu

2006

Front. Math. China

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The asymptotic expressions of the covariance matrices for both the least square estimates L α T and Markov (best linear) estimatesα T are obtained, based on a sample in a finite interval (0, T ) of the regression coefficients α = (α 1 , · · · , α m0 ) of a parameter-continuous process with a stationary residual. We assume that the regression variables ϕ ν (t), t 0, ν = 1, · · · , m 0 , are continuous in t, and satisfy conditions (3.1) − (3.3). For the residual, we assume that it is a stationary process that possesses a bounded continuous spectral density f (λ). Under these assumptions, it is proven thatwhere the matrices D T , B(0), α(λ) are defined in Section 3. Under the assumptions mentioned above, if, furthermore, there exist some positive integer m and a constant C such that g(λ)(1 + λ 2 ) m C > 0, where g(λ) is the spectral density of the residual, and for every N > 0,, 0 k, j m converge uniformly in h, l ∈ (−N, N ), then the following formula holds.The asymptotic equivalence of the least square estimates and the Markov estimates is also discussed.

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Some Problems in the Identification and Estimation of Continuous time Systems from Discrete time Series

Robinson

1976

Mathematics in Science and Engineering

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A regression problem concerning stationary processes

Abstract: 1. Introduction. Let y(t) =x(t) +m(t) be a continuous parameter stochastic process observed for 0 = i=P, the mean value being m(t)=Ey(t), and x(t) being weakly stationary (see [7, p. 33]) and continuous in the mean, with Ex(t) =0. The means m(t) are assumed to be of the form

Cited by 5 publications

References 8 publications

Linear regression in continuous time

Linear regression in continuous time

On the estimation of the regression coefficients of a continuous parameter process with stationary residual

Some Problems in the Identification and Estimation of Continuous time Systems from Discrete time Series

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