Density Fθ Research Articles

Local Whittle likelihood estimators and tests for non-Gaussian stationary processes

In this paper, we propose a local Whittle likelihood estimator for spectral densities of non-Gaussian processes and a local Whittle likelihood ratio test statistic for the problem of testing whether the spectral density of a non-Gaussian stationary process belongs to a parametric family or not. Introducing a local Whittle likelihood of a spectral density fθ (λ) around λ, we propose a local estimator \({\hat{\theta } = \hat{\theta } (\lambda ) }\) of θ which maximizes the local Whittle likelihood around λ, and use \({f_{\hat{\theta } (\lambda )} (\lambda )}\) as an estimator of the true spectral density. For the testing problem, we use a local Whittle likelihood ratio test statistic based on the local Whittle likelihood estimator. The asymptotics of these statistics are elucidated. It is shown that their asymptotic distributions do not depend on non-Gaussianity of the processes. Because our models include nonlinear stationary time series models, we can apply the results to stationary GARCH processes. Advantage of the proposed estimator is demonstrated by a few simulated numerical examples.

Prediction Problems for Square-Transformed Stationary Processes

This paper discusses the prediction problems for square-transformed process, Yt = Xt2, where Xt is a stationary process with spectral density g(λ). The square-transformation is important in prediction of the volatility of ARCH models. First, we evaluate the mean square prediction error for square-transformed process when the predictor is constructed from the true spectral density g(λ). However, it is often that the true structure g(λ) is not completely specified. Hence, we consider the problem of misspecified prediction when a conjectured spectral density fθ(λ), θ ∈ Θ, is fitted to g(λ). Then, constructing the best linear predictor based on fθ(λ), we can evaluate the prediction error for square-transformed process. Also, we consider a bias adjusted prediction problem for the above two cases. Furthermore, we may suppose that Xt is a non-Gaussian process. Then, we evaluate the mean square prediction errors when the best linear predictor is constructed by the true spectral density g(λ) and the conjectured spectral density fθ (λ), respectively. Since θ is usually unknown we estimate it by a quasi-MLE \(\hat \theta _Q\). The second-order asymptotic approximations of the mean square errors of the predictors based on g(λ) and f\(\hat \theta _Q\)(λ) are given. Finally, we provide some numerical examples, which show some unexpected features.

Rao–Cramer Type Inequalities for Mean Squared Error of Prediction

Let Y be an observable random vector and Z be an unobservable random variable with joint density fθ(y, z), where θ is an unknown parameter vector.Considering the problem of predicting Z based on Y, we derive a Rao–Cramer type lower bound for the mean squared error (MSE) of any given predictor of Z satisfying some regularity conditions. Under unbiasedness, the lower bound does not depend on the specific predictor, and the condition for attaining that bound can be used to easily identify the best unbiased predictors in some applications. Using the lower bound we develop a method for proving admissibility of predictors under squared error loss. Bhattacharyya type bounds and extensions of the results for vector Z are also discussed, and some examples illustrating the utility of the results are presented.

Misspecified prediction for time series

AbstractLet {Xt} be a stationary process with spectral density g(λ).It is often that the true structure g(λ) is not completely specified. This paper discusses the problem of misspecified prediction when a conjectured spectral density fθ(λ), θ∈Θ, is fitted to g(λ). Then, constructing the best linear predictor based on fθ(λ), we can evaluate the prediction error M(θ). Since θ is unknown we estimate it by a quasi‐MLE $\hat{\theta}_{Q}$. The second‐order asymptotic approximation of $M(\hat{\theta}_{Q})$ is given. This result is extended to the case when Xt contains some trend, i.e. a time series regression model. These results are very general. Furthermore we evaluate the second‐order asymptotic approximation of $M(\hat{\theta}_{Q})$ for a time series regression model having a long‐memory residual process with the true spectral density g(λ). Since the general formulae of the approximated prediction error are complicated, we provide some numerical examples. Then we illuminate unexpected effects from the misspecification of spectra. Copyright © 2001 John Wiley & Sons, Ltd.

On large deviation asymptotics of some tests in time series

Abstract Let {Xt} be a Gaussian stationary process with spectral density fθ(λ). The problem considered is that of testing a simple hypothesis H0:θ=θ0 against the alternative A:θ≠θ0. For this we investigate the Bahadur efficiency of the likelihood ratio, Rao, modified Wald and Wald tests. The Bahadur efficiency is based on the large deviation theory. Then it is shown that the asymptotics of the above tests are identical up to second-order in a certain sense. We show that this result makes a sharp contrast with the ordinary higher-order asymptotic theory for tests.

HIGHER‐ORDER ASYMPTOTIC PROPERTIES OF A WEIGHTED ESTIMATOR FOR GAUSSIAN ARMA PROCESSES

Abstract. Let {Xt} be a Gaussian ARMA process with spectral density fθ(Λ), where θ is an unknown parameter. To estimate θ, we propose an estimator θCw of the Bayes type. Since our standpoint in this paper is different from Bayes's original approach, we call it a weighted estimator. We then investigate various higher‐order asymptotic properties of θCw. It is shown that θCw is second‐order asymptotically efficient in the class of second‐order median unbiased estimators. Furthermore, if we confine our discussions to an appropriate class D of estimators, we can show that θCw is third‐order asymptotically efficient in D. We also investigate the Edgeworth expansion of a transformation of θCw. We can then give the transformation of θCw which makes the second‐order part of the Edgeworth expansion vanish. Finally we consider the problem of testing a simple hypothesis H:θ=θo against the alternative A:θ#θo. For this problem we propose a class of tests δA which are based on the weighted estimator. We derive the X2 type asymptotic expansion of the distribution of S (ζδA) under the sequence of alternatives An:θ=θo+εn1/2, ε > 0. We can then compare the local powers of various tests on the basis of their asymptotic expansions.

Minimax Estimation of the Mixing Proportion of Two Known Distributions

Abstract Let X 1 …, X n, be a sample from P ϑ = ϑP 1 + (1 − ϑ)P 0, with P 0 and P 1 known. Several estimators for ϑ have been proposed in the literature. The estimator proposed by Boes (1966) is minimax unbiased. In this article the unbiasedness is dropped and minimax estimators are derived with respect to the squared error loss function within the class of all moment-type estimators . For larger values of n, the minimax estimator coincides with a shrunken version of the Boes estimator. To give an example, let P ϑ have density fϑ(x) = ϑ(1 + 1.2x) + (1 − ϑ)(1 − .8x)(−.5 ≦ x ≦ .5). The Boes estimator is , with maximal risk equal to 3/n. For n ≧ 132 the minimax estimator is given by , with . Its maximal risk equals . For smaller values of n the minimax estimator is a bit harder to compute. Table 1 shows that for n = 20 the minimax estimator reduces the maximal risk from .150 to .088. In the last section minimax estimators are constructed with respect to the more general loss function L(ϑ, a) = w(ϑ)(ϑ − a)2. As an application it is shown how this estimator can be used in empirical Bayes classification.

Sequential Estimation in the Uniform Density

Abstract The main problem to be solved here may be described as follows: Let X1, X2, ··· be independent random variables, each with density fθ(x) = 1/θ over (0, θ) and zero elsewhere. It is desired to estimate the unknown parameter θ by an interval of length at most d units and with confidence at least 1 −α, for some specified d>0 and α in (0, 1). An exact solution and an asymptotic theory for a sequential procedure are given in Sections 1 and 2, respectively. The procedure proposed in this article satisfies an admissibility criterion which may be stated in terms of the maximum possible number of observations or, alternatively, the expected number of observations.