Limit Theory for High Frequency Sampled MCARMA Models

Empirical spectral processes for stationary state space models

The problem of estimating parameters of a multivariate normal p-dimensional random vector is considered for a banded covariance structure reflecting m-dependence. A simple non-iterative estimation procedure is suggested which gives an explicit, unbiased and consistent estimator of the mean and an explicit and consistent estimator of the covariance matrix for arbitrary p and m.

In this paper we propose a new procedure to determine the length of the subseries in the MBB bootstrap which takes into account the structure of the model and the strength of dependence in the observed series. It can be easily implemented and easily extended to much more complex structures: multivariate ARMA processes; non linear models; STARMA processes.KeywordsBootstrapTime seriesα-mixing processesSubseries

Chapter 7 - Estimation and Data Analysis

Asymptotic normal distributions of eigenvalue-normed eigenvectors (Pearson-Hotelling principal-component vectors) of sample variance and correlation matrices are derived. Each distribution follows from the asymptotic distribution of the whole matrix of eigenvectors, also obtained in the paper. The results are presented for the general case, where the existence of fourth-order finite population moments is assumed, and also specified for normal and elliptical populations. Population variance and correlation matrices are supposed to be nonsingular and without multiple eigenvalues. A comparison with unit-length eigenvectors (Anderson principal-component vectors) is being made. Special attention is being paid to asymptotic variances of eigenvectors of both sample variance and correlation matrices in the two approaches. It is shown that the asymptotic variance matrices of principal-component vectors for sample variance and correlation matrices are singular in the Anderson approach. In the Pearson-Hotelling approach, however, the asymptotic variance of eigenvectors of the sample variance matrix is nonsingular, whereas in the sample correlation matrix case the asymptotic variance matrix of eigenvectors is nonsingular under certain conditions.

In this paper we investigate a continuous-time MA (moving average) process (Xt)t≥0 sampled at an equally spaced time grid {Δ,2Δ, …, nΔ}, where the grid distance Δ > 0 is fixed and n denotes the number of observations, in the frequency domain. We derive for the process (XkΔ)k∈ℕ with finite second moments the asymptotic behavior of the periodogram and of the lag-window spectral density estimator. The periodogram is not a consistent estimator for the spectral density of (XkΔ)k∈ℕ. Different periodogram frequencies are asymptotically independent and exponentially distributed like for ARMA processes in discrete time. This result is basic for frequency bootstraps. In contrast, the lag-window spectral density estimator is a consistent estimator for the spectral density of (XkΔ)k∈ℕ and moreover, it is asymptotically normally distributed.

In this chapter we consider problems of statistical inference for time series based on frequency-domain properties of the series. The fundamental tool used is the periodogram, which is defined in Section 10.1 for any time series {x 1,..., x n }. Section 10.2 deals with statistical tests for the presence of “hidden periodicities” in the data. Several tests are discussed, corresponding to various different models and hypotheses which we may wish to test. Spectral analysis for stationary time series, and in particular the estimation of the spectral density, depends very heavily on the asymptotic distribution as n → ∞ of the periodogram ordinates of the series {X 1,..., X n }. The essential results are contained in Theorem 10.3.2. Under rather general conditions, the periodogram ordinates I n (λ i ) at any set of frequencies λ 1,..., λ m , 0 < λ 1 < ⋯ < λ m < π, are asymptotically independent exponential random variables with means 2πf(λ i ), were f is the spectral density of {X t }. Consequently the periodogram I n is not a consistent estimator of 2πf. Consistent estimators can however be constructed by applying linear smoothing filters to the periodogram. The asymptotic behaviour of the resulting discrete spectral average estimators can be derived from the asymptotic behaviour of the periodogram as shown in Section 10.4. Lag-window estimators of the form \({\left( {2\pi } \right)^{ - 1}}\sum\nolimits_{\left| h \right| \leqslant r} {w\left( {h/r} \right)\hat \gamma \left( h \right)} {e^{ - ih\omega }}\), where w(x), −1 ≤ x ≤ 1, is a suitably chosen weight function, are also discussed in Section 10.4 and compared with discrete spectral average estimators. Approximate confidence intervals for the spectral density are given in Section 10.5. An alternative approach to spectral density estimation, based on fitting an AR MA model to the data and computing the spectral density of the fitted process, is discussed in Section 10.6. An important role in the development of spectral analysis has been played by the fast Fourier transform algorithm, which makes possible the rapid calculation of the periodogram for very large data sets. An introduction to the algorithm and its application to the computation of autocovariances is given in Section 10.7. The chapter concludes with a discussion of the asymptotic behaviour of the maximum likelihood estimators of the coefficients of an ARMA(p, q) process.

Maximum likelihood estimation is used widely in classical statistics. However, except in a few cases, it does not have a closed form. Furthermore, it takes time to derive the maximum likelihood estimator (MLE) owing to the use of iterative methods such as Newton–Raphson. Nonetheless, this estimation method has several advantages, chief among them being the invariance property and asymptotic normality. Based on the first approximation to the solution of the likelihood equation, we obtain an estimator that has the same asymptotic behavior as the MLE for multivariate gamma distribution. The newly proposed estimator, denoted as , is also in closed form as long as the ‐consistent initial estimator is in the closed form. Hence, we develop some closed‐form ‐consistent estimators for multivariate gamma distribution to improve the small‐sample property. is an alternative to MLE and performs better compared to MLE in terms of computation time, especially for large datasets, and stability. For the bivariate gamma distribution, the is over 130 times faster than the MLE, and as the sample size increasing, the is over 200 times faster than the MLE. Owing to the instant calculation of the proposed estimator, it can be used in state–space modeling or real‐time processing models.

New robust confidence intervals for the mean under dependence

xftanghotmail.com SUMMARY In many applications, researchers are interested in estimating the mean of a multivariate normal random vector whose components are subject to order restrictions. Various authors have demonstrated that the likelihood-based methodology may perform poorly under certain conditions for such problems. The problem is much harder when the underlying covariance matrix is nondiagonal. In this paper a simple iterative algorithm is introduced that can be used for estimating the mean of a multivariate normal population when the components are subject to any order restriction. The proposed methodology is illustrated through an application to human reproductive hormone data.

The cross-validation error rates of quadratic discriminant classifiers were substantially reduced by weighting the off-diagonal elements of the covariance matrices by a constant less than one. The improvement was found both when real chromosome data was classified, and also when simulated multivariate normal random vectors were classified. It is shown empirically that the optimum value of the weighting constant depended on the size of the training set, and that its relative benefit was greatest when the training set was small. The optimum weight appeared to be largely independent of the number of features. The relationship of this heuristic to Stein's paradox (1956) was explored, and it is shown that near-optimal values of weight could be predicted directly, thereby avoiding the need for an expensive empirical search.

A proper method of monitoring a stochastic system is to use the control charts of statistical process control in which a drift in characteristics of output may be due to one or several assignable causes. In the establishment of X charts in statistical process control, an assumption is made that there is no correlation within the samples. However, in practice, there are many cases where the correlation does exist within the samples. It would be more appropriate to assume that each sample is a realization of a multivariate normal random vector. Using three di erent loss functions in the concept of quality control charts with economic and economic statistical design leads to better decisions in the industry. Although some research works have considered the economic design of control charts under single assignable cause and correlated data, the economic statistical design of X control chart for multiple assignable causes and correlated data under Weibull shock model with three di erent loss functions have not been presented yet. Based on the optimization of the average cost per unit of time and taking into account the di erent combination values of Weibull distribution parameters, optimal design values of sample size, sampling interval and control limit coecient were derived and calculated. Then the cost models under non-uniform and uniform sampling scheme were compared. The results revealed that the model under multiple assignable causes with correlated samples with non-uniform sampling integrated with three di erent loss functions has a lower cost than the model with uniform sampling.

It is often of interest to understand how the structure of a genetic network differs between two conditions. In this paper, each condition-specific network is modeled using the precision matrix of a multivariate normal random vector, and a method is proposed to directly estimate the difference of the precision matrices. In contrast to other approaches, such as separate or joint estimation of the individual matrices, direct estimation does not require those matrices to be sparse, and thus can allow the individual networks to contain hub nodes. Under the assumption that the true differential network is sparse, the direct estimator is shown to be consistent in support recovery and estimation. It is also shown to outperform existing methods in simulations, and its properties are illustrated on gene expression data from late-stage ovarian cancer patients.

Using Blackwell's definition for comparison of experiments, it is shown that some sets of positively dependent random variables are less informative than similar sets of independent random variables. It is also shown that the information content of symmetric multivariate normal random vectors with a common known variance increases as the common correlation coefficient decreases. Some results which compare members of two‐parameter exponential families are also included.

Estimation for nonhomogeneous Poisson processes from aggregated data