Integrated marked Poisson processes with application to image correlation spectroscopy

Image correlation spectroscopy (ICS) is an experimental technique used in the study of cell tissue, specifically to obtain information about protein aggregates. This paper discusses a filtered Poisson process model of ICS, and uses this to obtain the sampling distribution of an estimator of interest, using spectral methods. Integrals of third- and fourth-order spectra are involved, and methods proposed in the literature are studied numerically. They are found to be infeasible methods, requiring months of computing time. An alternative method is proposed and is illustrated on some actual ICS experiments. A novel feature in this study is the use of a SIMD-class parallel computer, a MasPar MP-2 machine, to obtain some statistical properties of the estimation method.     

On Fan's adaptive Neyman tests for comparing two spectral densities

In this paper, we consider tests for assessing whether two stationary and independent time series have the same spectral densities (or same autocovariance functions). Both frequency domain and time domain test statistics for this purpose are reviewed. The adaptive Neyman tests are then introduced and their performances are investigated. Our tests are adaptive, that is, they are constructed completely by the data and do not involve any unknown smoothing parameters. Simulation studies show that our proposed tests are at least comparable to the current tests in most cases. Furthermore, our tests are much more powerful in some cases, such as against the long orders of autoregressive moving average (ARMA) models such as seasonal ARMA series.     

Exact simulation of Gaussian Time Series from Nonparametric Spectral Estimates with Application to Bootstrapping

The circulant embedding method for generating statistically exact simulations of time series from certain Gaussian distributed stationary processes is attractive because of its advantage in computational speed over a competitive method based upon the modified Cholesky decomposition. We demonstrate that the circulant embedding method can be used to generate simulations from stationary processes whose spectral density functions are dictated by a number of popular nonparametric estimators, including all direct spectral estimators (a special case being the periodogram), certain lag window spectral estimators, all forms of Welch's overlapped segment averaging spectral estimator and all basic multitaper spectral estimators. One application for this technique is to generate time series for bootstrapping various statistics. When used with bootstrapping, our proposed technique avoids some – but not all – of the pitfalls of previously proposed frequency domain methods for simulating time series.     

Testing Semiparametric Hypotheses in Locally Stationary Processes

In this paper, we investigate the problem of testing semiparametric hypotheses in locally stationary processes. The proposed method is based on an empirical version of the L₂‐distance between the true time varying spectral density and its best approximation under the null hypothesis. As this approach only requires estimation of integrals of the time varying spectral density and its square, we do not have to choose a smoothing bandwidth for the local estimation of the spectral density – in contrast to most other procedures discussed in the literature. Asymptotic normality of the test statistic is derived both under the null hypothesis and the alternative. We also propose a bootstrap procedure to obtain critical values in the case of small sample sizes. Additionally, we investigate the finite sample properties of the new method and compare it with the currently available procedures by means of a simulation study. Finally, we illustrate the performance of the new test in two data examples, one regarding log returns of the S&P 500 and the other a well‐known series of weekly egg prices.     

A Note on a Specification Test for Time Series Models Based on Spectral Density Estimation

In a recent paper, Paparoditis [Scand. J. Statist. 27 (2000) 143] proposed a new goodness‐of‐fit test for time series models based on spectral density estimation. The test statistic is based on the distance between a kernel estimator of the ratio of the true and the hypothesized spectral density and the expected value of the estimator under the null and provides a quantification of how well the parametric density fits the sample spectral density. In this paper, we give a detailed asymptotic analysis of the corresponding procedure under fixed alternatives.     

Spectral Density Based Goodness-of-Fit Tests for Time Series Models

A new goodness-of-fit test for time series models is proposed. The test statistic is based on the distance between a kernel estimator of the ratio between the true and the hypothesized spectral density and the expected value of the estimator under the null. It provides a quantification of how well a parametric spectral density model fits the sample spectral density (periodogram). The asymptotic distribution of the statistic proposed is derived and its power properties are discussed. To improve upon the large sample (Gaussian) approximation of the distribution of the test statistic under the null, a bootstrap procedure is presented and justified theoretically. The finite sample performance of the test is investigated through a simulation experiment and applications to real data sets are given.     

Semiparametric estimation of spatial long-range dependence

Estimation of the long-range dependence parameter in spatial processes using a semiparametric approach is studied. An extended formulation of the averaged periodogram method proposed in Robinson [1994. Semiparametric analysis of long memory time series. Ann. Statist. 22, 515–539] is derived, considering a certain homogeneous and isotropic behaviour of the spectral distribution in the low frequencies. The weak consistency of the estimator proposed is proved.     

A generalization of a Gaussian semiparametric estimator on multivariate long-range dependent processes

In this paper, we propose and study a general class of Gaussian semiparametric estimators (GSE) of the fractional differencing parameter in the context of long-range dependent multivariate time series. We establish large sample properties of the estimator without assuming Gaussianity. The class of models considered here satisfies simple conditions on the spectral density function, restricted to a small neighbourhood of the zero frequency and includes important class of vector autoregressive fractionally integrated moving average processes. We also present a simulation study to assess the finite sample properties of the proposed estimator based on a smoothed version of the GSE which supports its competitiveness.     

Frequency Domain Tests of Semiparametric Hypotheses for Locally Stationary Processes

Abstract.  Many time series in applied sciences obey a time-varying spectral structure. In this article, we focus on locally stationary processes and develop tests of the hypothesis that the time-varying spectral density has a semiparametric structure, including the interesting case of a time-varying autoregressive moving-average (tvARMA) model. The test introduced is based on a L ₂ -distance measure of a kernel smoothed version of the local periodogram rescaled by the time-varying spectral density of the estimated semiparametric model. The asymptotic distribution of the test statistic under the null hypothesis is derived. As an interesting special case, we focus on the problem of testing for the presence of a tvAR model. A semiparametric bootstrap procedure to approximate more accurately the distribution of the test statistic under the null hypothesis is proposed. Some simulations illustrate the behaviour of our testing methodology in finite sample situations.     

Spectral analysis with replicated time series

A doubly stochastic process {x(b,t);b?B,t?Z} is considered, with (B,β,P_β) being a probability space so that for each b, {X(b,t);t ? Z} is a stationary process with an absolutely continuous spectral distribution. The population spectrum is defined as f(ω) = E_B[Q(b,ω)] with Q(b,ω) being the spectral density function of X(b,t). The aim of this paper is to estimate f(ω) by means of a random sample b₁,…,b_r from (B,β,P_β). For each b₁? B, the processes X(b₁,t) are observed at the same times t=1,…,N. Thus, r time series (x(b₁,t)} are available in order to estimate f(ω). A model for each individual periodogram, which involves f(ω), is formulated. It has been proven that a certain family of linear stationary processes follows the above model In this context, a kernel estimator is proposed in order to estimate f(ω). The bias, variance and asymptotic distribution of this estimator are investigated under certain conditions.     

Semiparametric estimation for seasonal long-memory time series using generalized exponential models

We consider a generalized exponential (GEXP) model in the frequency domain for modeling seasonal long-memory time series. This model generalizes the fractional exponential (FEXP) model [Beran, J., 1993. Fitting long-memory models by generalized linear regression. Biometrika 80, 817–822] to allow the singularity in the spectral density occurring at an arbitrary frequency for modeling persistent seasonality and business cycles. Moreover, the short-memory structure of this model is characterized by the Bloomfield [1973. An exponential model for the spectrum of a scalar time series. Biometrika 60, 217–226] model, which has a fairly flexible semiparametric form. The proposed model includes fractionally integrated processes, Bloomfield models, FEXP models as well as GARMA models [Gray, H.L., Zhang, N.-F., Woodward, W.A., 1989. On generalized fractional processes. J. Time Ser. Anal. 10, 233–257] as special cases. We develop a simple regression method for estimating the seasonal long-memory parameter. The asymptotic bias and variance of the corresponding long-memory estimator are derived. Our methodology is applied to a sunspot data set and an Internet traffic data set for illustration.     

Posterior consistency for the spectral density of non-Gaussian stationary time series

Various nonparametric approaches for Bayesian spectral density estimation of stationary time series have been suggested in the literature, mostly based on the Whittle likelihood approximation. A generalization of this approximation involving a nonparametric correction of a parametric likelihood has been proposed in the literature with a proof of posterior consistency for spectral density estimation in combination with the Bernstein–Dirichlet process prior for Gaussian time series. In this article, we will extend the posterior consistency result to non-Gaussian time series by employing a general consistency theorem for dependent data and misspecified models. As a special case, posterior consistency for the spectral density under the Whittle likelihood is also extended to non-Gaussian time series. Small sample properties of this approach are illustrated with several examples of non-Gaussian time series.     

Asymptotic confidence interval of power spectrum of a continuous time process through progressively faster sampling

If the power spectral density of a continuous time stationary stochastic process is not limited to a finite bandwidth, data sampled from that process at any uniform sampling rate leads to biased and inconsistent spectrum estimators, which are unsuitable for constructing confidence intervals. In this paper, we use the smoothed periodogram estimator to construct asymptotic confidence intervals shrinking to the true spectra, by allowing the sampling rate to go to infinity suitably fast as the sample size goes to infinity. The proposed method requires minimal computation, as it does not involve bootstrap or other resampling. The method is illustrated through a Monte-Carlo simulation study, and its performance is compared with that of the corresponding method based on uniform sampling at a fixed rate.     

Estimation of the frequency in cyclical long-memory series

Time series with cyclical long memory are characterized by a spectral pole at some frequency ω between 0 and π such that the series has a persistent cycle of period 2π/ω, implying a quasi-periodic behaviour that slightly evolves with time. Accurate estimation of ω is needed for a precise determination of the characteristic of the series (e.g. for business cycle determination or signal estimation). We propose a simple iterative algorithm of estimation of ω based on the maximizer of the periodogram evaluated at an increasingly finer grid of frequencies and compare its performance with more usual methods of estimation restricted to Fourier frequencies. We also apply this technique to the estimation of the frequency of the sunspot index and the business cycle of the differenced unemployment level of the USA.     

Automatic Local Smoothing for Spectral Density Estimation

This article uses local polynomial techniques to fit Whittle's likelihood for spectral density estimation. Asymptotic sampling properties of the proposed estimators are derived, and adaptation of the proposed estimator to the boundary effect is demonstrated. We show that the Whittle likelihood-based estimator has advantages over the least-squares based log-periodogram. The bandwidth for the Whittle likelihood-based method is chosen by a simple adjustment of a bandwidth selector proposed in Fan & Gijbels (1995). The effectiveness of the proposed procedure is demonstrated by a few simulated and real numerical examples. Our simulation results support the asymptotic theory that the likelihood based spectral density and log-spectral density estimators are the most appealing among their peers     

A Spectral Estimator of Arma Parameters from Thresholded Data

We consider computationally-fast methods for estimating parameters in ARMA processes from binary time series data, obtained by thresholding the latent ARMA process. All methods involve matching estimated and expected autocorrelations of the binary series. In particular, we focus on the spectral representation of the likelihood of an ARMA process and derive a restricted form of this likelihood, which uses correlations at only the first few lags. We contrast these methods with an efficient but computationally-intensive Markov chain Monte Carlo (MCMC) method. In a simulation study we show that, for a range of ARMA processes, the spectral method is more efficient than variants of least squares and much faster than MCMC. We illustrate by fitting an ARMA(2,1) model to a binary time series of cow feeding data.     

A joint modelling approach for clustered recurrent events and death events

In dental implant research studies, events such as implant complications including pain or infection may be observed recurrently before failure events, i.e. the death of implants. It is natural to assume that recurrent events and failure events are correlated to each other, since they happen on the same implant (subject) and complication times have strong effects on the implant survival time. On the other hand, each patient may have more than one implant. Therefore these recurrent events or failure events are clustered since implant complication times or failure times within the same patient (cluster) are likely to be correlated. The overall implant survival times and recurrent complication times are both interesting to us. In this paper, a joint modelling approach is proposed for modelling complication events and dental implant survival times simultaneously. The proposed method uses a frailty process to model the correlation within cluster and the correlation within subjects. We use Bayesian methods to obtain estimates of the parameters. Performance of the joint models are shown via simulation studies and data analysis.     

The periodogram regression:correction and comments

Ratios of periodogram and spectral density at different harmonic frequencies are independent if the frequency zero is approached slowly enough. This is an asymptotically relevant condition for the periodogram regression to work with fractionally integrated series. In finite samples, however, this theoretical condition should be ignored as we illustrate experimentally.     

Estimation of covariance matrix via the sparse Cholesky factor with lasso

In this paper, we discuss a parsimonious approach to estimation of high-dimensional covariance matrices via the modified Cholesky decomposition with lasso. Two different methods are proposed. They are the equi-angular and equi-sparse methods. We use simulation to compare the performance of the proposed methods with others available in the literature, including the sample covariance matrix, the banding method, and the L₁-penalized normal loglikelihood method. We then apply the proposed methods to a portfolio selection problem using 80 series of daily stock returns. To facilitate the use of lasso in high-dimensional time series analysis, we develop the dynamic weighted lasso (DWL) algorithm that extends the LARS-lasso algorithm. In particular, the proposed algorithm can efficiently update the lasso solution as new data become available. It can also add or remove explanatory variables. The entire solution path of the L₁-penalized normal loglikelihood method is also constructed.     

Frequency domain bootstrap for ratio statistics under long-range dependence

A frequency domain bootstrap (FDB) is a common technique to apply Efron’s independent and identically distributed resampling technique (Efron, 1979) to periodogram ordinates – especially normalized periodogram ordinates – by using spectral density estimates. The FDB method is applicable to several classes of statistics, such as estimators of the normalized spectral mean, the autocorrelation (but not autocovariance), the normalized spectral density function, and Whittle parameters. While this FDB method has been extensively studied with respect to short-range dependent time processes, there is a dearth of research on its use with long-range dependent time processes. Therefore, we propose an FDB methodology for ratio statistics under long-range dependence, using semi- and nonparametric spectral density estimates as a normalizing factor. It is shown that the FDB approximation allows for valid distribution estimation for a broad class of stationary, long-range (or short-range) dependent linear processes, without any stringent assumptions on the distribution of the underlying process. The results of a large simulation study show that the FDB approximation using a semi- or nonparametric spectral density estimator is often robust for various values of a long-memory parameter reflecting magnitude of dependence. We apply the proposed procedure to two data examples.