A New Approach for Testing Periodicity

This article describes testing for periodicity in the presence of FD processes. We propose two approaches for testing the periodicity based on Fisher's test. The first one is performed using the periodogram which has been divided into different parts. The second one is based on the discrete wavelet transform. Properties of the tests are illustrated by means of Monte Carlo simulations.     

Some Properties of the Normalized Periodogram of a Fractionally Integrated Separable Spatial ARMA (FISSARMA) Model

In this article, we study the properties of the normalized periodogram of the Fractionally Integrated Separable Spatial ARMA (FISSARMA) models. In particular, we establish the asymptotic mean of the normalised periodogram and the asymptotic second-order moments of the normalised Fourier coefficients. We also establish the asymptotic distribution of the normalised periodogram. Some numerical results are also provided.     

Testing for climate warming in Sweden during 1850-1999, using wavelets analysis

This paper describes an alternative approach for testing for the existence of trend among time series. The test method has been constructed using wavelet analysis which has the ability of decomposing a time series into low frequencies (trend) and high-frequency (noise) components. Under the normality assumption, the test is distributed as F. However, using generated empirical critical values, the properties of the test statistic have been investigated under different conditions and different types of wavelet. The Harr wavelet has shown to exhibit the highest power among the other wavelet types.

The methodology here has been applied to real temperature data in Sweden for the period 1850-1999. The results indicate a significant increasing trend which agrees with the 'global warming' hypothesis during the last 100 years.     

Testing in mixed-effects FANOVA models

We consider the testing problem in the mixed-effects functional analysis of variance models. We develop asymptotically optimal (minimax) testing procedures for testing the significance of functional global trend and the functional fixed effects based on the empirical wavelet coefficients of the data. Wavelet decompositions allow one to characterize various types of assumed smoothness conditions on the response function under the nonparametric alternatives. The distribution of the functional random-effects component is defined in the wavelet domain and captures the sparseness of wavelet representation for a wide variety of functions. The simulation study presented in the paper demonstrates the finite sample properties of the proposed testing procedures. We also applied them to the real data from the physiological experiments.     

Wavelet processes and adaptive estimation of the evolutionary wavelet spectrum

This paper defines and studies a new class of non-stationary random processes constructed from discrete non-decimated wavelets which generalizes the Cramér (Fourier) representation of stationary time series. We define an evolutionary wavelet spectrum (EWS) which quantifies how process power varies locally over time and scale. We show how the EWS may be rigorously estimated by a smoothed wavelet periodogram and how both these quantities may be inverted to provide an estimable time-localized autocovariance. We illustrate our theory with a pedagogical example based on discrete non-decimated Haar wavelets and also a real medical time series example.     

Percentile-t Bootstrap Confidence Intervals for Period Using Periodogram

The magnitude of light intensity of many stars varies over time in a periodic way. Therefore, estimation of period and making inference about this parameter are of great interest in astronomy. The periodogram can be used to estimate period, properly. Bootstrap confidence intervals for period suggested here, are based on using the periodogram and constructed by percentile-t methods. We prove that the equal-tailed percentile-t bootstrap confidence intervals for period have an error of order n ^?1. We also show that the symmetric percentile-t bootstrap confidence intervals reduce the error to order n ^?2, and hence have a better performance. Finally, we assess the theoretical results by conducting a simulation study, compare the results with the coverages of percentile bootstrap confidence intervals for period and then analyze a real data set related to the eclipsing system R Canis Majoris collected by Shiraz Biruni Observatory.     

Non-parametric Curve Estimation by Wavelet Thresholding with Locally Stationary Errors

An important aspect in the modelling of biological phenomena in living organisms, whether the measurements are of blood pressure, enzyme levels, biomechanical movements or heartbeats, etc., is time variation in the data. Thus, the recovery of a 'smooth' regression or trend function from noisy time-varying sampled data becomes a problem of particular interest. Here we use non-linear wavelet thresholding to estimate a regression or a trend function in the presence of additive noise which, in contrast to most existing models, does not need to be stationary. (Here, non-stationarity means that the spectral behaviour of the noise is allowed to change slowly over time). We develop a procedure to adapt existing threshold rules to such situations, e.g. that of a time-varying variance in the errors. Moreover, in the model of curve estimation for functions belonging to a Besov class with locally stationary errors, we derive a near-optimal rate for the -risk between the unknown function and our soft or hard threshold estimator, which holds in the general case of an error distribution with bounded cumulants. In the case of Gaussian errors, a lower bound on the asymptotic minimax rate in the wavelet coefficient domain is also obtained. Also it is argued that a stronger adaptivity result is possible by the use of a particular location and level dependent threshold obtained by minimizing Stein's unbiased estimate of the risk. In this respect, our work generalizes previous results, which cover the situation of correlated, but stationary errors. A natural application of our approach is the estimation of the trend function of non-stationary time series under the model of local stationarity. The method is illustrated on both an interesting simulated example and a biostatistical data-set, measurements of sheep luteinizing hormone, which exhibits a clear non-stationarity in its variance.     

Spectral estimation for locally stationary time series with missing observations

Time series arising in practice often have an inherently irregular sampling structure or missing values, that can arise for example due to a faulty measuring device or complex time-dependent nature. Spectral decomposition of time series is a traditionally useful tool for data variability analysis. However, existing methods for spectral estimation often assume a regularly-sampled time series, or require modifications to cope with irregular or ‘gappy’ data. Additionally, many techniques also assume that the time series are stationary, which in the majority of cases is demonstrably not appropriate. This article addresses the topic of spectral estimation of a non-stationary time series sampled with missing data. The time series is modelled as a locally stationary wavelet process in the sense introduced by Nason et al. (J. R. Stat. Soc. B 62(2):271–292, 2000) and its realization is assumed to feature missing observations. Our work proposes an estimator (the periodogram) for the process wavelet spectrum, which copes with the missing data whilst relaxing the strong assumption of stationarity. At the centre of our construction are second generation wavelets built by means of the lifting scheme (Sweldens, Wavelet Applications in Signal and Image Processing III, Proc. SPIE, vol. 2569, pp. 68–79, 1995), designed to cope with irregular data. We investigate the theoretical properties of our proposed periodogram, and show that it can be smoothed to produce a bias-corrected spectral estimate by adopting a penalized least squares criterion. We demonstrate our method with real data and simulated examples.     

Wavelet goodness-of-fit test for dependent data

We introduce a new goodness-of-fit test which can be applied to hypothesis testing about the marginal distribution of dependent data. We derive a new test for the equivalent hypothesis in the space of wavelet coefficients. Such properties of the wavelet transform as orthogonality, localisation and sparsity make the hypothesis testing in wavelet domain easier than in the domain of distribution functions. We propose to test the null hypothesis separately at each wavelet decomposition level to overcome the problem of bi-dimensionality of wavelet indices and to be able to find the frequency where the empirical distribution function differs from the null in case the null hypothesis is rejected. We suggest a test statistic and state its asymptotic distribution under the null and under some of the alternative hypotheses.     

Wavelet Improvement of the Over-Rejection of Unit Root Test Under GARCH Errors: An Application to Swedish Immigration Data

In this article, we use the wavelet technique to improve the over-rejection problem of the traditional Dickey–Fuller tests for unit root when the data is associated with volatility like the GARCH(1, 1) effect. The logic of this technique is based on the idea that the wavelet spectrum decomposition can separate out information of different frequencies in the data series. We prove that the asymptotic distribution of the test in the wavelet environment is still the same as the traditional Dickey–Fuller type of tests. The finite sample property is improved when the data suffers from GARCH error. The investigation of the size property and the finite sample distribution of the test is carried out by Monte Carlo experiment. An empirical example with data on the net immigration to Sweden during the period 1950–2000 is used to illustrate the performance of the wavelet improved test under GARCH errors. The results reveal that using the traditional Dickey–Fuller type of tests, the unit root hypothesis is rejected while our wavelet improved test do not reject as it is more robust to GARCH errors in finite samples.     

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.     

Solving Noisy ICA Using Multivariate Wavelet Denoising with an Application to Noisy Latent Variables Regression

A novel approach to solve the independent component analysis (ICA) model in the presence of noise is proposed. We use wavelets as natural denoising tools to solve the noisy ICA model. To do this, we use a multivariate wavelet denoising algorithm allowing spatial and temporal dependency. We propose also using a statistical approach, named nested design of experiments, to select the parameters such as wavelet family and thresholding type. This technique helps us to select more convenient combination of the parameters. This approach could be extended to many other problems in which one needs to choose parameters between many choices. The performance of the proposed method is illustrated on the simulated data and promising results are obtained. Also, the suggested method applied in latent variables regression in the presence of noise on real data. The good results confirm the ability of multivariate wavelet denoising to solving noisy ICA.     

Some partially sequential nonparametric tests for detecting linear trend

In the present study, we develop two nonparametric partially sequential tests for detecting possible presence of linear trend among the incoming series of observations. We assume that a sample of fixed size is available a priori from some unknown univariate continuous population and there is no sign of trend among these historical observations. Our proposed tests can be viewed as the sequential type tests for monitoring structural changes. We use partial sequential sampling schemes based on usual ranks as well as on sequential ranks. We provide detailed discussion on asymptotic studies related to the proposed tests. We compare the two tests under various situations. We also present some numerical results based on simulation studies. Proposed tests are extremely important in profit making in volatile market through Margin Trading. We illustrate the mechanism with a detailed analysis of a stock price data.     

On approximate least squares estimators of parameters of one-dimensional chirp signal

Chirp signals are quite common in many natural and man-made systems such as audio signals, sonar, and radar. Estimation of the unknown parameters of a signal is a fundamental problem in statistical signal processing. Recently, Kundu and Nandi [Parameter estimation of chirp signals in presence of stationary noise. Stat Sin. 2008;75:187–201] studied the asymptotic properties of least squares estimators (LSEs) of the unknown parameters of a simple chirp signal model under the assumption of stationary noise. In this paper, we propose periodogram-type estimators called the approximate least squares estimators (ALSEs) to estimate the unknown parameters and study the asymptotic properties of these estimators under the same error assumptions. It is observed that the ALSEs are strongly consistent and asymptotically equivalent to the LSEs. Similar to the periodogram estimators, these estimators can also be used as initial guesses to find the LSEs of the unknown parameters. We perform some numerical simulations to see the performance of the proposed estimators and compare them with the LSEs and the estimators proposed by Lahiri et al. [Efficient algorithm for estimating the parameters of two dimensional chirp signal. Sankhya B. 2013;75(1):65–89]. We have analysed two real data sets for illustrative purposes.     

Wavelet‐based estimators for mixture regression

We consider a process that is observed as a mixture of two random distributions, where the mixing probability is an unknown function of time. The setup is built upon a wavelet‐based mixture regression. Two linear wavelet estimators are proposed. Furthermore, we consider three regularizing procedures for each of the two wavelet methods. We also discuss regularity conditions under which the consistency of the wavelet methods is attained and derive rates of convergence for the proposed estimators. A Monte Carlo simulation study is conducted to illustrate the performance of the estimators. Various scenarios for the mixing probability function are used in the simulations, in addition to a range of sample sizes and resolution levels. We apply the proposed methods to a data set consisting of array Comparative Genomic Hybridization from glioblastoma cancer studies.     

Expansion estimation by Bayes rules

In the problem of estimating a location parameter in any symmetric unimodal location parameter model, we demonstrate that Bayes rules with respect to squared error loss can be expanders for some priors that belong to the family of all symmetric priors. That generalizes the results obtained by DasGupta and Rubin for the one dimensional case. We also consider symmetric priors which either have an appropriate point mass at 0 or are unimodal, and prove that under the latter condition all Bayes rules are shrinkers. Results of such nature are important, for example, in wavelet based function estimation and data denoising, where shrinkage of wavelet coefficients is associated with smoothing the data. We illustrate the results using FIAT stock market data.     

Estimation of a nonparametric regression spectrum for multivariate time series

Estimation of a nonparametric regression spectrum based on the periodogram is considered. Neither trend estimation nor smoothing of the periodogram is required. Alternatively, for cases where spectral estimation of phase shifts fails and the shift does not depend on frequency, a time domain estimator of the lag-shift is defined. Asymptotic properties of the frequency and time domain estimators are derived. Simulations and a data example illustrate the methods.     

Evaluation of Linear Trend Tests Using Resampling Techniques

A number of parametric and non-parametric linear trend tests for time series are evaluated in terms of test size and power, using also resampling techniques to form the empirical distribution of the test statistics under the null hypothesis of no linear trend. For resampling, both bootstrap and surrogate data are considered. Monte Carlo simulations were done for several types of residuals (uncorrelated and correlated with normal and nonnormal distributions) and a range of small magnitudes of the trend coefficient. In particular for AR(1) and ARMA(1, 1) residual processes, we investigate the discrimination of strong autocorrelation from linear trend with respect to the sample size. The correct test size is obtained for larger data sizes as autocorrelation increases and only when a randomization test that accounts for autocorrelation is used. The overall results show that the type I and II errors of the trend tests are reduced with the use of resampled data. Following the guidelines suggested by the simulation results, we could find significant linear trend in the data of land air temperature and sea surface temperature.     

A Non‐Parametric Estimator of the Spectral Density of a Continuous‐Time Gaussian Process Observed at Random Times

Abstract. In numerous applications data are observed at random times and an estimated graph of the spectral density may be relevant for characterizing and explaining phenomena. By using a wavelet analysis, one derives a non‐parametric estimator of the spectral density of a Gaussian process with stationary increments (or a stationary Gaussian process) from the observation of one path at random discrete times. For every positive frequency, this estimator is proved to satisfy a central limit theorem with a convergence rate depending on the roughness of the process and the moment of random durations between successive observations. In the case of stationary Gaussian processes, one can compare this estimator with estimators based on the empirical periodogram. Both estimators reach the same optimal rate of convergence, but the estimator based on wavelet analysis converges for a different class of random times. Simulation examples and an application to biological data are also provided.     

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.