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.     

Dependent SiZer: Goodness-of-Fit Tests for Time Series Models

In this paper, we extend SiZer (SIgnificant ZERo crossing of the derivatives) to dependent data for the purpose of goodness-of-fit tests for time series models. Dependent SiZer compares the observed data with a specific null model being tested by adjusting the statistical inference using an assumed autocovariance function. This new approach uses a SiZer type visualization to flag statistically significant differences between the data and a given null model. The power of this approach is demonstrated through some examples of time series of Internet traffic data. It is seen that such time series can have even more burstiness than is predicted by the popular, long- range dependent, Fractional Gaussian Noise model.     

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     

Chi-Squared Goodness-of-Fit Tests for Parametric Accelerated Failure Time Models

We give chi-squared goodness-of fit tests for parametric regression models such as accelerated failure time, proportional hazards, generalized proportional hazards, frailty models, transformation models, and models with cross-effects of survival functions. Random right censored data are used. Choice of random grouping intervals as data functions is considered.     

Goodness-of-Fit Tests for Multiplicative Models with Dependent Data

Abstract.  Several classical time series models can be written as a regression model between the components of a strictly stationary bivariate process. Some of those models, such as the ARCH models, share the property of proportionality of the regression function and the scale function, which is an interesting feature in econometric and financial models. In this article, we present a procedure to test for this feature in a non-parametric context. The test is based on the difference between two non-parametric estimators of the distribution of the regression error. Asymptotic results are proved and some simulations are shown in the paper in order to illustrate the finite sample properties of the procedure.     

Tests for Multiple Peaks in the Spectra of Categorical Time Series

A Fisher's-type test for the significance of peaks in the spectra of categorical time series is developed. This test extends to peaks other than the maximum. The test was based on the Walsh–Fourier periodogram, which is more suitable for data which exhibit sharp jumps rather than smooth curves. The test is applied to a data set of neonatal sleep patterns, which have previously been shown to contain at least two clinically interesting periods.     

Bispectral-Based Goodness-of-Fit Tests of Gaussianity and Linearity of Stationary Time Series

Spectral domain tests for time series linearity typically suffer from a lack of power compared to time domain tests. We present two tests for Gaussianity and linearity of a stationary time series. The tests are two-stage procedures applying goodness-of-fit techniques to the estimated normalized bispectrum. We illustrate the performances of the tests are competitive with time domain tests. The new tests typically outperform Hinich's (1982 Hinich , M. J. ( 1982 ). Testing for Gaussianity and linearity of a stationary time series . J. Time Ser. Anal. 3 : 169 – 176 .[Crossref] , [Google Scholar]) bispectral based test, especially when the length of the time series is not large.     

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.     

The Whittle Estimator for Strongly Dependent Stationary Gaussian Fields

In this article we generalize results on the asymptotic behaviour of the Whittle estimator for certain stationary Gaussian long range dependent fields. These results have been established in the one-dimensional case under very general conditions. They require controlling the estimation bias and also giving convergence theorems for certain quadratic forms of the observations. In the multidimensional setting, our main interest will be controlling the bias. This can be done for d ≤ 3 using taper functions, and, depending on the shape of the singularity, also introducing certain regularizing functions. In this last case, however, the estimator will no longer be efficient. We also present certain partial results concerning the convergence to a limiting Gaussian distribution of the associated quadratic forms.     

Simple Robust Tests for Autocorrelated Errors in Time Series Design Intervention Models

A simple, robust test for the autocorrelation parameter in an intervention time-series model (AB design) is proposed. It is analogous to the traditional tests and can easily be computed by using the freeware R. In the same way as traditional tests of autocorrelation are based on least squares (LS) fits of a linear model, our robust test is based on the highly efficient Wilcoxon fit of the linear model. We present the results of a Monte Carlo study which show that our robust test inherits the good efficiency properties of this Wilcoxon fit. Its empirical power is only slightly less than the empirical power of the least squares test over situations with normally distributed errors while it exhibited much more power over situations with error distributions having tails heavier than those of a normal distribution. It also showed robustness of validity over all null situations simulated. We also present the results of the application of our test to a real data set which illustrates the robustness of our test.     

Time Series Classification Based on Spectral Analysis

For time series data with obvious periodicity (e.g., electric motor systems and cardiac monitor) or vague periodicity (e.g., earthquake and explosion, speech, and stock data), frequency-based techniques using the spectral analysis can usually capture the features of the series. By this approach, we are able not only to reduce the data dimensions into frequency domain but also utilize these frequencies by general classification methods such as linear discriminant analysis (LDA) and k-nearest-neighbor (KNN) to classify the time series. This is a combination of two classical approaches. However, there is a difficulty in using LDA and KNN in frequency domain due to excessive dimensions of data. We overcome the obstacle by using Singular Value Decomposition to select essential frequencies. Two data sets are used to illustrate our approach. The classification error rates of our simple approach are comparable to those of several more complicated methods.     

Periodogram Analysis of Censored Time Series from a Designed Experiment

Growth hormone plasma concentrations vary rhythmically between high and low values. Radioimmunoassay measurements of low values are often indistinguishable from low controls, and are reported as a censored value, the 'minimum detectable dose'. This paper reports such a dataset from a designed experiment with about 60% of the values censored but large distinct signals for the remainder of the data. The ordinates of the average periodogram for each treatment group are independently gamma distributed, with distribution depending on the underlying spectrum and the replication for that group. This situation can lead to an analysis for common spectral shape using a gamma generalized linear model with log link, and the hypothesis of common spectral shape is rejected here. Since such a level of censoring reduces the variance of each profile, the periodogram, which is a partition of the variance, is also reduced in overall magnitude. A simulation study shows that this reduction is not necessarily uniform over the frequency domain, but may be more pronounced at lower or higher ordinates depending on the underlying model. Therefore it is possible that the rejection of common spectral shape is an artefact of the censoring.     

Goodness-of-fit Tests Based on the Kernel Density Estimator

Abstract.  Given an i.i.d. sample drawn from a density f on the real line, the problem of testing whether f is in a given class of densities is considered. Testing procedures constructed on the basis of minimizing the L ₁-distance between a kernel density estimate and any density in the hypothesized class are investigated. General non-asymptotic bounds are derived for the power of the test. It is shown that the concentration of the data-dependent smoothing factor and the 'size' of the hypothesized class of densities play a key role in the performance of the test. Consistency and non-asymptotic performance bounds are established in several special cases, including testing simple hypotheses, translation/scale classes and symmetry. Simulations are also carried out to compare the behaviour of the method with the Kolmogorov-Smirnov test and an L ₂ density-based approach due to Fan [ Econ. Theory 10 (1994) 316].     

A Goodness-of-Fit Test for a Class of Autoregressive Conditional Duration Models

This article develops a method for testing the goodness-of-fit of a given parametric autoregressive conditional duration model against unspecified nonparametric alternatives. The test statistics are functions of the residuals corresponding to the quasi maximum likelihood estimate of the given parametric model, and are easy to compute. The limiting distributions of the test statistics are not free from nuisance parameters. Hence, critical values cannot be tabulated for general use. A bootstrap procedure is proposed to implement the tests, and its asymptotic validity is established. The finite sample performances of the proposed tests and several other competing ones in the literature, were compared using a simulation study. The tests proposed in this article performed well consistently throughout, and they were either the best or close to the best. None of the tests performed uniformly the best. The tests are illustrated using an empirical example.     

Goodness-of-Fit Tests for Multivariate Distributions

We propose three new statistics, Z _p , C _p , and R _p for testing a p-variate (p ≥ 2) normal distribution and compare them with the prominent test statistics. We show that C _p is overall most powerful and is effective against skew, long-tailed as well as short-tailed symmetric alternatives. We show that Z _p and R _p are most powerful against skew and long-tailed alternatives, respectively. The Z _p and R _p statistics can also be used for testing an assumed p-variate nonnormal distribution.     

New Goodness-of-Fit Tests Based on Sample Quantiles

In this article power divergences statistics based on sample quantiles are transformed in order to introduce new goodness-of-fit tests. Quantiles of the distribution of proposed statistics are calculated under uniformity, normality, and exponentiality. Several power comparisons are performed to show that the new tests are generally more powerful than the original ones.     

Goodness-of-Fit Tests Based on Correcting Moments of Entropy Estimators

In this article, we consider the entropy estimator introduced by Alizadeh Noughabi and Arghami (2010) and derive the nonparametric distribution function corresponding to our estimator as a piece-wise uniform distribution. We use the results to introduce goodness-of-fit tests for the normal and the exponential distributions. The critical values and powers for some alternatives are obtained by simulation. The powers of the proposed tests under various alternatives are compared with the competitors.     

Goodness-of-Fit Methods for Probabilistic Index Models

A class of semiparametric regression models, called probabilistic index models, has been recently proposed. Because these models are semiparametric, inference is only valid when the proposed model is consistent with the underlying data-generating model. However, no formal goodness-of-fit methods for these probabilistic index models exist yet. We propose a test and a graphical tool for assessing the model adequacy. Simulation results indicate that both methods succeed in detecting lack-of-fit. The methods are also illustrated on a case study.     

Asymptotic Theory of Outlier Detection Algorithms for Linear Time Series Regression Models

Outlier detection algorithms are intimately connected with robust statistics that down‐weight some observations to zero. We define a number of outlier detection algorithms related to the Huber‐skip and least trimmed squares estimators, including the one‐step Huber‐skip estimator and the forward search. Next, we review a recently developed asymptotic theory of these. Finally, we analyse the gauge, the fraction of wrongly detected outliers, for a number of outlier detection algorithms and establish an asymptotic normal and a Poisson theory for the gauge.     

Goodness-of-Fit Tests for the Gamma Distribution Based on the Empirical Laplace Transform

We propose a class of goodness-of-fit tests for the gamma distribution that utilizes the empirical Laplace transform. The consistency of the tests as well as their asymptotic distribution under the null hypothesis are investigated. As the decay of the weight function tends to infinity, the test statistics approach limit values related to the first non zero component of Neyman's smooth test for the gamma law. The new tests are compared with other omnibus tests for the gamma distribution.