The difference-sign runs length distribution in testing for serial independence

We investigate the sequence of difference-sign runs length of a time series in the context of non-parametric tests for serial independence. This sequence is, under suitable conditioning, a stationary sequence and we prove that the normalized correlation of two consecutive runs length is small (≈0.0427). We use this result in a test based on the relative entropy of the empirical distribution of the runs length. We investigate the performance of the test in simulated series and test serial independence of cardiac data series in atrial fibrillation.     

Tests for randomness of observations that involve two factors

Tests for randomness of observations that involve one factor have been considered by many authors among them are Mosteller [4], Bateman [21], Barton and David [1] and Shaughnessy [7]. However, on many occasions, data involve two different factors such as time and location, temperature and pressure, or levels of doses and responses of patients and so on. In this paper, we consider tests for randomness of observations that involve two factors for which data are given in a matrix form. Some new definitions of runs of a matrix of data are defined and discussed. A special kind of run is proposed for the test for randomness. Distributions and properties of this run are studied. Some critical regions are tabulated.     

Adaptive positive and negative runs test

As a nonparametric randomness test, the positive and negative runs test is widely used in practice due to the simplicity of its procedures. The test can lose efficiency if the alternative distribution is symmetrical at 0.5. In addition, the test can only be applied to test the randomness of a sequence from the uniform distribution. In this paper, we introduce an adaptive positive and negative runs test method to maximize the power function by choosing the optimal cut point. Also, the test is extended to check the randomness of a sequence generated from any other given distributions. Furthermore, we derive the exact distribution and obtain the asymptotical critical values of the proposed test statistics. Compared with the existed test, the efficiency of the proposed adaptive positive and negative runs test is competitive through simulation study.     

Bootstrap entropy test for general location-scale time series models with heteroscedasticity

This study considers a goodness-of-fit test for location-scale time series models with heteroscedasticity, including a broad class of generalized autoregressive conditional heteroscedastic-type models. In financial time series analysis, the correct identification of model innovations is crucial for further inferences in diverse applications such as risk management analysis. To implement a goodness-of-fit test, we employ the residual-based entropy test generated from the residual empirical process. Since this test often shows size distortions and is affected by parameter estimation, its bootstrap version is considered. It is shown that the bootstrap entropy test is weakly consistent, and thereby its usage is justified. A simulation study and data analysis are conducted by way of an illustration.     

Tests for comparing time series of unequal lengths

This paper deals with hypothesis testing for independent time series with unequal length. It proposes a spectral test based on the distance between the periodogram ordinates and a parametric test based on the distance between the parameter estimates of fitted autoregressive moving average models. Both tests are compared with a likelihood ratio test based on the pooled spectra. In all cases, the null hypothesis is that the two series under consideration are generated by the same stochastic process. The performance of the three tests is investigated by a Monte Carlo simulation study.     

Dynamic structural models with covariates for short-term forecasting of time series with complex seasonal patterns

This work presents a framework of dynamic structural models with covariates for short-term forecasting of time series with complex seasonal patterns. The framework is based on the multiple sources of randomness formulation. A noise model is formulated to allow the incorporation of randomness into the seasonal component and to propagate this same randomness in the coefficients of the variant trigonometric terms over time. A unique, recursive and systematic computational procedure based on the maximum likelihood estimation under the hypothesis of Gaussian errors is introduced. The referred procedure combines the Kalman filter with recursive adjustment of the covariance matrices and the selection method of harmonics number in the trigonometric terms. A key feature of this method is that it allows estimating not only the states of the system but also allows obtaining the standard errors of the estimated parameters and the prediction intervals. In addition, this work also presents a non-parametric bootstrap approach to improve the forecasting method based on Kalman filter recursions. The proposed framework is empirically explored with two real time series.     

Detecting Randomness: A Review of Existing Tests with New Comparisons

In this article we review and compare a number of existing tests for detecting randomness in time series data, with emphasis on stock market index data. By comparing variance ratio tests with traditional statistical tests, we have the most extensive simulation comparison of such procedures. The investigated tests are compared over a diverse group of distributions, models, and stock market applications. In our stock market data analysis, the choice of data transformation can have a noticeable effect on test results. This study provides the reader with a guide as to which test and transformation is most appropriate for their use.     

Properties of the Cox–Stuart Test for Trend in Application to Hydrological Series: The Simulation Study

The size and power properties of the Cox–Stuart test for detection of a monotonic deterministic trend in hydrological time series are analyzed using the Monte Carlo method. The influence of distribution properties, lengths of series, and trend slopes is studied. Results indicate good size in all cases. The power is high for: length over 60 and strong trend slope, low or medium variation, and medium slope. The power declines if slope and length decrease and if variability increases. The properties are better for skewed distributions than for symmetrical. The test is slightly weaker in comparison to the Mann–Kendall test.     

On runs of length exceeding a threshold: normal approximation

Run statistics denoting number of runs and sum of run lengths are defined on binary sequences and their asymptotic normality is established by a simple unified way for Bernoulli sequences. All the considered statistics share a common feature; they refer to runs of length exceeding a specific length (a threshold). Asymptotic results of associated statistics denoting run lengths and waiting times are derived as well. Specific probabilities of the examined statistics are used in applications in the fields of system reliability and molecular biology. The study is illustrated by an extensive numerical experimentation.     

Methods for testing subblock patterns

A number of statistical tests have been recommended over the last twenty years for assessing the randomness of long binary strings used in cryptographic algorithms. Several of these tests include methods of examining subblock patterns. These tests are the uniformity test, the universal test and the repetition test. The effectiveness of these tests are compared based on the subblock length, the limitations on data requirements, and on their power in detecting deviations from randomness. Due to the complexity of the test statistics, the power functions are estimated by simulation methods. The results show that for small subblocks the uniformity test is more powerful than the universal test, and that there is some doubt about the parameters of the hypothesised distribution for the universal test statistic. For larger subblocks the results show that the repetition test is the most effective test, since it requires far less data than either of the other two tests and is an efficient test in detecting deviations from randomness in binary strings.     

Testing Periodicity in Short Series and Application to Gene Expression Data

Testing for periodicity in microarray time series encounters the challenges of short series length, missing values and presence of non-Fourier frequencies. In this article, a test method for such series has been proposed. The method is completely simulation based and finds p-values for test of periodicity through fitting Pearson Type VI distribution. The simulation results compare and reveal the excellence of this method over Fisher's g test for varying series length, frequencies, and error variance. This approach is applied to Caulobacter crescentus cell cycle data in order to demonstrate the practical performance.     

Nonparametric analysis of treatment effects with missing observations

This paper develops a test for comparing treatment effects when observations are missing at random for repeated measures data on independent subjects. It is assumed that missingness at any occasion follows a Bernoulli distribution. It is shown that the distribution of the vector of linear rank statistics depends on the unknown parameters of the probability law that governs missingness, which is absent in the existing conditional methods employing rank statistics. This dependence is through the variance–covariance matrix of the vector of linear ranks. The test statistic is a quadratic form in the linear rank statistics when the variance–covariance matrix is estimated. The limiting distribution of the test statistic is derived under the null hypothesis. Several methods of estimating the unknown components of the variance–covariance matrix are considered. The estimate that produces stable empirical Type I error rate while maintaining the highest power among the competing tests is recommended for implementation in practice. Simulation studies are also presented to show the advantage of the proposed test over other rank-based tests that do not account for the randomness in the missing data pattern. Our method is shown to have the highest power while also maintaining near-nominal Type I error rates. Our results clearly illustrate that even for an ignorable missingness mechanism, the randomness in the pattern of missingness cannot be ignored. A real data example is presented to highlight the effectiveness of the proposed method.     

Testing for Equal Predictability of Stationary ARMA Processes

In this work we use a measure of predictability of a time series following a stationary ARMA process to develop a test of equal predictability of two or more time series. The test is derived by a set of propositions which links the structure of the AR and MA coefficients to the predictability measure. A particular case of this general approach is constituted by time series having a Wold decomposition with weights having the same sign; in this framework the equal predictability is equivalent to parallelism among ARMA models and the null hypothesis of equal predictability is simply a set of linear restrictions. The ARMA representation of the GARCH models presents non-negative weights, so that this test can be extended to verify the equal predictability of squared time series following GARCH structures.     

The local fractional bootstrap

We introduce a bootstrap procedure for high‐frequency statistics of Brownian semistationary processes. More specifically, we focus on a hypothesis test on the roughness of sample paths of Brownian semistationary processes, which uses an estimator based on a ratio of realized power variations. Our new resampling method, the local fractional bootstrap, relies on simulating an auxiliary fractional Brownian motion that mimics the fine properties of high‐frequency differences of the Brownian semistationary process under the null hypothesis. We prove the first‐order validity of the bootstrap method, and in simulations, we observe that the bootstrap‐based hypothesis test provides considerable finite‐sample improvements over an existing test that is based on a central limit theorem. This is important when studying the roughness properties of time series data. We illustrate this by applying the bootstrap method to two empirical data sets: We assess the roughness of a time series of high‐frequency asset prices and we test the validity of Kolmogorov's scaling law in atmospheric turbulence data.     

Nonparametric testing for time series: A bibliography

This bibliography brings together and classifies a wide variety of nonparametric methods which are pocentially useful for the analysis of time series data, in particular for testing randomness. Inference on Markov chain models is also extensively surveyed.     

On- and offline detection of structural breaks in thermal spraying processes

We investigate and develop methods for structural break detection, considering time series from thermal spraying process monitoring. Since engineers induce technical malfunctions during the processes, the time series exhibit structural breaks at known time points, giving us valuable information to conduct the investigations. First, we consider a recently developed robust online (also real-time) filtering (i.e. smoothing) procedure that comprises a test for local linearity. This test rejects when jumps and trend changes are present, so that it can also be useful to detect such structural breaks online. Second, based on the filtering procedure we develop a robust method for the online detection of ongoing trends. We investigate these two methods as to the online detection of structural breaks by simulations and applications to the time series from the manipulated spraying processes. Third, we consider a recently developed fluctuation test for constant variances that can be applied offline, i.e. after the whole time series has been observed, to control the spraying results. Since this test is not reliable when jumps are present in the time series, we suggest data transformation based on filtering and demonstrate that this transformation makes the test applicable.     

熵权模糊综合评判法在人口综合发展评价模型中的应用

统筹解决人口问题是中国实现经济发展、社会进步和可持续发展面临的重大而紧迫的战略任务。从人口综合发展指标体系出发，针对人口综合发展评价中客观存在的模糊性与随机性，将信息论汇总的熵值理论引入人口综合发展评价模型中，利用熵权法与模糊综合评判对全国及四个直辖市的人口综合发展状况进行评价检验，证明把该方法应用于人口综合发展评价模型具有较高的可信度。     

Further Evidence on the Great Crash,the Oil-Price Shock,and the Unit-Root Hypothesis

Recently, Perron has carried out tests of the unit-root hypothesis against the alternative hypothesis of trend stationarity with a break in the trend occurring at the Great Crash of 1929 or at the 1973 oil-price shock. His analysis covers the Nelson–Plosser macroeconomic data series as well as a postwar quarterly real gross national product (GNP) series. His tests reject the unit-root null hypothesis for most of the series. This article takes issue with the assumption used by Perron that the Great Crash and the oil-price shock can be treated as exogenous events. A variation of Perron's test is considered in which the breakpoint is estimated rather than fixed. We argue that this test is more appropriate than Perron's because it circumvents the problem of data-mining. The asymptotic distribution of the estimated breakpoint test statistic is determined. The data series considered by Perron are reanalyzed using this test statistic. The empirical results make use of the asymptotics developed for the test statistic as well as extensive finite-sample corrections obtained by simulation. The effect on the empirical results of fat-tailed and temporally dependent innovations is investigated, in brief, by treating the breakpoint as endogenous, we find that there is less evidence against the unit-root hypothesis than Perron finds for many of the data series but stronger evidence against it for several of the series, including the Nelson-Plosser industrial-production, nominal-GNP, and real-GNP series.     

Simple detection of outlying short time series

Sets of relatively short time series arise in many situations. One aspect of their analysis may be the detection of outlying series. We examine the performance of standard normal outlier tests applied to the means, or to simple functions of the means, of AR(1) series, not necessarily of equal lengths. Although unequal lengths of series implies that the means have unequal variances, that are only known approximately, it is shown that nominal significance levels hold good under most circumstances. Thus a standard outlier test can usefully be applied, avoiding the complication of estimating the time series' parameters. The test's power is affected by unequal lengths, being higher when the slippage occurs in one of the longer series     

Tests for noncorrelation of two multivariate ARMA time series

In many situations, we want to verify the existence of a relationship between multivariate time series. In this paper, we generalize the procedure developed by Haugh (1976) for univariate time series in order to test the hypothesis of noncorrelation between two multivariate stationary ARMA series. The test statistics are based on residual cross-correlation matrices. Under the null hypothesis of noncorrelation, we show that an arbitrary vector of residual cross-correlations asymptotically follows the same distribution as the corresponding vector of cross-correlations between the two innovation series. From this result, it follows that the test statistics considered are asymptotically distributed as chi-square random variables. Two test procedures are described. The first one is based on the residual cross-correlation matrix at a particular lag, whilst the second one is based on a portmanteau type statistic that generalizes Haugh's statistic. We also discuss how the procedures for testing noncorrelation can be adapted to determine the directions of causality in the sense of Granger (1969) between the two series. An advantage of the proposed procedures is that their application does not require the estimation of a global model for the two series. The finite-sample properties of the statistics introduced were studied by simulation under the null hypothesis. It led to modified statistics whose upper quantiles are much better approximated by those of the corresponding chi-square distribution. Finally, the procedures developed are applied to two different sets of economic data.