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We estimate the probability that random noise, of several plausible standard distributions, creates a false alarm that a periodicity (or log-periodicity) is found in a time series. The solution of this problem is already known for independent Gaussian distributed noise. We investigate more general situations with non-Gaussian correlated noises and present synthetic tests on the detectability and statistical significance of periodic components. A periodic component of a time series is usually detected by some sort of Fourier analysis. Here, we use the Lomb periodogram analysis, which is suitable and outperforms Fourier transforms for unevenly sampled time series. We examine the false-alarm probability of the largest spectral peak of the Lomb periodogram in the presence of power-law distributed noises, of short-range and of long-range fractional-Gaussian noises. Increasing heavy-tailness (respectively correlations describing persistence) tends to decrease (respectively increase) the false-alarm probability of finding a large spurious Lomb peak. Increasing anti-persistence tends to decrease the false-alarm probability. We also study the interplay between heavy-tailness and long-range correlations. In order to fully determine if a Lomb peak signals a genuine rather than a spurious periodicity, one should in principle characterize the Lomb peak height, its width and its relations to other peaks in the complete spectrum. As a step towards this full characterization, we construct the joint-distribution of the frequency position (relative to other peaks) and of the height of the highest peak of the power spectrum. We also provide the distributions of the ratio of the highest Lomb peak to the second highest one. Using the insight obtained by the present statistical study, we re-examine previously reported claims of "log-periodicity" and find that the credibility for log-periodicity in 2D-freely decaying turbulence is weakened while it is strengthened for fracture, for the ion-signature prior to the Kobe earthquake and for financial markets.

The stochastic properties of the pseudo-random number generator are discussed, with emphasis on the average period. Within a factor 2 this turns out to be the root of the maximally possible period. The actual set of periods depends on minor details of the algorithm, and the system settles down in one of only a few different cycles. These features are in perfect agreement with absolute random motion in phase space, to the extent allowed by deterministic dynamics.

The recent rapid spread of deadly epidemics have precipitated an urgent need to speed up the development of different treatments, as well as methods of evaluating their efficacy. The first step towards this is the collection of data relating to the cure rate in groups of patients who have had different treatments applied to them. As most of the available data in these cases is random, it is now the role of statisticians to provide efficient statistical tests to assess the treatment methods through the data. This research aims to provide a new statistical test with high efficiency to reach the right decision with accurate results as quickly as possible using parallel computing algorithms through Wolfram Mathematica software.

Although recurrence quantification analysis provides useful information about underlying dynamics, it cannot be used to obtain significance levels on its own. Therefore, we propose methods to determine significance levels of serial dependence and laminarity from plotted patterns of recurrence plots. We evaluate the probability of two recurrences with an assumption that these recurrences are statistically independent. By modeling the number of simultaneous recurrences with a binomial distribution, we obtain the significance levels of serial dependence and laminarity without generating surrogate data. Further, we apply the proposed methods to a real wind dataset and find that it has serial dependence and laminarity.

We update the previous analysis of correlation between ultra-high energy cosmic rays (UHECR) and active galactic nuclei (AGN), using 69 UHECR events with energy E ≥ 55 EeV released in 2010 by Pierre Auger observatory and 862 AGN within the distance d ≤ 100 Mpc listed in the 13th edition of Véron-Cetty and Véron AGN catalog. To make the test hypothesis definite, we use the simple AGN source model in which UHECR are originated both from AGN, with the fraction f_A, and from the isotropic background. We treat all AGN as equal sources of UHECR, and introduce the smearing angle θ_s to incorporate the effects of intervening magnetic fields. We compare the arrival direction distributions observed by Pierre Auger Observatory (PAO) and expected from the model by the correlational angular distance distribution (CADD) method. CADD method rules out the AGN dominance model with a small smearing angle (f_A ≳ 0.7 and θ_s ≲ 6°). Concerning the isotropy, CADD shows that the distribution of PAO data is marginally consistent with isotropy. The best fit model lies around the AGN fraction f_A = 0.4 and the moderate smearing angle θ_s = 10°. For the fiducial value f_A = 0.7, the best probability of CADD was obtained at a rather large smearing angle θ_s = 46°. Our results imply that for the whole AGN to be viable sources of UHECR, either an appreciable amount of additional isotropic background or the large smearing effect is required. Thus, we try to bin the distance range of AGN to narrow down the UHECR sources and found that the AGN residing in the distance range 60–80 Mpc have good correlation with the updated PAO data. It is an indication that further study on the subclass of AGN as the UHECR source may be quite interesting.

The price and return time series are two distinct features of any financial asset. Hence, examining the evolution of multiscale characteristics of price and returns sequential data in time domain would be helpful in gaining a better understanding of the dynamical evolution mechanism of the financial asset as a complex system. In fact, this is important to understand their respective dynamics and to design their appropriate predictive models. The main purpose of the current work is to investigate the multiscale fractals of price and return high frequency data in Turkish stock market. In this regard, the wavelet leaders computational method is applied to each high frequency data to reveal its multi-fractal behavior. In particular, the method is applied to a large set of Turkish stocks and statistical results are performed to check for (i) presence of multi-fractals in price and return series and (ii) differences between prices and returns in terms of multi-fractals. Our statistical results show strong evidence that high frequency price and return data exhibit multi-fractal dynamics. In addition, they show evidence of distinct fractal characteristics on different scales between price and return series. Furthermore, our statistical results show evidence of differences in local fluctuation characteristics of price and return time series. Therefore, differences in local characteristics are useful to build specific predictive models for each type of data for better modeling and prediction to generate profits. Besides, we found evidence that both long-range correlations and fat-tail distributions contribute to the multifractality in Turkish stocks. This finding can be attributed to the major role played by international investors in increasing the volatility of Turkish stocks.

In the context of quantitative analysis of the arbitrage pricing theory (APT) model, conventional factor analytic approaches such as maximum likelihood factor analysis (MLFA) cannot provide satisfactory answers to two important questions. The first one concerns the correct identification of factor number while the second one is related to the rotation indeterminacy of factor loadings. In the literature, MLFA followed by likelihood ratio (LR) test and the analysis of eigenvalues of sample covariance matrix were two popular approaches used to determine the appropriate number of factors. However, it was shown empirically that both of them suffered from different kinds of biases. We find the recently developed non-gaussian factor analysis (NFA) model by Xu [24] provides a new perspective for the determination of the appropriate factor number k in APT, with promising empirical results demonstrated.

This paper studies the model risk; the risk of selecting a model for estimating the Value-at-Risk (VaR). By considering four GARCH-type volatility processes exponentially weighted moving average (EWMA), generalized autoregressive conditional heteroskedasticity (GARCH), exponential GARCH (EGARCH), and fractionally integrated GARCH (FIGARCH), we evaluate the performance of the estimated VaRs using statistical tests including the Kupiec's likelihood ratio (LR) test, the Christoffersen's LR test, the CHI (Christoffersen, Hahn, and Inoue) specification test, and the CHI nonnested test. The empirical study based on Shanghai Stock Exchange A Share Index indicates that both EGARCH and FIGARCH models perform much better than the other two in VaR computation and that the two CHI tests are more suitable for analyzing model risk.

In this paper, we propose a high performance computing toolbox implementing efficient statistical methods for the study of phylogenies. This toolbox, which implements logit models and LASSO-type penalties, gives a way to better understand, measure, and compare the impact of each gene on a global phylogeny. As an application, we study the Echinococcus phylogeny, which is often considered as a particularly difficult example. Mitochondrial and nuclear genomes (19 coding sequences) of nine Echinococcus species are considered in order to investigate the molecular phylogeny of this genus. First, we check that the 19 gene trees lead to 19 totally different unsupported topologies (a topology is the sister relationship when both branch lengths and supports are ignored in a phylogenetic tree), while using the 19 genes as a whole are not sufficient for estimating the phylogeny. In order to circumvent this issue and understand the impact of the genes, we computed 43,796 trees using combinations ranging from 13 to 19 genes. By doing so, 15 topologies are obtained. Four particular topologies, appearing more robust and frequent, are then selected for more precise investigation. Refining further our statistical analysis, a particularly robust topology is extracted. We also carefully demonstrate the influence of nuclear genes on the likelihood of the phylogeny.