Goodness-of-fit test for $$\alpha$$-stable distribution based on the quantile conditional variance statistics

The class of $$\alpha$$ α -stable distributions is ubiquitous in many areas including signal processing, finance, biology, physics, and condition monitoring. In particular, it allows efficient noise modeling and incorporates distributional properties such as asymmetry and heavy-tails. Despite the popularity of this modeling choice, most statistical goodness-of-fit tests designed for $$\alpha$$ α -stable distributions are based on a generic distance measurement methods. To be efficient, those methods require large sample sizes and often do not efficiently discriminate distributions when the corresponding $$\alpha$$ α -stable parameters are close to each other. In this paper, we propose a novel goodness-of-fit method based on quantile (trimmed) conditional variances that is designed to overcome these deficiencies and outperforms many benchmark testing procedures. The effectiveness of the proposed approach is illustrated using extensive simulation study with focus set on the symmetric case. For completeness, an empirical example linked to plasma physics is provided.

Analysis of intra-day fluctuations in the Mexican financial market index

In this paper, a statistical analysis of high frequency fluctuations of the IPC, the Mexican Stock Market Index, is presented. A sample of tick--to--tick data covering the period from January 1999 to December 2002 was analyzed, as well as several other sets obtained using temporal aggregation. Our results indicates that the highest frequency is not useful to understand the Mexican market because almost two thirds of the information corresponds to inactivity. For the frequency where fluctuations start to be relevant, the IPC data does not follows any $\alpha$-stable distribution, including the Gaussian, perhaps because of the presence of autocorrelations. For a long range of lower-frequencies, but still in the intra-day regime, fluctuations can be described as a truncated L\'evy flight, while for frequencies above two-days, a Gaussian distribution yields the best fit. Thought these results are consistent with other previously reported for several markets, there are significant differences in the details of the corresponding descriptions.

Robust Signals Detection Algorithm Based on Cyclostationarity in Impulsive Noise

A robust method for detecting the communication signals impinging on an antenna with interference and non-Gaussian impulsive noise is introduced in this paper. Degradation of the conventional cyclic detector which based on max-output-SNR criterion in impulsive noise is shown both theoretically and experimentally. By fusing second-order cyclostationarity and fractional lower-order statistics, a type of cyclic fractional lower-order statistics is developed which is defined for exploiting cyclostationarity property. Then, a new robust type of detection algorithm is developed using the theory of optimal filtering based on max-output-SNR criterion and alpha-stable distribution, including the fractional lower-order cyclic matched filter, which is formulated for detecting the communication signals in the presence of interference and non-Gaussian alpha-stable distribution impulsive noise. It is shown that the new method is robust to Gaussian and non-Gaussian impulsive noises, and is immune to the interfering signals which occupy the same spectral band as that of the received signal. Simulation results show the robustness and effectiveness of the proposed algorithm.