New fat-tail normality test based on conditional second moments with applications to finance

Abstract

In this paper we introduce an efficient fat-tail measurement framework that is based on the conditional second moments. We construct a goodness-of-fit statistic that has a direct interpretation and can be used to assess the impact of fat-tails on central data conditional dispersion. Next, we show how to use this framework to construct a powerful normality test. In particular, we compare our methodology to various popular normality tests, including the Jarque-Bera test that is based on third and fourth moments, and show that in many cases our framework outperforms all others, both on simulated and market stock data. Finally, we derive asymptotic distributions for conditional mean and variance estimators, and use this to show asymptotic normality of the proposed test statistic.

Local Defect Detection in Bearings in the Presence of Heavy-Tailed Noise and Spectral Overlapping of Informative and Non-Informative Impulses

“…For the idea of the selector based on the conditional variance statistic raised for technical (the local fault) diagnosis in the industrial crushing machine with heavy-tailed background noise, which is an inherent drawback of the machine’s operation, see [ 34 ]. The conditional variance statistic is related to the so-called Rule [ 52 , 53 ]. According to this rule, if the sorted (in ascending order) Gaussian random sample is divided into three groups, the first one containing of the largest values, the second one with of the middle, and the last one with of the smallest values, there is a particular relationship between some empirical characteristics corresponding to these groups, see [ 52 ].…”

“…In the literature [ 34 ], the CVB selector of the IFB has been defined as follows: where n is the sample size. The index 7 denotes the number of to the partitions applied to the vector while is the i-th set of this partition [ 53 ]: where, denotes the empirical quantile of order q for the vector . The in ( 3 ) is the sample standard deviation in the set and is the sample standard deviation of the whole vector x .…”

“…The main property of the division is that the variances corresponding to the appropriate sets are equal. If we assume the Gaussian distribution of the random vector, then the following is satisfied [ 53 ]: …”

Research on fault diagnosis of rolling bearings based on conditional variance statistic and cross-correlation spectrum

“…In other words, we can determine whether the sample belongs to Gaussian distribution by the comparation on the value of conditional variance and the conditional central variance [14]. On this basis, literature [20] proposed a test statistic to measure the degree of heavy tail of the data. Given a set of data = ( , … , ), there is a statistical test:…”

“…where equals to 0.2, ∈ is a constant, is the length of the signal, denotes sample variance and is the conditional sample variance on set . Literature [20] pointed out that the asymptotic distribution of is standard normal if the sample is independent and identically distributed. Moreover, if the sample belongs to (symmetric) heavy tailed distribution, the values of statistic should be larger than 0 because of the high values of conditional tail variances on sets and .…”