Mod-phi convergence I: Normality zones and precise deviations

In this paper, we use the framework of mod-$\phi$ convergence to prove precise large or moderate deviations for quite general sequences of real valued random variables $(X_{n})_{n \in \mathbb{N}}$, which can be lattice or non-lattice distributed. We establish precise estimates of the fluctuations $P[X_{n} \in t_{n}B]$, instead of the usual estimates for the rate of exponential decay $\log( P[X_{n}\in t_{n}B])$. Our approach provides us with a systematic way to characterise the normality zone, that is the zone in which the Gaussian approximation for the tails is still valid. Besides, the residue function measures the extent to which this approximation fails to hold at the edge of the normality zone. The first sections of the article are devoted to a proof of these abstract results and comparisons with existing results. We then propose new examples covered by this theory and coming from various areas of mathematics: classical probability theory, number theory (statistics of additive arithmetic functions), combinatorics (statistics of random permutations), random matrix theory (characteristic polynomials of random matrices in compact Lie groups), graph theory (number of subgraphs in a random Erd\H{o}s-R\'enyi graph), and non-commutative probability theory (asymptotics of random character values of symmetric groups). In particular, we complete our theory of precise deviations by a concrete method of cumulants and dependency graphs, which applies to many examples of sums of "weakly dependent" random variables. The large number as well as the variety of examples hint at a universality class for second order fluctuations.Comment: 103 pages. New (final) version: multiple small improvements ; a new section on mod-Gaussian convergence coming from the factorization of the generating function ; the multi-dimensional results have been moved to a forthcoming paper ; and the introduction has been reworke

Test for Infinite Variance in Stock Returns

The existence of second order moment or the finite variance is a commonly used assumption in financial time series analysis. We examine the validation of this condition for main stock index return series by applying the extreme value theory. We compare the performances of the adaptive Hill’s estimator and the Smith’s estimator for the tail index using Monte Carlo simulations for both i.i.d data and dependent data. The simulation results show that the Hill’s estimator with adaptive data-based truncation number performs better in both cases. It has not only smaller bias but also smaller MSE when the true tail index α is not more than 2. Moreover, the Hill’s estimator shows precise results for the hypothesis test of infinite variance. Applying the adaptive Hill’s estimator to main stock index returns over the world, we find that for most indices, the second moment does exist for daily, weekly and monthly returns. However, an additional test for the existence of the fourth moment shows that generally the fourth moment does not exist, especially for daily returns. And these results don’t change when a Gaussian-GARCH effect is removed from the original return series

Generalized Error Exponents For Small Sample Universal Hypothesis Testing

The small sample universal hypothesis testing problem is investigated in this paper, in which the number of samples $n$ is smaller than the number of possible outcomes $m$. The goal of this work is to find an appropriate criterion to analyze statistical tests in this setting. A suitable model for analysis is the high-dimensional model in which both $n$ and $m$ increase to infinity, and $n=o(m)$. A new performance criterion based on large deviations analysis is proposed and it generalizes the classical error exponent applicable for large sample problems (in which $m=O(n)$). This generalized error exponent criterion provides insights that are not available from asymptotic consistency or central limit theorem analysis. The following results are established for the uniform null distribution: (i) The best achievable probability of error $P_e$ decays as $P_e=\exp\{-(n^2/m) J (1+o(1))\}$ for some $J>0$. (ii) A class of tests based on separable statistics, including the coincidence-based test, attains the optimal generalized error exponents. (iii) Pearson's chi-square test has a zero generalized error exponent and thus its probability of error is asymptotically larger than the optimal test.Comment: 43 pages, 4 figure

Symmetric and Asymmetric Rounding

If rounded data are used in estimating moments and regression coefficients, the estimates are typically more or less biased. The purpose of the paper is to study the bias inducing effect of rounding, which is also seen when population moments instead of their estimates are considered. Under appropriate conditions this effect can be approximately specified by versions of Sheppard's correction formula. We discuss the conditions under which these approximations are valid. We also investigate the efficiency loss that comes along with rounding. The rounding error, which corresponds to the measurement error of a measurement error model, has a marginal distribution which can be approximated by the uniform distribution. We generalize the concept of simple rounding to that of asymmetric rounding and study its effect on the mean and variance of a distribution under similar circumstances as with simple rounding