Heavy-Tailed Features and Empirical Analysis of the Limit Order Book Volume Profiles in Futures Markets

This paper poses a few fundamental questions regarding the attributes of the volume profile of a Limit Order Books stochastic structure by taking into consideration aspects of intraday and interday statistical features, the impact of different exchange features and the impact of market participants in different asset sectors. This paper aims to address the following questions: 1. Is there statistical evidence that heavy-tailed sub-exponential volume profiles occur at different levels of the Limit Order Book on the bid and ask and if so does this happen on intra or interday time scales ? 2.In futures exchanges, are heavy tail features exchange (CBOT, CME, EUREX, SGX and COMEX) or asset class (government bonds, equities and precious metals) dependent and do they happen on ultra-high (<1sec) or mid-range (1sec -10min) high frequency data? 3.Does the presence of stochastic heavy-tailed volume profile features evolve in a manner that would inform or be indicative of market participant behaviors, such as high frequency algorithmic trading, quote stuffing and price discovery intra-daily? 4. Is there statistical evidence for a need to consider dynamic behavior of the parameters of models for Limit Order Book volume profiles on an intra-daily time scale ? Progress on aspects of each question is obtained via statistically rigorous results to verify the empirical findings for an unprecedentedly large set of futures market LOB data. The data comprises several exchanges, several futures asset classes and all trading days of 2010, using market depth (Type II) order book data to 5 levels on the bid and ask

Approximating predictive probabilities of Gibbs-type priors

Gibbs-type random probability measures, or Gibbs-type priors, are arguably the most "natural" generalization of the celebrated Dirichlet prior. Among them the two parameter Poisson-Dirichlet prior certainly stands out for the mathematical tractability and interpretability of its predictive probabilities, which made it the natural candidate in several applications. Given a sample of size $n$, in this paper we show that the predictive probabilities of any Gibbs-type prior admit a large $n$ approximation, with an error term vanishing as $o(1/n)$, which maintains the same desirable features as the predictive probabilities of the two parameter Poisson-Dirichlet prior.Comment: 22 pages, 6 figures. Added posterior simulation study, corrected typo

The use of the comprehensive family of distributions for the regime switching ACD framework

In recent methodological work the well known ACD approach, originally introduced by Engle and Russell (1998), has been supplemented by the involvement of an unobservable stochastic process which accompanies the underlying process of durations via a discrete mixture of distributions. The Mixture ACD model, emanating from the specialized proposal of De Luca and Gallo (2004), has proved to be a moderate tool for description of financial duration data. The use of one and the same family of ordinary distributions has been common practice until now. Our contribution incites to use the rich parameterized comprehensive family of distributions which allows for interacting different distributional idiosyncrasies. JEL classification: C41, C22, C25, C51, G14

Generalized Mittag-Leffler Distributions and Processes for Applications in Astrophysics and Time Series Modeling

Geometric generalized Mittag-Leffler distributions having the Laplace transform $\frac{1}{1+\beta\log(1+t^\alpha)},00$ is introduced and its properties are discussed. Autoregressive processes with Mittag-Leffler and geometric generalized Mittag-Leffler marginal distributions are developed. Haubold and Mathai (2000) derived a closed form representation of the fractional kinetic equation and thermonuclear function in terms of Mittag-Leffler function. Saxena et al (2002, 2004a,b) extended the result and derived the solutions of a number of fractional kinetic equations in terms of generalized Mittag-Leffler functions. These results are useful in explaining various fundamental laws of physics. Here we develop first-order autoregressive time series models and the properties are explored. The results have applications in various areas like astrophysics, space sciences, meteorology, financial modeling and reliability modeling.Comment: 12 pages, LaTe

Generalized gamma approximation with rates for urns, walks and trees

We study a new class of time inhomogeneous P\'olya-type urn schemes and give optimal rates of convergence for the distribution of the properly scaled number of balls of a given color to nearly the full class of generalized gamma distributions with integer parameters, a class which includes the Rayleigh, half-normal and gamma distributions. Our main tool is Stein's method combined with characterizing the generalized gamma limiting distributions as fixed points of distributional transformations related to the equilibrium distributional transformation from renewal theory. We identify special cases of these urn models in recursive constructions of random walk paths and trees, yielding rates of convergence for local time and height statistics of simple random walk paths, as well as for the size of random subtrees of uniformly random binary and plane trees.Comment: Published at http://dx.doi.org/10.1214/15-AOP1010 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Distance statistics in random media: high dimension and/or high neighborhood order cases

Consider an unlimited homogeneous medium disturbed by points generated via Poisson process. The neighborhood of a point plays an important role in spatial statistics problems. Here, we obtain analytically the distance statistics to $k$th nearest neighbor in a $d$-dimensional media. Next, we focus our attention in high dimensionality and high neighborhood order limits. High dimensionality makes distance distribution behavior as a delta sequence, with mean value equal to Cerf's conjecture. Distance statistics in high neighborhood order converges to a Gaussian distribution. The general distance statistics can be applied to detect departures from Poissonian point distribution hypotheses as proposed by Thompson and generalized here.Comment: 5 pages and 2 figure

Generalized Negative Binomial Processes and the Representation of Cluster Structures

The paper introduces the concept of a cluster structure to define a joint distribution of the sample size and its exchangeable random partitions. The cluster structure allows the probability distribution of the random partitions of a subset of the sample to be dependent on the sample size, a feature not presented in a partition structure. A generalized negative binomial process count-mixture model is proposed to generate a cluster structure, where in the prior the number of clusters is finite and Poisson distributed and the cluster sizes follow a truncated negative binomial distribution. The number and sizes of clusters can be controlled to exhibit distinct asymptotic behaviors. Unique model properties are illustrated with example clustering results using a generalized Polya urn sampling scheme. The paper provides new methods to generate exchangeable random partitions and to control both the cluster-number and cluster-size distributions.Comment: 30 pages, 8 figure