15 research outputs found
Approximation and Error Analysis of Forward-Backward SDEs driven by General L\'evy Processes using Shot Noise Series Representations
We consider the simulation of a system of decoupled forward-backward
stochastic differential equations (FBSDEs) driven by a pure jump L\'evy process
and an independent Brownian motion . We allow the L\'evy process to
have an infinite jump activity. Therefore, it is necessary for the simulation
to employ a finite approximation of its L\'evy measure. We use the generalized
shot noise series representation method by Rosinski (2001) to approximate the
driving L\'evy process . We compute the error, , between the
true and the approximated FBSDEs which arises from the finite truncation of the
shot noise series (given sufficient conditions for existence and uniqueness of
the FBSDE). We also derive the error between the true solution and the
discretization of the approximated FBSDE using an appropriate backward Euler
scheme
Parametric Estimation of Tempered Stable Laws
Tempered stable distributions are frequently used in financial applications
(e.g., for option pricing) in which the tails of stable distributions would be
too heavy. Given the non-explicit form of the probability density function,
estimation relies on numerical algorithms which typically are time-consuming.
We compare several parametric estimation methods such as the maximum likelihood
method and different generalized method of moment approaches. We study large
sample properties and derive consistency, asymptotic normality, and asymptotic
efficiency results for our estimators. Additionally, we conduct simulation
studies to analyze finite sample properties measured by the empirical bias,
precision, and asymptotic confidence interval coverage rates and compare
computational costs. We cover relevant subclasses of tempered stable
distributions such as the classical tempered stable distribution and the
tempered stable subordinator. Moreover, we discuss the normal tempered stable
distribution which arises by subordinating a Brownian motion with a tempered
stable subordinator. Our financial applications to log returns of asset indices
and to energy spot prices illustrate the benefits of tempered stable models
Student't mixture models for stock indices. A comparative study
We perform a comparative study for multiple equity indices of different
countries using different models to determine the best fit using the
Kolmogorov-Smirnov statistic, the Anderson-Darling statistic, the Akaike
information criterion and the Bayesian information criteria as goodness-of-fit
measures. We fit models both to daily and to hourly log-returns. The main
result is the excellent performance of a mixture of three Student's
distributions with the numbers of degrees of freedom fixed a priori (3St). In
addition, we find that the different components of the 3St mixture with
small/moderate/high degree of freedom parameter describe the
extreme/moderate/small log-returns of the studied equity indices
On the parametric description of log-growth rates of cities' sizes of four European countries and the USA
We have studied the parametric description of the distribution of the
log-growth rates of the sizes of cities of France, Germany, Italy, Spain and
the USA. We have considered several parametric distributions well known in the
literature as well as some others recently introduced. There are some models
that provide similar excellent performance, for all studied samples. The normal
distribution is not the one observed empirically
Composite distributions in the social sciences: A comparative empirical study of firms' sales distribution for France, Germany, Italy, Japan, South Korea, and Spain
We study 17 different statistical distributions for sizes obtained {}from the
classical and recent literature to describe a relevant variable in the social
sciences and Economics, namely the firms' sales distribution in six countries
over an ample period. We find that the best results are obtained with mixtures
of lognormal (LN), loglogistic (LL), and log Student's (LSt) distributions.
The single lognormal, in turn, is strongly not selected. We then find that the
whole firm size distribution is better described by a mixture, and there exist
subgroups of firms. Depending on the method of measurement, the best fitting
distribution cannot be defined by a single one, but as a mixture of at least
three distributions or even four or five. We assess a full sample analysis, an
in-sample and out-of-sample analysis, and a doubly truncated sample analysis.
We also provide the formulation of the preferred models as solutions of the
Fokker--Planck or forward Kolmogorov equation
Effects of Early Warning Emails on Student Performance
We use learning data of an e-assessment platform for an introductory
mathematical statistics course to predict the probability of passing the final
exam for each student. Subsequently, we send warning emails to students with a
low predicted probability to pass the exam. We detect a positive but
imprecisely estimated effect of this treatment, suggesting the effectiveness of
such interventions only when administered more intensively.Comment: arXiv admin note: text overlap with arXiv:1906.0986
Approximation and error analysis of forward–backward SDEs driven by general Lévy processes using shot noise series representations
We consider the simulation of a system of decoupled forward–backward stochastic differential equations (FBSDEs) driven by a pure jump Lévy process L and an independent Brownian motion B. We allow the Lévy process L to have an infinite jump activity. Therefore, it is necessary for the simulation to employ a finite approximation of its Lévy measure. We use the generalized shot noise series representation method by [26] to approximate the driving Lévy process L. We compute the Lp error, p ≥ 2, between the true and the approximated FBSDEs which arises from the finite truncation of the shot noise series (given sufficient conditions for existence and uniqueness of the FBSDE). We also derive the Lp error between the true solution and the discretization of the approximated FBSDE using an appropriate backward Euler scheme
E-Assessment Using Variable-Content Exercises in Mathematical Statistics
Computer-assisted assessment (CAA) is widely used in modern university courses in many different fields. It can be used both in formative and summative assessments and with different emphasis on (self-)training, grading, and feedback generation. This article reports on experiences from using the CAA tool “JACK” to support a university lecture on mathematical statistics for exercises, tests, and exams. We show and discuss, among others, a positive relationship between usage intensity of JACK and final grades. Moreover, students generally report to invest more time into studying when offered a CAA, and to be satisfied with such a setup