1,624 research outputs found
Asymptotics of the discrete log-concave maximum likelihood estimator and related applications
The assumption of log-concavity is a flexible and appealing nonparametric
shape constraint in distribution modelling. In this work, we study the
log-concave maximum likelihood estimator (MLE) of a probability mass function
(pmf). We show that the MLE is strongly consistent and derive its pointwise
asymptotic theory under both the well- and misspecified setting. Our asymptotic
results are used to calculate confidence intervals for the true log-concave
pmf. Both the MLE and the associated confidence intervals may be easily
computed using the R package logcondiscr. We illustrate our theoretical results
using recent data from the H1N1 pandemic in Ontario, Canada.Comment: 21 pages, 7 Figure
On a flexible construction of a negative binomial model
This work presents a construction of stationary Markov models with
negative-binomial marginal distributions. A simple closed form expression for
the corresponding transition probabilities is given, linking the proposal to
well-known classes of birth and death processes and thus revealing interesting
characterizations. The advantage of having such closed form expressions is
tested on simulated and real data.Comment: Forthcoming in "Statistics & Probability Letters
Approximation of ruin probabilities via Erlangized scale mixtures
In this paper, we extend an existing scheme for numerically calculating the
probability of ruin of a classical Cram\'er--Lundberg reserve process having
absolutely continuous but otherwise general claim size distributions. We employ
a dense class of distributions that we denominate Erlangized scale mixtures
(ESM) and correspond to nonnegative and absolutely continuous distributions
which can be written as a Mellin--Stieltjes convolution of a
nonnegative distribution with an Erlang distribution . A distinctive
feature of such a class is that it contains heavy-tailed distributions.
We suggest a simple methodology for constructing a sequence of distributions
having the form to approximate the integrated tail distribution of
the claim sizes. Then we adapt a recent result which delivers an explicit
expression for the probability of ruin in the case that the claim size
distribution is modelled as an Erlangized scale mixture. We provide simplified
expressions for the approximation of the probability of ruin and construct
explicit bounds for the error of approximation. We complement our results with
a classical example where the claim sizes are heavy-tailed
The MM Alternative to EM
The EM algorithm is a special case of a more general algorithm called the MM
algorithm. Specific MM algorithms often have nothing to do with missing data.
The first M step of an MM algorithm creates a surrogate function that is
optimized in the second M step. In minimization, MM stands for
majorize--minimize; in maximization, it stands for minorize--maximize. This
two-step process always drives the objective function in the right direction.
Construction of MM algorithms relies on recognizing and manipulating
inequalities rather than calculating conditional expectations. This survey
walks the reader through the construction of several specific MM algorithms.
The potential of the MM algorithm in solving high-dimensional optimization and
estimation problems is its most attractive feature. Our applications to random
graph models, discriminant analysis and image restoration showcase this
ability.Comment: Published in at http://dx.doi.org/10.1214/08-STS264 the Statistical
Science (http://www.imstat.org/sts/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Algorithms for operations on probability distributions in a computer algebra system
In mathematics and statistics, the desire to eliminate mathematical tedium and facilitate exploration has lead to computer algebra systems. These computer algebra systems allow students and researchers to perform more of their work at a conceptual level. The design of generic algorithms for tedious computations allows modelers to push current modeling boundaries outward more quickly.;Probability theory, with its many theorems and symbolic manipulations of random variables is a discipline in which automation of certain processes is highly practical, functional, and efficient. There are many existing statistical software packages, such as SPSS, SAS, and S-Plus, that have numeric tools for statistical applications. There is a potential for a probability package analogous to these statistical packages for manipulation of random variables. The software package being developed as part of this dissertation, referred to as A Probability Programming Language (APPL) is a random variable manipulator and is proposed to fill a technology gap that exists in probability theory.;My research involves developing algorithms for the manipulation of discrete random variables. By defining data structures for random variables and writing algorithms for implementing common operations, more interesting and mathematically intractable probability problems can be solved, including those not attempted in undergraduate statistics courses because they were deemed too mechanically arduous. Algorithms for calculating the probability density function of order statistics, transformations, convolutions, products, and minimums/maximums of independent discrete random variables are included in this dissertation
Extended Object Tracking: Introduction, Overview and Applications
This article provides an elaborate overview of current research in extended
object tracking. We provide a clear definition of the extended object tracking
problem and discuss its delimitation to other types of object tracking. Next,
different aspects of extended object modelling are extensively discussed.
Subsequently, we give a tutorial introduction to two basic and well used
extended object tracking approaches - the random matrix approach and the Kalman
filter-based approach for star-convex shapes. The next part treats the tracking
of multiple extended objects and elaborates how the large number of feasible
association hypotheses can be tackled using both Random Finite Set (RFS) and
Non-RFS multi-object trackers. The article concludes with a summary of current
applications, where four example applications involving camera, X-band radar,
light detection and ranging (lidar), red-green-blue-depth (RGB-D) sensors are
highlighted.Comment: 30 pages, 19 figure
Capacity Upper Bounds for Deletion-Type Channels
We develop a systematic approach, based on convex programming and real
analysis, for obtaining upper bounds on the capacity of the binary deletion
channel and, more generally, channels with i.i.d. insertions and deletions.
Other than the classical deletion channel, we give a special attention to the
Poisson-repeat channel introduced by Mitzenmacher and Drinea (IEEE Transactions
on Information Theory, 2006). Our framework can be applied to obtain capacity
upper bounds for any repetition distribution (the deletion and Poisson-repeat
channels corresponding to the special cases of Bernoulli and Poisson
distributions). Our techniques essentially reduce the task of proving capacity
upper bounds to maximizing a univariate, real-valued, and often concave
function over a bounded interval. We show the following:
1. The capacity of the binary deletion channel with deletion probability
is at most for , and, assuming the capacity
function is convex, is at most for , where
is the golden ratio. This is the first nontrivial
capacity upper bound for any value of outside the limiting case
that is fully explicit and proved without computer assistance.
2. We derive the first set of capacity upper bounds for the Poisson-repeat
channel.
3. We derive several novel upper bounds on the capacity of the deletion
channel. All upper bounds are maximums of efficiently computable, and concave,
univariate real functions over a bounded domain. In turn, we upper bound these
functions in terms of explicit elementary and standard special functions, whose
maximums can be found even more efficiently (and sometimes, analytically, for
example for ).
Along the way, we develop several new techniques of potentially independent
interest in information theory, probability, and mathematical analysis.Comment: Minor edits, In Proceedings of 50th Annual ACM SIGACT Symposium on
the Theory of Computing (STOC), 201
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