72 research outputs found
Optimal design of dilution experiments under volume constraints
The paper develops methods to construct a one-stage optimal design of
dilution experiments under the total available volume constraint typical for
bio-medical applications. We consider various design criteria based on the
Fisher information both is Bayesian and non-Bayasian settings and show that the
optimal design is typically one-atomic meaning that all the dilutions should be
of the same size. The main tool is variational analysis of functions of a
measure and the corresponding steepest descent type numerical methods. Our
approach is generic in the sense that it allows for inclusion of additional
constraints and cost components, like the cost of materials and of the
experiment itself.Comment: 29 pages, 10 figure
Bit flipping and time to recover
We call `bits' a sequence of devices indexed by positive integers, where
every device can be in two states: (idle) and (active). Start from the
`ground state' of the system when all bits are in -state. In our first
Binary Flipping (BF) model, the evolution of the system is the following: at
each time step choose one bit from a given distribution on the
integers independently of anything else, then flip the state of this bit to the
opposite. In our second Damaged Bits (DB) model a `damaged' state is added:
each selected idling bit changes to active, but selecting an active bit changes
its state to damaged in which it then stays forever.
In both models we analyse the recurrence of the system's ground state when no
bits are active. We present sufficient conditions for both BF and DB models to
show recurrent or transient behaviour, depending on the properties of
. We provide a bound for fractional moments of the return time to
the ground state for the BF model, and prove a Central Limit Theorem for the
number of active bits for both models
Stability for random measures, point processes and discrete semigroups
Discrete stability extends the classical notion of stability to random
elements in discrete spaces by defining a scaling operation in a randomised
way: an integer is transformed into the corresponding binomial distribution.
Similarly defining the scaling operation as thinning of counting measures we
characterise the corresponding discrete stability property of point processes.
It is shown that these processes are exactly Cox (doubly stochastic Poisson)
processes with strictly stable random intensity measures. We give spectral and
LePage representations for general strictly stable random measures without
assuming their independent scattering. As a consequence, spectral
representations are obtained for the probability generating functional and void
probabilities of discrete stable processes. An alternative cluster
representation for such processes is also derived using the so-called Sibuya
point processes, which constitute a new family of purely random point
processes. The obtained results are then applied to explore stable random
elements in discrete semigroups, where the scaling is defined by means of
thinning of a point process on the basis of the semigroup. Particular examples
include discrete stable vectors that generalise discrete stable random
variables and the family of natural numbers with the multiplication operation,
where the primes form the basis.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ301 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Nonparametric estimation of infinitely divisible distributions based on variational analysis on measures
The paper develops new methods of non-parametric estimation a compound
Poisson distribution. Such a problem arise, in particular, in the inference of
a Levy process recorded at equidistant time intervals. Our key estimator is
based on series decomposition of functionals of a measure and relies on the
steepest descent technique recently developed in variational analysis of
measures. Simulation studies demonstrate applicability domain of our methods
and how they positively compare and complement the existing techniques. They
are particularly suited for discrete compounding distributions, not necessarily
concentrated on a grid nor on the positive or negative semi-axis. They also
give good results for continuous distributions provided an appropriate
smoothing is used for the obtained atomic measure
On the capacity functional of the infinite cluster of a Boolean model
Consider a Boolean model in with balls of random, bounded radii with
distribution , centered at the points of a Poisson process of intensity
. The capacity functional of the infinite cluster is given by
\theta_L(t) = \BP\{ Z_\infty\cap L\ne\emptyset\}, defined for each compact
.
We prove for any fixed and that is infinitely
differentiable in , except at the critical value ; we give a
Margulis-Russo type formula for the derivatives. More generally, allowing the
distribution to vary and viewing as a function of the measure
, we show that it is infinitely differentiable in all directions with
respect to the measure in the supercritical region of the cone of positive
measures on a bounded interval.
We also prove that grows at least linearly at the critical
value. This implies that the critical exponent known as is at most 1
(if it exists) for this model. Along the way, we extend a result of H.~Tanemura
(1993), on regularity of the supercritical Boolean model in with
fixed-radius balls, to the case with bounded random radii.Comment: 23 pages, 24 references, 1 figure in Annals of Applied Probability,
201
Branching-stable point processes
The notion of stability can be generalised to point processes by defining the scaling operation in a randomised way: scaling a configuration by t corresponds to letting such a configuration evolve according to a Markov branching particle system for -log(t) time. We prove that these are the only stochastic operations satisfying basic associativity and distributivity properties and we thus introduce the notion of branching-stable point processes. For scaling operations corresponding to particles that branch but do not diffuse, we characterise stable distributions as thinning stable point processes with multiplicities given by the quasi stationary (or Yaglom) distribution of the branching process under consideration. Finally we extend branching-stability to continuous random variables with the help of continuous branching (CB) processes, and we show that, at least in some frameworks, branching-stable integer random variables are exactly Cox (doubly stochastic Poisson) random variables driven by corresponding CB-stable continuous random variables
Populations in environments with a soft carrying capacity are eventually extinct
Consider a population whose size changes stepwise by its members reproducing
or dying (disappearing), but is otherwise quite general. Denote the initial
(non-random) size by and the size of the th change by , . Population sizes hence develop successively as $Z_1=Z_0+C_1,\
Z_2=Z_1+C_2Z_n=0Z_{n+1}=0, without there being any other finite absorbing class
of population sizes. We make no assumptions about the time durations between
the successive changes. In the real world, or more specific models, those may
be of varying length, depending upon individual life span distributions and
their interdependencies, the age-distribution at hand and intervening
circumstances. Changes may have quite varying distributions. The basic
assumption is that there is a {\em carrying capacity}, i.e. a non-negative
number KC_n$ equals -1 (one individual dying) with a conditional (given the
past) probability uniformly bounded away from 0. It is a simple and not very
restrictive way to avoid parity phenomena, it is related to irreducibility in
Markov settings. The straightforward, but in contents and implications
far-reaching, consequence is that all such populations must die out.
Mathematically, it follows by a submartingale convergence property and positive
probability of reaching the absorbing extinction state.Comment: To appear in J.Math.Bio
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