1.
Nonparametric estimation of the chord length distribution - Hansen, Martin B.; van Zwet, Erik W.
The distribution of the length of a typical chord of a stationary random set is an
interesting feature of the set's whole distribution. We give a nonparametric estimator
of the chord length distribution and prove its strong consistency. We report on a
simulation experiment in which our estimator compared favourably to a reduced sample
estimator. Both estimators are illustrated by applying them to an image sample from a
yoghurt ferment. We briefly discuss the closely related problem of estimation of the
linear contact distribution. We show by a simulation experiment that a transformation
of our estimator of the chord length distribution is more efficient than a
Kaplan-Meier type...
2.
Estimation of orientation characteristic of fibrous material - Kärkkäinnen, Salme; Penttinen, Antti; Ushakov, Nikolai G.; Ushakova, Alexandra P.
A new statistical method for estimating the orientation distribution of fibres in a
fibre process is suggested where the process is observed in the form of a degraded
digital greyscale image. The method is based on line transect sampling of the image in
a few fixed directions. A well-known method based on stereology is available if the
intersections between the transects and fibres can be counted. We extend this to the
case where, instead of the intersection points, only scaled variograms of grey levels
along the transects are observed. The nonlinear estimation equations for a parametric
orientation distribution as well as a numerical algorithm are given. The...
3.
The specific connectivity number of random networks - Mecke, Joseph; Stoyan, Dietrich
A network is a system of segments or edges in Rd which
intersect only in the segment endpoints, which are called vertices. An
example is the system of edges of a tessellation. It is possible to
give formulas for the specific connectivity number of a random network;
in the stationary case, the intensity of the 0-curvature measure is
equal to the difference of the intensities of the point processes of
vertices and edge centres.
4.
On the rose of intersections of stationary flat processes - Spodarev, Eugene
The paper yields retrieval formulae of the directional distribution of
a stationary k-flat process in Rd if its rose of intersections with
all r-flats is known. Cases k = d -1, 1 ? r ? d - 1 for
arbitrary d and d = 4, k = 2, r = 2 are considered. Some
generalizations to manifold processes in Rd are made. The proofs use
the methods of harmonic analysis on higher Grassmannians (spherical
harmonics, integral transforms).
5.
Post-data inference of coalescence times and segregating-site distribution in a two-island model with symmetric migration - Gorroochurn, P.
In this paper, we present the distribution of the coalescence time of two DNA
sequences (or genes) subject to symmetric migration between two islands, and
conditional on the observed number of segregating sites in the sequences. The
distribution for the segregating-site pattern is also obtained. Some surprising
results emerge when both genes are initially on the same island. First, the post-data
mean coalescence time is shown to be dependent on the migration parameter, as opposed
to the pre-data mean. Second, both the post-data density and expectation for the
coalescence time are shown to converge, in the weak-migration limit, to the
corresponding panmictic results, as opposed to the pre-data...
6.
Analogies and correspondences between variograms and covariance functions - Gneiting, Tilmann; Sasvári, Zoltán; Schlather, Martin
Variograms and covariance functions are key tools in
geostatistics. However, various properties, characterizations, and
decomposition theorems have been established for covariance functions
only. We present analogous results for variograms and explore the
connections with covariance functions. Our findings include criteria
for covariance functions on intervals, and we apply them to
exponential models, fractional Brownian motion, and locally polynomial
covariances. In particular, we characterize isotropic locally
polynomial covariance functions of degree 3.
7.
Shot noise on cluster processes with cluster marks, and studies of long range dependence - Ramirez-Perez, Filemon; Serfling, Robert
With the aim of providing greater flexibility in developing and applying shot noise
models, this paper studies shot noise on cluster point processes with both pointwise
and cluster marks. For example, in financial modelling, responses to events in the
financial market may occur in clusters, with random amplitudes including a `cluster
component' reflecting a degree of commonness among responses within a cluster. For
such shot noise models, general formulae for the characteristic functional are
developed and specialized to the case of Neyman-Scott clustering with cluster marks.
For several general forms of response function, long range dependence of the
corresponding equilibrium shot noise models is investigated. It is shown,...
8.
Asymptotic expansions in multidimensional Markov renewal theory and first passage times for Markov random walks - Fuh, Cheng-Der; Lai, Tze Leung
We prove a d-dimensional renewal theorem, with an estimate on the rate of
convergence, for Markov random walks. This result is applied to a variety of boundary
crossing problems for a Markov random walk {(Xn,Sn) , n ?0}, in which Xn
takes values in a general state space and Sn takes values in Rd. In
particular, for the case d = 1, we use this result to derive an asymptotic formula for
the variance of the first passage time when Sn exceeds a high threshold b,
generalizing Smith's classical formula in the case of i.i.d. positive increments for
Sn. For d > 1, we apply this result...
9.
Lundberg inequalities for renewal equations - Willmot, Gordon E.; Cai, Jun; Lin, X. Sheldon
Sharp upper and lower bounds are derived for the solution of renewal equations. These
include as special cases exponential inequalities, some of which have been derived for
specific renewal equations. Together with the well-known Cramér-Lundberg
asymptotic estimate, these bounds give additional information about the behaviour of
the solution. Nonexponential bounds, which are of use in connection with defective
renewal equations, are also obtained. The results are then applied in examples
involving the severity of insurance ruin, age-dependent branching processes, and a
generalized type II Geiger counter.
10.
On random motions with velocities alternating at Erlang-distributed random times - Di Crescenzo, Antonio
We analyse a non-Markovian generalization of the telegrapher's random process. It
consists of a stochastic process describing a motion on the real line characterized by
two alternating velocities with opposite directions, where the random times separating
consecutive reversals of direction perform an alternating renewal process. In the case
of Erlang-distributed interrenewal times, explicit expressions of the transition
densities are obtained in terms of a suitable two-index pseudo-Bessel function. Some
results on the distribution of the maximum of the process are also disclosed.
11.
Calculation of noncrossing probabilities for Poisson processes and its corollaries - Khmaladze, Estate; Shinjikashvili, Eka
The paper describes a new numerical method for the calculation of noncrossing
probabilities for arbitrary boundaries by a Poisson process. We find the method to be
simple in implementation, quick and efficient - it works reliably for Poisson
processes of very high intensity n, up to several thousand. Hence, it can be used to
detect unusual features in the finite-sample behaviour of empirical process and trace
it down to very high sample sizes. It also can be used as a good approximation for
noncrossing probabilities for Brownian motion and Brownian bridge, in particular when
the boundaries are not regular. As a numerical example we demonstrate the divergence
of...
12.
On the estimation of a star-shaped set - Baíllo, Amparo; Cuevas, Antonio
The estimation of a star-shaped set S from a random sample
of points X1,...,Xn is considered.
We show that S can be consistently approximated (with respect to both
the Hausdorff metric and the `distance in measure' between sets) by an estimator
?Sn defined as a union of balls centered at the
sample points with a common radius which can be chosen in such a way that
?Sn is also star-shaped. We also prove that,
under some mild conditions, the topological boundary of the estimator
?Sn converges, in the Hausdorff sense, to that
of S; this has a particular interest when the proposed estimation problem
is considered from the point of...
13.
The number of two-dimensional maxima - Barbour, A. D.; Xia, A.
Let n points be placed uniformly at random in a subset A of the plane. A point is
said to be maximal in the configuration if no other point is larger in both coordinates.
We show that, for large n and for many sets A, the number of maximal points is
approximately normally distributed. The argument uses Stein's method, and is also
applicable in higher dimensions.
14.
Central limit theorem for germination-growth models in R
d
with non-Poisson locations - Chiu, S. N.; Quine, M. P.
Seeds are randomly scattered in Rd according to an m-dependent point
process. Each seed has its own potential germination time. From each seed that succeeds
in germinating, a spherical inhibited region grows to prohibit germination of any seed
with later potential germination time. We show that under certain conditions on the
distribution of the potential germination time, the number of germinated seeds in a large
region has an asymptotic normal distribution.
15.
On the almost sure coverage property of Voronoi tessellation: the R1 case - Khmaladze, Estate; Toronjadze, N.
This paper raises the following question: let
{?n(A), A ? Rd}
be a Poisson process with intensity nf(x),
x ? Rd and let
c(Xi | ?n) be a Voronoi tile
with nucleus Xi (a jump point of ?n).
Let ?(.) denote Lebesgue measure in Rd. Is it true that,
for any bounded measurable subset B of Rd,
?Xi?B?(c(Xi
| ?n)) ? ?(B) almost surely
as n ? ? only if f > 0 almost everywhere? This
statement can be viewed as the strong law of large numbers for Voronoi tessellation.
Though the positive answer may seem `obvious', we could not find any such statement,
especially for arbitrary measurable B and nonhomogeneous Poisson processes....
16.
Estimation variances for Poisson processes of compact sets - Mrkvi?ka, Tomá
A complete and sufficient statistic is found for various stationary Poisson processes of
compact sets with known primary grain. In the particular case of a segment process, the
uniformly best unbiased estimator for the length density is the number of segments
hitting the sampling window divided by a certain constant and multiplied by the mean
segment length.
17.
Testing for signals with unknown location and scale in a
?2 random field, with an application to fMRI - Worsley, Keith J.
Siegmund and Worsley (1995) considered the problem of testing for
signals with unknown location and scale in a Gaussian random field defined on
RN. The test statistic was the maximum of a Gaussian
random field in an N+1 dimensional `scale space', N dimensions
for location and 1 dimension for the scale of a smoothing filter. Scale space
is identical to a continuous wavelet transform with a kernel smoother as the
wavelet, though the emphasis here is on signal detection rather than image
compression or enhancement. Two methods were used to derive an approximate null
distribution for N=2 and N=3: one based on the method of volumes
of tubes, the...
18.
On filtering in Markovian term structure - Chiarella, Carl; Pasquali, Sara; Runggaldier, Wolfgang J.
We consider a parametrization of the Heath-Jarrow-Morton (HJM)
family of term structure of interest rate models that allows a
finite-dimensional Markovian representation of the stochastic
dynamics. This parametrization results from letting the volatility
function depend on time to maturity and on two factors: the
instantaneous spot rate and one fixed-maturity forward rate. Our
main purpose is an estimation methodology for which we have to
model the observations under the historical probability measure.
This leads us to consider as an additional third factor the market
price of interest rate risk, that connects the historical and the
HJM martingale measures. Assuming that the information comes from
noisy observations of the fixed-maturity forward rate,...
19.
Lagrangian observations of homogeneous random environments - Zirbel, Craig L.
This article deals with the distribution of the view of a random environment as seen
by an observer whose location at each moment is determined by the environment. The
main application is in statistical fluid mechanics, where the environment consists of
a random velocity field and the observer is a particle moving in the velocity field,
possibly subject to molecular diffusion. Several results on such Lagrangian
observations of the environment have appeared in the literature, beginning with the
1957 dissertation of J. L. Lumley. This article unites these results into a simple
unified framework and rounds out the theory with new results in several directions.
When the environment...
20.
On the asymptotic relationship between the overflow probability and the loss ratio - Kim, Han S.; Shroff, Ness B.
In this paper we study the asymptotic relationship between the
loss ratio in a finite buffer system and the overflow probability (the tail
of the queue length distribution) in the corresponding infinite buffer system.
We model the system by a fluid queue which consists of a server with constant
rate c and a fluid input. We provide asymptotic upper and lower bounds
on the difference between log P{Q > x} and
logPL(x) under different conditions. The
conditions for the upper bound are simple and are satisfied by a very large
class of input processes. The conditions on the lower bound are more complex
but we show that various classes...
1.
