2,325 research outputs found
Constrained realizations and minimum variance reconstruction of non-Gaussian random fields
With appropriate modifications, the Hoffman--Ribak algorithm that constructs
constrained realizations of Gaussian random fields having the correct ensemble
properties can also be used to construct constrained realizations of those
non-Gaussian random fields that are obtained by transformations of an
underlying Gaussian field. For example, constrained realizations of lognormal,
generalized Rayleigh, and chi-squared fields having degrees of freedom
constructed this way will have the correct ensemble properties. The lognormal
field is considered in detail. For reconstructing Gaussian random fields,
constrained realization techniques are similar to reconstructions obtained
using minimum variance techniques. A comparison of this constrained realization
approach with minimum variance, Wiener filter reconstruction techniques, in the
context of lognormal random fields, is also included. The resulting
prescriptions for constructing constrained realizations as well as minimum
variance reconstructions of lognormal random fields are useful for
reconstructing masked regions in galaxy catalogues on smaller scales than
previously possible, for assessing the statistical significance of small-scale
features in the microwave background radiation, and for generating certain
non-Gaussian initial conditions for -body simulations.Comment: 12 pages, gzipped postscript, MNRAS, in pres
The invariances of power law size distributions
Size varies. Small things are typically more frequent than large things. The
logarithm of frequency often declines linearly with the logarithm of size. That
power law relation forms one of the common patterns of nature. Why does the
complexity of nature reduce to such a simple pattern? Why do things as
different as tree size and enzyme rate follow similarly simple patterns? Here I
analyze such patterns by their invariant properties. For example, a common
pattern should not change when adding a constant value to all observations.
That shift is essentially the renumbering of the points on a ruler without
changing the metric information provided by the ruler. A ruler is shift
invariant only when its scale is properly calibrated to the pattern being
measured. Stretch invariance corresponds to the conservation of the total
amount of something, such as the total biomass and consequently the average
size. Rotational invariance corresponds to pattern that does not depend on the
order in which underlying processes occur, for example, a scale that additively
combines the component processes leading to observed values. I use tree size as
an example to illustrate how the key invariances shape pattern. A simple
interpretation of common pattern follows. That simple interpretation connects
the normal distribution to a wide variety of other common patterns through the
transformations of scale set by the fundamental invariances.Comment: Added appendix discussing the lognormal distribution, updated to
match version 2 of published version at F1000Researc
Higher Moments of the Claims Development Result in General Insurance
The claims development result (CDR) is one of the major risk drivers in the profit and loss statement of a general insurance company. Therefore, the CDR has become a central object of interest under new solvency regulation. In current practice, simple methods based on the first two moments of the CDR are implemented to find a proxy for the distribution of the CDR. Such approximations based on the first two moments are rather rough and may fail to appropriately describe the shape of the distribution of the CDR. In this paper we provide an analysis of higher moments of the CDR. Within a Bayes chain ladder framework we consider two different models for which it is possible to derive analytical solutions for the higher moments of the CDR. Based on higher moments we can e.g. calculate the skewness and the excess kurtosis of the distribution of the CDR and obtain refined approximations. Moreover, a case study investigates and answers questions raised in IAS
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