2,963 research outputs found
A two-step approach to model precipitation extremes in California based on max-stable and marginal point processes
In modeling spatial extremes, the dependence structure is classically
inferred by assuming that block maxima derive from max-stable processes.
Weather stations provide daily records rather than just block maxima. The point
process approach for univariate extreme value analysis, which uses more
historical data and is preferred by some practitioners, does not adapt easily
to the spatial setting. We propose a two-step approach with a composite
likelihood that utilizes site-wise daily records in addition to block maxima.
The procedure separates the estimation of marginal parameters and dependence
parameters into two steps. The first step estimates the marginal parameters
with an independence likelihood from the point process approach using daily
records. Given the marginal parameter estimates, the second step estimates the
dependence parameters with a pairwise likelihood using block maxima. In a
simulation study, the two-step approach was found to be more efficient than the
pairwise likelihood approach using only block maxima. The method was applied to
study the effect of El Ni\~{n}o-Southern Oscillation on extreme precipitation
in California with maximum daily winter precipitation from 35 sites over 55
years. Using site-specific generalized extreme value models, the two-step
approach led to more sites detected with the El Ni\~{n}o effect, narrower
confidence intervals for return levels and tighter confidence regions for risk
measures of jointly defined events.Comment: Published at http://dx.doi.org/10.1214/14-AOAS804 in the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Statistics for the Luria-Delbr\"uck distribution
The Luria-Delbr\"uck distribution is a classical model of mutations in cell
kinetics. It is obtained as a limit when the probability of mutation tends to
zero and the number of divisions to infinity. It can be interpreted as a
compound Poisson distribution (for the number of mutations) of exponential
mixtures (for the developing time of mutant clones) of geometric distributions
(for the number of cells produced by a mutant clone in a given time). The
probabilistic interpretation, and a rigourous proof of convergence in the
general case, are deduced from classical results on Bellman-Harris branching
processes. The two parameters of the Luria-Delbr\"uck distribution are the
expected number of mutations, which is the parameter of interest, and the
relative fitness of normal cells compared to mutants, which is the heavy tail
exponent. Both can be simultaneously estimated by the maximum likehood method.
However, the computation becomes numerically unstable as soon as the maximal
value of the sample is large, which occurs frequently due to the heavy tail
property. Based on the empirical generating function, robust estimators are
proposed and their asymptotic variance is given. They are comparable in
precision to maximum likelihood estimators, with a much broader range of
calculability, a better numerical stability, and a negligible computing time
Modeling Asset Prices
As an asset is traded, its varying prices trace out an interesting time series. The price, at least in a general way, reflects some underlying value of the asset. For most basic assets, realistic models of value must involve many variables relating not only to the individual asset, but also to the asset class, the industrial sector(s) of the asset, and both the local economy and the general global economic conditions. Rather than attempting to model the value, we will confine our interest to modeling the price. The underlying assumption is that the price at which an asset trades is a "fair market price" that reflects the actual value of the asset. Our initial interest is in models of the price of a basic asset, that is, not the price of a derivative asset. Usually instead of the price itself, we consider the relative change in price, that is, the rate of return, over some interval of time. The purpose of asset pricing models is not for prediction of future prices; rather the purpose is to provide a description of the stochastic behavior of prices. Models of price changes have a number of uses, including, for investors, optimal construction of portfolios of assets and, for market regulators, maintaining a fair and orderly market. A major motivation for developing models of price changes of given assets is to use those models to develop models of fair value of derivative assets that depend on the given assets.Discrete time series models, continuous time diffusion models, models with jumps, stochastic volatility, GARCH
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