Krzysztof Podgórski

Velocities for moving random surfaces

For a stationary two-dimensional random field evolving in time, we derive statistical distributions of appropriately defined velocities. The results are based on a generalization of the Rice formula. We discuss importance of identifying the correct form of the distribution which accounts for the sampling bias. The theory can be applied to practical problems where evolving random fields are considered to be adequate models. Examples include changes of atmospheric pressure, variation of air pollution, or dynamical models of the sea surface elevation. We study the last application in more detail by applying the derived results to Gaussian fields representing irregular sea surfaces. In particular, we study statistical properties of velocities both for the sea surface and for the envelope field based on this surface. The latter is better fitted to study wave group velocities and is of particular interest for engineering applications. For wave and wave group velocities, numerical computations of distributions are presented and illustrated graphically.

Exact distributions for apparent waves in irregular seas

We discuss the long-run distributions of several characteristics for the apparent waves in a Gaussian sea. Three types of one-dimensional wave records are considered: 1) the seaway in time at a fixed position; 2) the instantaneous profile along a horizontal line; 3) the encountered seaway. Exact integral forms of the joint long run distributions are derived for the apparent periods, lengths, and heights. Results of numerical approximations of these distributions are presented in examples. For the computations we considered, as the input spectra, empirical estimates of the frequency spectra as well as JONSWAP type spectra. Effective algorithms are discussed and utilized in the form of a comprehensive computer package of numerical routines.

Maximum Likelihood Estimation of Asymmetric Laplace Parameters

Maximum likelihood estimators (MLEs) are presented for the parameters of a univariate asymmetric Laplace distribution for all possible situations related to known or unknown parameters. These estimators admit explicit form in all but two cases. In these exceptions effective algorithms for computing the estimators are provided. Asymptotic distributions of the estimators are given. The asymptotic normality and consistency of the MLEs for the scale and location parameters are derived directly via representations of the relevant random variables rather than from general sufficient conditions for asymptotic normality of the MLEs.

Asymmetric Multivariate Laplace Distribution

We present a class of multivariate laws which is an extension of the symmetric multivariate Laplace distributions and of the univariate asymmetric Laplace distributions. The extension retains natural, asymmetric and multivariate, properties characterizing these two subclasses. The results include characterizations, mixture representations, formulas for densities and moments, and a simulation algorithm. The new family can be viewed as a subclass of hyperbolic distributions.

Asymmetric Laplace Distributions

Chapter 3 is devoted to asymmetric Laplace distributions — a skewed family of distributions that in our opinion is the most appropriate skewed generalization of the classical Laplace law. In the last several decades, various forms of skewed Laplace distributions have sporadically appeared in the literature. One of the earliest is due to McGill (1962), who considers distributions with p.d.f. n n

Classical Symmetric Laplace Distribution

f(x) = left{ {begin{array}{*{20}c} {frac{{varphi _1 }} {2}e^{ - varphi _1 |x - theta |} , x leqslant theta ,} {frac{{varphi _2 }} {2}e^{ - varphi _2 |x - theta |} , x > theta ,} end{array} } right.

STATISTICAL PROPERTIES OF ENVELOPE FIELD FOR GAUSSIAN SEA SURFACE

n n(3.0.1) n nwhile Holla and Bhattacharya (1968) study the distribution with p.d.f. n n