Variability of distributions of wave set-up heights along a shoreline with complicated geometry

The phenomenon of wave set-up may substantially contribute to the formation of devastating coastal flooding in certain coastal areas. We study the appearance and properties of empirical probability density distributions of the occurrence of different set-up heights on an approximately 80 km long section of coastline near Tallinn in the Gulf of Finland, eastern Baltic Sea. The study area is often attacked by high waves propagating from various directions, and the typical approach angle of high waves varies considerably along the shore. The distributions in question are approximated by an exponential distribution with a quadratic polynomial as the exponent. Even though different segments of the study area have substantially different wave regimes, the leading term of this polynomial is usually small (between −0.005 and 0.005) and varies insignificantly along the study area. Consequently, the distribution of wave set-up heights substantially deviates from a Rayleigh or Weibull distribution (that usually reflect the distribution of different wave heights). In about three-quarters of the occasions, it is fairly well approximated by a standard exponential distribution. In about 25 % of the coastal segments, it qualitatively matches a Wald (inverse Gaussian) distribution. The Kolmogorov–Smirnov test (D value) indicates that the inverse Gaussian distribution systematically better matches the empirical probability distributions of set-up heights than the Weibull, exponential, or Gaussian distributions.

The Transmuted Odd Fréchet-G Family of Distributions: Theory and Applications

The last years, the odd Fréchet-G family has been considered with success in various statistical applications. This notoriety can be explained by its simple and flexible exponential-odd structure quite different to the other existing families, with the use of only one additional parameter. In counter part, some of its statistical properties suffer of a lack of adaptivity in the sense that they really depend on the choice of the baseline distribution. Hence, efforts have been made to relax this subjectivity by investigating extensions or generalizations of the odd transformation at the heart of the construction of this family, with the aim to reach new perspectives of applications as well. This study explores another possibility, based on the transformation of the whole cumulative distribution function of this family (while keeping the odd transformation intact), through the use of the quadratic rank transmutation that has proven itself in other contexts. We thus introduce and study a new family of flexible distributions called the transmuted odd Fréchet-G family. We show how the former odd Fréchet-G family is enriched by the proposed transformation through theoretical and practical results. We emphasize the special distribution based on the standard exponential distribution because of its desirable features for the statistical modeling. In particular, different kinds of monotonic and nonmonotonic shapes for the probability density and hazard rate functions are observed. Then, we show how the new family can be used in practice. We discuss in detail the parametric estimation of a special model, along with a simulation study. Practical data sets are handle with quite favorable results for the new modeling strategy.

Variability of distributions of wave set-up heights along a shoreline with complicated geometry

The phenomenon of wave set-up may substantially contribute to the formation of devastating coastal flooding in certain coastal areas. We study the appearance and properties of empirical probability distributions of the occurrence of different set-up heights in about 80 km long section of coastline near Tallinn in the Gulf of Finland, the eastern Baltic Sea. The study area is often attacked by high waves propagating from various directions and the approach angle of waves varies largely along the shore. The distribution in question is approximated by an exponential distribution with a quadratic polynomial as the exponent. Even though different segments of the study area host substantially different wave regimes, the leading term of this polynomial is usually small (between −0.005 and 0.005) and varies insignificantly along the study area. Consequently, the distribution of wave set-up heights substantially deviates from a Rayleigh or Weibull distribution (that usually reflect the distribution of different wave heights). In about 3/4 of occasions it is fairly well approximated by a standard exponential distribution. In about 25 % of coastal segments it matches a Wald (inverse Gaussian) distribution. This property signals that very high extreme set-up events may in some locations occur substantially more frequently than it could be expected from the probability of occurrence of severe seas.

QMLE OF A STANDARD EXPONENTIAL ACD MODEL: ASYMPTOTIC DISTRIBUTION AND RESIDUAL CORRELATION

Since the seminal work by Engle and Russell, (1998), numerous studies have applied their standard/linear ACD(m,q) model (autoregressive conditional duration model of orders m and q) to fit the irregular spaced transaction data. Recently, Araichi et al. (2013) also applied the ACD model to claims in insurance. Many of these papers assume that the standardized error follows a standard exponential distribution. In this paper, we derive the asymptotic distribution of the quasi-maximum likelihood estimator (QMLE) when a standard exponential distribution is used. In other words, we provide robust standard errors for an ACD model. Applying this asymptotic theory, we then derive the asymptotic distribution of the corresponding residual autocorrelation.

On the comparison of fisher information of some probability distributions

In 1998 Gupta, R. C., Gupta, P. L. and Gupta, R. D. have introduced the exponentiated exponential distribution (or the generalized exponential distribution) as a generalization of the standard exponential distribution. The mathematical properties of this distribution have been studied in detail by Gupta and Kundu (2001). The aim of this paper is to establish some relations concerning the Fisher’s information of the generalized exponential distribution and the similar information corresponding in the case of the weighted version.

Linearity of regressions inside top-k-lists and related characterizations

López-Blázquez and Weso lowski [6] introduced the top-k-lists sequence of random vectors and elaborated the usefulness of such data. They also developed the distribution of top-k-lists and their properties arising from various probability distributions, such as standard exponential distribution and uniform distribution on (0, 1). In this paper, we study the linearity of regressions inside top-k-lists and then based on this study we present characterizations of certain distributions.

Order Statistics and Benford's Law

Fix a baseB>1and letζhave the standard exponential distribution; the distribution of digits ofζbaseBis known to be very close to Benford's law. If there exists aCsuch that the distribution of digits ofCtimes the elements of some set is the same as that ofζ, we say that set exhibits shifted exponential behavior baseB. LetX1,…,XNbe i.i.d.r.v. If theXi's are Unif, then asN→∞the distribution of the digits of the differences between adjacent order statistics converges to shifted exponential behavior. If insteadXi's come from a compactly supported distribution with uniformly bounded first and second derivatives and a second-order Taylor series expansion at each point, then the distribution of digits of anyNδconsecutive differencesandallN−1normalized differences of the order statistics exhibit shifted exponential behavior. We derive conditions on the probability density which determine whether or not the distribution of the digits of all the unnormalized differences converges to Benford's law, shifted exponential behavior, or oscillates between the two, and show that the Pareto distribution leads to oscillating behavior.

The lifetimes of configuration states in statistical physics

The moments and probability distribution of the lifetime of a configuration state relative to m disjoint regions in ℝ d for particles under stochastic motion are expressed in terms of the derivatives at the origin of the probability after-effects for the m regions and for the union of the regions, together with the single integrated and the m randomized first-passage-time distributions, relative to the union of the m regions and to the separate complements of the m regions, respectively. The lifetime, suitably normed in terms of the mean lifetime, is shown to have a limiting standard exponential distribution. Finally, the distributions of lifetime when the motion of the particles is either Brownian or such as to generate a persistent generalized Smoluchowski process are discussed; in the first case, the distribution of lifetime reduces to a standard problem in heat conduction, and in the second case the distribution is expressed in terms of an exponential function and the m probability after-effects for the m regions.

The lifetimes of configuration states in statistical physics

The moments and probability distribution of the lifetime of a configuration state relative to m disjoint regions in ℝd for particles under stochastic motion are expressed in terms of the derivatives at the origin of the probability after-effects for the m regions and for the union of the regions, together with the single integrated and the m randomized first-passage-time distributions, relative to the union of the m regions and to the separate complements of the m regions, respectively. The lifetime, suitably normed in terms of the mean lifetime, is shown to have a limiting standard exponential distribution. Finally, the distributions of lifetime when the motion of the particles is either Brownian or such as to generate a persistent generalized Smoluchowski process are discussed; in the first case, the distribution of lifetime reduces to a standard problem in heat conduction, and in the second case the distribution is expressed in terms of an exponential function and the m probability after-effects for the m regions.