Rinya Takahashi

The maximum size of the planar sections of random spheres and its application to metallurgy

A theorem of this paper proves that if the size distribution of random spheres is generalized gamma, its Wicksell transform and other related distributions belong to the domain of attraction of the Gumbel distribution. The theorem also shows the attraction coefficients of the distributions. The fatigue strength of high-strength steel is closely related to the maximum size of nonmetallic inclusions in the region of maximum stress of the steel. Murakami and others developed a method, making use of the Gumbel QQ-plot, for predicting the maximum size from the size distribution of inclusion circles in microscopic view-fields. Based on the Gumbel approximation of the maximum of wicksell transforms, a modified and extended version of Murakamis method is justified, and its performance is evaluated by simulation.

Prediction of the Maximum Size in Wicksell's Corpuscle Problem

In the Wicksell corpuscle problem, the maximum size of random spheres in a volume part is to be predicted from the sectional circular distribution of spheres cut by a plane. The size of the spheres is assumed to follow the generalized gamma distribution. Some prediction methods according to measurement methods on the sectional plane are proposed, and their performances are evaluated by simulation. The prediction method based on the r largest sizes and the total number of the sectional circles is recommended, because of its satisfactory performance.

Normalizing constants of a distribution which belongs to the domain of attraction of the Gumbel distribution

Simple sufficient conditions that a distribution function belongs to the domain of attraction of the Gumbel distribution and a method to determine the normalizing constants are shown. The results are applied to some specific distribution functions.

Some properties of multivariate extreme value distributions and multivariate tail equivalence

SummaryDenote byH ak-dimensional extreme value distribution with marginal distributionHi(x)=Λ(x)=exp(−e−x),x∈R1. Then it is proved thatH(x)=Λ(x1)...Λ(xk) for anyx=(x1, ...,xk) ∈Rk, if and only if the equation holds forx=(0,...,0). Next some multivariate extensions of the results by Resnick (1971,J. Appl. Probab.,8, 136–156) on tail equivalence and asymptotic distributions of extremes are established.

Maximum Size Prediction in Wicksell's Corpuscle Problem for the Exponential Tail Data

In the Wicksell corpuscle problem, the maximum size of random spheres in a reference volume is to be predicted from the size distribution of circles which are planar sections of spheres cut by a plane. If the area of the great circle of spheres have the exponential tail, simple prediction methods are applied. Performance of the methods is evaluated by simulation and they are applied to a dataset of graphite nodule sizes in spheroidal graphite cast iron. The effect of left-truncation in Wicksell transform is discussed in a general framework.

Asymptotic independence and perfect dependence of vector components of multivariate extreme statistics

Simple necessary and sufficient conditions for asymptotic independence and perfect dependence of the vector components of multivariate extreme statistics are shown. Related results for a multivariate tail equivalence are also shown.

126. DEGREE OF EXPERIENCE FOR EXTREME WAVE STATISTICS

The confidence interval for the extreme wave height of longer return period is good for nothing. Unreasonable uncertainty of estimation has been accepted. We were helpless against this difficulty. We engineers believe in a certain limitation of extrapolation , but they have been bothered with a problem: how much wide interval is no worth being considered. This study proposes a new index to use for this judgement.

DIFFRACTIVE UNCERTAINTY TOWARD THE FUTURE ESTIMATION OF RETURN WAVE HEIGHT

Freshness is very important factor, which is not exceptional for the observation data. The data ready for statistical analyses, is observed in the past, while the return period is laid on the future for the predicting return levels. It is natural that the farther ahead in the future the return level is predicted, the larger the statistical variety becomes. Thus, the freshness of the record is falling down as the pass of time. In the point of view, the statistical uncertainty becomes wide, as entering into the shadow area distinguished by the straight lines of confidence regions by the conventional method. The proposed method will give a potential trend even for the stationary model. It will be useful tool because in many cases it is very difficult to detect the faint trend in short records of wave heights and sea levels due to the climate change, as one of the applications.