Masaaki Sibuya

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.

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.

ORDERED SAMPLE FROM TWO-PARAMETER GEM DISTRIBUTION

Pitmans two-parameter GEM distribution is characterized by the product moments. Based on that fact, sampling distributions of some statistics of a random sample from the two-parameter GEM distribution are derived. The main statistics are the frequencies of categories in a sample in the order of appearance, the intervals between new categories, and number of distinct categories.

Extended stirling family of discrete probability distributions

In this paper. introduced is a family of discrete probability distributions. whose probability function includes explieitly the Striling-Carlitz polynomial of the first or the second kind. The new family extends the stirling family of distributions. Sibuya (1988) includes the conditional distributions of the orginal ones and enlarges the application area.

Optimal Threshold for the k-Out-Of-n Monitor with Dual Failure Modes

A monitor consists of n identical sensors working independently. Each sensor measures a variate of output or environment of a system, and is activated if a variate is over a threshold specified in advance for each sensor. The monitor alarms if at least k out of n sensors are activated. The performance of the monitor, the probabilities of failure to alarm and false alarming, depends on the number k, the threshold values and the probability distributions of the variate at normal and abnormal states of the system. In this paper, a sufficient condition on the pair of the distributions is given under which the same threshold values for all the sensors are optimal. The condition motivates new orders between probability distributions. Solving an optimization problem an explicit condition is obtained for maximizing or minimizing a symmetric function with the constraint of another symmetric function.

The Most Random Partition of a Finite Set and its Application to Classification

The notion of ‘the most random partition of a finite set’ is defined and proposed as a null model in classification. A Hamming-type distance between two independent and most random partitions is used for justifying its randomness, and is used for testing this null hypothesis. The probability distribution of the distance is studied for the latter purpose.