On two random search problems

The paper is concerned with static search on a finite set. An unknown subset of cardinality k of the finite set is to be found by testing its subsets. We investigate two problems: in the first, the number of common elements of the tested and the unknown subset is given; in the second, only the information whether the tested and the unknown subset are disjoint or not is given. Both problems correspond to problems on false coins. If the unknown subset is taken from the family of k-element sets with uniform distribution, we determine the minimum of the lengths of the strategies that find the unknown element with small error probability. The strategies are constructed by probabilistic means.     

随机测度的收敛性

本文研究了随机共轭算子的值域及随机测度空间的度量性     

ON THE HILDRETH-HOUCK ESTIMATOR FOR RANDOM COEFFICIENT REGRESSION MODELS

This paper is concerned with the linear regression model in which the coefficients are random variables. The Hildreth-Houck method is considered for estimating the model. Sufficient conditions for the consistency of the Hildreth-Houck estimator are discussed and its asymptotic normality is established.     

The case of the random telephone numbers

An unusual pattern of telephone numbers recorded on suspect credit-card purchases led to the development of a generalized multinomial distribution. One usage of credit cards included 26 telephone numbers with a pattern of 7 unique numbers: 16 of one kind, 4 of another, 2 of another, and 4 different ones. With the assumption that the probability of any digit appearing in a telephone number is 1/10, the pattern of the 26 telephone numbers is found to have a probability of occurrence on the order of 1/10⁶⁵. That is, the chance of such a pattern occurring at random is roughly equivalent to winning nine times in a row the popular ottery, LOTO 6/49.     

Some asymptotic properties of point estimation with n-dimensional fuzzy data

Some asymptotic properties of point estimation with n-dimensional fuzzy data with respect to a special L ₂-metric ρ are investigated in this article. It is shown that the collection of all n-dimensional fuzzy data endowed with the ρ-metric is a complete and separable space. Some criterions for point estimation in such fuzzy environments are proposed, and the sample mean, variance and covariance with n-dimensional fuzzy data under these criteria are further studied.     

Additive time-dependent hazard model with doubly truncated data

For doubly truncated data, i.e. the variables of interest are only observable if they lie in a certain random interval, an additive hazard model with time-dependent regression coefficients is investigated. Consistency and asymptotic normality are proven under mild assumptions. A simulation study investigates the finite sample properties and the influence of the truncation distribution on the estimation error. Finally, the method is applied to a doubly truncated data set of German companies, where the age at insolvency is of interest.     

Between group variance component interval estimation for the unbalanced heteroscedastic one-way random effects model

The studied topic is motivated by the problem of interlaboratory comparisons. This paper focuses on the confidence interval estimation of the between group variance in the unbalanced heteroscedastic one-way random effects model. Several interval estimators are proposed and compared by means of the simulation study. The most recommended (safest) is the confidence interval based on Bonferroni's inequality.     

On qualitative multi-attribute group decision making and its consensus measure: A probability based perspective

This paper focuses on qualitative multi-attribute group decision making (MAGDM) with linguistic information in terms of single linguistic terms and/or flexible linguistic expressions. To do so, we propose a new linguistic decision rule based on the concepts of random preference and stochastic dominance, by a probability based interpretation of weight information. The importance weights and the concept of fuzzy majority are incorporated into both the multi-attribute and collective decision rule by the so-called weighted ordered weighted averaging operator with the input parameters expressed as probability distributions over a linguistic term set. Moreover, a probability based method is proposed to measure the consensus degree between individual and collective overall random preferences based on the concept of stochastic dominance, which also takes both the importance weights and the fuzzy majority into account. As such, our proposed approaches are based on the ordinal semantics of linguistic terms and voting statistics. By this, on one hand, the strict constraint of the uniform linguistic term set in linguistic decision making can be released; on the other hand, the difference and variation of individual opinions can be captured. The proposed approaches can deal with qualitative MAGDM with single linguistic terms and flexible linguistic expressions. Two application examples taken from the literature are used to illuminate the proposed techniques by comparisons with existing studies. The results show that our proposed approaches are comparable with existing studies.     

Gambler’s Ruin: The Duration of Play

We study the gambler’s ruin problem with a general distribution of the payoffs in each game. Assuming the expected value of the payoff distribution is negative, so that eventual ruin occurs with probability 1, we are interested in the distribution of the duration to ruin, also known as the first-passage time distribution. A generating function for this distribution is obtained. Exact expressions for the expected value and variance of this distribution, as well as asymptotic expressions for the case of large initial wealth, are derived.     

Analysis of short-term systematic measurement error variance for the difference of paired data without repetition of measurement

The variance of short-term systematic measurement errors for the difference of paired data is estimated. The difference of paired data is determined by subtracting the measurement results of two methods, which measure the same item only once without measurement repetition. The unbiased estimators for short-term systematic measurement error variances based on the one-way random effects model are not fit for practical purpose because they can be negative. The estimators, which are derived for balanced data as well as for unbalanced data, are always positive but biased. The basis of these positive estimators is the one-way random effects model. The biases, variances, and the mean squared errors of the positive estimators are derived as well as their estimators. The positive estimators are fit for practical purpose.