Abstract We propose a new method, the confidence interval (CI) to select valid instruments from larger set of potential for instrumental variable (IV) estimation causal effect an exposure on outcome. Invalid are such that they fail exclusion conditions and enter model as explanatory variables. The CI method is based CIs per instrument effects estimates selects largest group with all overlapping...

This paper considers an extension of probability space based on interval random variables. In this approach, first, a notion of interval random variable is introduced. Then, based on a family of continuous distribution functions with interval parameters, a concept of probability space of an interval random variable is proposed. Then, the mean and variance of an interval random variable are intr...

The purpose of this paper is to propose “interval scale,” a new concept defined as a set of ordered pitch-class intervals. Unlike an ordinary scale, this concept restricts usable intervals and doesn’t restrict pitch classes directly. This provides possibilities that interval scales can be used in atonal music that uses all pitch classes and can be used to express some differences in a similar w...

Scale free graphs have attracted attention by their non-uniform structure that can be used as a model for various social and physical networks. In this paper, we propose a natural and simple random model for generating scale free interval graphs. The model generates a set of intervals randomly under a certain distribution, which defines a random interval graph. The main advantage of the model i...

Determining the set of all optimal solutions of a linear program with interval data is one of the main problems discussed in interval optimization. We review two methods based on duality in linear programming, which are used to approximate the optimal set. Additionally, another decomposition method based on complementary slackness is proposed. This method provides the exact description of the o...

Conventional data envelopment analysis (DEA) for measuring the relative efficiency of a set of decisionmaking units (DMUs) requires the observations to have precise values. When observations are imprecise and represented by interval values, the efficiencies are also expected to reflect interval values. Several methods exist to calculate the interval overall and technical efficiencies, but such ...