Choosing Sample Sizes for Statistical Measures on Interval-Valued Data

Abstract

Intervals have frequently been used in the literature to represent uncertainty in data, from eliciting uncertain judgements from experts to representing uncertainty in sensor measurements. This widespread use of intervals has led to research on interval statistics to help understand the data. However, even seemingly trivial statistics (such as variance) cannot be calculated on interval-valued data using the same approach as for point data without incurring substantial loss of precision to a level which can make results close to useless. This loss of precision makes it challenging for decision makers to appropriately interpret interval-valued data using familiar statistics. Although there exist several approaches to computing statistics such as variance, these are all developed for specific properties of the data, and there is no general-case method. In addition, there are many statistical measures for which no efficient and accurate method exist. For such cases, we can use a Monte Carlo sampling approach to generate approximate statistics. While sampling does not generally produce exact solutions, it can provide a useful and efficient approximation to a desired degree of accuracy given sufficient computational resources. In this paper, we focus on the application of Monte Carlo sampling to generate statistics for interval-valued data. Specifically, we explore the optimum sample size required to calculate statistics on interval-valued data for a given degree of accuracy desired. We compare different sizes of data and different sampling methods to demonstrate how these affect the choice of an optimum sample size.