Abstract
The need for building and generating statistically dependent random variables arises in various fields of study where simulation has proven to be a useful tool.In this work, we present an approach for constructing ordinal variables with arbitrarily assigned marginal distributions and value of association or correlation, expressed in terms of either Goodman and Kruskal's gamma or Pearson's linear correlation. The approach first constructs a class of bivariate copula-based distributions matching the assigned margins, and then, within this class, identifies the distribution matching the assigned association or correlation, by calibrating the copula parameter. A numerical example and a possible application are illustrated.
Highlights
The need for building and drawing samples from statistically dependent random variables emerges in various fields of study where simulation has proven to be a powerful tool
We present an approach for constructing ordinal variables with arbitrary marginal distributions and assigned value of association, expressed in terms of either Goodman and Kruskal’s gamma or Pearson’s linear correlation
Interval; given two marginal distributions and a value ρ ∈ [−1, +1], it is not always possible to construct a joint distribution with those assigned margins, whose correlation is equal to the assigned ρ (McNeil, Frey, and Embrechts 2005)
Summary
The need for building and drawing samples from statistically dependent random variables emerges in various fields of study where simulation has proven to be a powerful tool. The ability to simulate data resembling the observed data is fundamental to compare and investigate the behaviour of statistical procedures when analytical results are not derivable or are cumbersome to derive. Many datasets, especially those arising in the social sciences, often contain ordinal variables. There are several statistical models and techniques that can be employed for handling multivariate ordinal data without trying to quantify their ordered categories: Agresti (2010) gives a thorough treatment. We present an approach for constructing ordinal variables with arbitrary marginal distributions and assigned value of association, expressed in terms of either Goodman and Kruskal’s gamma or Pearson’s linear correlation.
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