Abstract
We propose a method to generate random distributions with known quantile distribution, or, more generally, with known distribution for some form of generalized quantile. The method takes inspiration from the random Sequential Barycenter Array distributions (SBA) proposed by Hill and Monticino (1998) which generates a Random Probability Measure (RPM) with known expected value. We define the Sequential Quantile Array (SQA) and show how to generate a random SQA from which we can derive RPMs. The distribution of the generated SQA-RPM can have full support and the RPMs can be both discrete, continuous and differentiable. We face also the problem of the efficient implementation of the procedure that ensures that the approximation of the SQA-RPM by a finite number of steps stays close to the SQA-RPM obtained theoretically by the procedure. Finally, we compare SQA-RPMs with similar approaches as Polya Tree.
Highlights
Random probability measures (RPM) find their applications in several different fields, such as statistics, mathematical finance, stochastic processes.In nonparametric Bayesian statistics, for example, the construction of random probability distributions permits to draw a prior at random from the space of probability measures
Inspired by the Sequential Barycenter Array distributions (SBA) of [17], in this paper we propose an alternative to the SBA procedure, which we denote as the Sequential Quantile Array (SQA), that allows to construct random distribution functions with the τ -quantile following a given distribution (for some τ ∈ (0, 1))
Theorem 3.4 proves the large support of the SQA-RPM in the weak topology; while, in applications, it is often desirable that the random probability measures have full support in a stronger sense
Summary
Random probability measures (RPM) find their applications in several different fields, such as statistics, mathematical finance, stochastic processes. Methods that permit to generate continuous prior distributions have been studied: for example the Polya Tree models (including DP as particular cases) and Bernstein processes (see [26] for a more detailed review on RPM used in Bayesian data analysis). This method can be used to generate random probability measures whose associated Value at Risk (VaR), that is a risk measure largely used, follows a specified distribution By exploiting their link with the quantiles, it is possible to generate RPMs with specified distribution for alternative risk measures based on the general notion of M -quantiles defined by [5].
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