Using Imperfectly Elicited Fractiles for the Estimation of Probability Distribution Parameters

We develop direct and indirect analytical approaches to determine the parameters of a distribution using elicited fractile values in the presence of elicitation errors. Both approaches seek to minimize the variance on the errors in the estimation of the parameters of the distribution. In the indirect approach we obtainweights for the elicited fractile values to estimate the moments of the distribution; estimates for the probability distribution parameters can then be obtained indirectly from the moment estimates. The direct approach provides weights to estimate the parameters directly from the elicited fractile values. For both approaches, we show that the weights are independent of the actual parameter values and depend only on the fractile probabilities being elicited when the distribution is a location-scale distribution. We show numerically that both these approaches should be preferred over approaches that ignore elicitation error or elicit only a specific set of fractiles. The parameter invariant weights for an arbitrary set of fractile probabilities provide for a flexible elicitation of probability distributions. Subsequently, we extend the results to other non location-scale distributions including the Johnson family of distributions.