Estimating mixed logit with non-parametric random variables

The estimation of random parameters by means of mixed logit models is becoming current practice amongst discrete choice analysts, one of the most straightforward applications being the derivation of willingness to pay distribution over a heterogeneous population. In many practical cases, parametric distributions are a priori specified and the parameters for these distributions are estimated. This approach can however lead to many practical problems. Firstly, it is difficult to assess which is the more appropriate analytical distribution. Secondly, unbounded distributions often produce values ranges with difficult behavioral interpretation. Thirdly, little is known about the tails and their effects on the mean of the estimates. (Hess et al, 2005; Cirillo and Axhausen, 2006) This paper extends the nonparametric methods in a classical context of mixed logit models. The random variables of the objective functions are assumed to be continuous, bounded, and independent, and we are interested by the inverse cumulating distribution functions. These functions are modeled by means of cubic B-splines with strictly increasing base coefficients, a sufficient condition to construct monotonic (increasing) functions. As a result, the number of parameters that have to be estimated increases; the information on the tails and on the shape of the random variables however should help the analyst to find the right parametric distribution for the random parameters (if this exists). This technique is applied to simulated data and the ability to recover both parametric and nonparametric random vectors is tested. The non-parametric mixed logit model is also used on real data derived from a survey on electric car, whose prototype has been realized and tested in a number of cities in Europe. The data set, which is part of a European study called “Cybercar” is a Stated Preference experiment conducted in Brussels in 2002. The model presents multiple choices and is estimated on repeated observations.