Extreme value data with a high clump-at-zero occur in many domains. Moreover,
it might happen that the observed data are either truncated below a given
threshold and/or might not be reliable enough below that threshold because of
the recording devices. These situations occur, in particular, with radio
audience data measured using personal meters that record environmental noise
every minute, that is then matched to one of the several radio programs. There
are therefore genuine zeros for respondents not listening to the radio, but
also zeros corresponding to real listeners for whom the match between the
recorded noise and the radio program could not be achieved. Since radio
audiences are important for radio broadcasters in order, for example, to
determine advertisement price policies, possibly according to the type of
audience at different time points, it is essential to be able to explain not
only the probability of listening to a radio but also the average time spent
listening to the radio by means of the characteristics of the listeners. In
this paper we propose a generalized linear model for zero-inflated truncated
Pareto distribution (ZITPo) that we use to fit audience radio data. Because it
is based on the generalized Pareto distribution, the ZITPo model has nice
properties such as model invariance to the choice of the threshold and from
which a natural residual measure can be derived to assess the model fit to the
data. From a general formulation of the most popular models for zero-inflated
data, we derive our model by considering successively the truncated case, the
generalized Pareto distribution and then the inclusion of covariates to explain
the nonzero proportion of listeners and their average listening time. By means
of simulations, we study the performance of the maximum likelihood estimator
(and derived inference) and use the model to fully analyze the audience data of
a radio station in a certain area of Switzerland.Comment: Published in at http://dx.doi.org/10.1214/10-AOAS358 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org