Semi-Parametric Estimation with Laguerre Polynomials in Pandemic Models and Efficient Quantile Regression under Censoring

We will use Laguerre polynomials to formulate approximating classes of densities by extending the Laplace distribution. These classes are flexible enough to approximate many continuous distributions, as long as the number of parameters in this Enriched Laplace distribution grows to infinity. Moreover, the densities can be constructed in a way such that desired properties, like the location of a quantile, are fulfilled. These classes can be used to construct sieve estimators in semi-parametric models in which densities appear as nuisance parameters. This yields a novel methodology to estimate linear quantile regression models when the response is randomly right censored. We will show that with this new quantile regression model, we can obtain novel estimators of the quantile function, which are shown to be consistent and asymptotically normal. We also establish the asymptotic efficiency bound. But we will also illustrate how the approximating classes can be used in a different context in epidemics where many interesting quantities, like the reproduction number, depend on the incubation period (time from infection to symptom onset) and/or the generation time (time until a new person is infected from another infected person). The estimation, however, is challenging because it is normally not possible to obtain observations of them. Instead, in the beginning of a pandemic, it is possible to observe for transmission pairs the time of symptom onset for both people as well as a window for infection of the first person (e.g. because of travel to a risk area). We show that our approximating density classes can be used to construct consistent sieve estimators also in this context.