Conference Agenda

A new approach for nonparametric estimation of quantile density functions based on
local polynomial estimation is presented. Estimation of the quantile density is important
because it appears for example in the expression for the asymptotic variance of empirical
and kernel type estimators of the quantile function.

The new approach uses a local polynomial regression on (F_n(X_i), Q_n(F_n(X_i))), where F_n
and Q_n represent the empirical distribution function and the empirical quantile function
respectively.

The new approach has more advantageous properties at the boundary than classical quan-
tile density estimators. We present a result on the asymptotic normality of the proposed
estimator. In this context we also get the leading bias term and the asymptotic variance,
which we use to compare the new estimator to classical estimators, especially focusing on
the boundary.

Keywords: asymptotic normality, bias rates, boundary adaptation, empirical quan-
tile function, nonparametric function estimator

There is a great variety of statistical models expressing relations between response variables of interest and explanatory variables, ranging from classical conditional mean regression to fully distributional regression models. We are particularly interested in expressing regression models by means of copulas which are a valuable tool to separate marginal distributions and dependencies. New goodness-of-fit tests and new measures of deviation can be developed based on such copula representations. These tests are desirable since regression models often impose parametric or semiparametric assumptions to overcome the curse of dimensionality, running a risk of misspecification. We present a new goodness-of-fit test for the classical mean regression model. More importantly, we also introduce a new measure of deviation between the true regression function and the imposed parametric assumption. By self-normalization, we develop pivotal inference for this measure including tests for relevant hypotheses. These inference tools are illustrated via simulated and empirical data.

Rank-based association measures, including Spearman’s rho, Gini’s gamma and Spearman’s footrule, are well established in continuous settings, but become problematic when ties are present. We investigate these measures in context of zero-inflated data, where continuous random variables have an increased probability mass at zero and there is a substantial number of ties. Such data is commonly found in fields such as insurance, health care and weather forecasting. Traditional rank-based estimators exhibit a large bias in these settings. To overcome this problem, we derive new formulations of the association measures and propose plug-in estimators. In a simulation study, we show that these outperform state-of-the-art estimators. Additionally, we make the estimator interpretable by deriving its achievable bounds.