SEMI-PARAMETRIC ESTIMATION OF WEIGHTED AVERAGE DERIVATIVES
OF MODAL REGRESSION
HIROFUMI OHTA
Graduate School of Economics, Department of Statistics, University of Tokyo
Abstract. In this talk, I will discuss a single index model for modal regression function, andpropose a novel estimator of its weighted average derivatives. I construct the estimator based on
sample-splitting/cross-fitting techniques. (cf. [1]). In general semi-parametric estimation, theseapproaches are expected to reduce “own-estimating bias” from estimating nuisance functions
and parameters of interest based on full sample. Under the sample-splitting scheme, I derivesome asymptotic properties of the estimator and a valid inference method.

Keywords: single index model, conditional mode, average derivatives, U -statistics, cross-fitting.
This paper studies a regression model which estimates the conditional mode of an outcome Y
given regressors X, called modal regression. I consider a single index model of the form
Mode(Y |X) = m0 (X),
m0 (X) = G(X ⊤ β),
where X is a d-dimensional vector of continuous regressors, Y is a continuous scalar outcome
variable, m0 is an unknown regression function, and G is an unknown link function. Let f (y|x) bea conditional density function of Y given X, and assume the global mode of f (y|x), denoted byMode(Y |X) = arg maxy f (y|x) is unique. The focus of this paper is on semi-parametric estimationof the index coeﬃcient β ∈ Rd , when m0 , G and f (y|x) are unspecified.
Compared to mean regression, the modal regression has particularly useful features when thedata distribution is highly skewed or has fat tails. In such situations, the mean regression, which
estimates the conditional mean of the distribution, fails to capture the major trend underlying
the data. This is because the conditional mean is not necessarily the point where the data points
distribute densely. Then, the conditional mode is a convenient alternative in this situation as it cancapture the majority of the data.
Recently, several modal regression models and their estimation methods, including fully nonparametric model and linear model, have been developed. On the other hand, there is no study on
estimating the single index model for the modal regression. The single index model is known as asemi-parametric model, which restricts the regression function to be the transformation G of thelinear term X ⊤ β. Define ε := Y − G(X ⊤ β) as a disturbance, then, Mode(ε|X) = 0 is the restriction
on the noise. The single index model is semi-parametric because G is unknown function and theconditional density of ε given X is not specified. Thus, the lack of information of G makes the estimation of β diﬃcult and nontrivial. I construct the estimator based on [2] and the sample-splitting
allows to use asymptotic analysis for U -statistics.
References[1] V. Chernozhukov, D. Chetverikov, M. Demirer, E. Duflo, C. Hansen, W. Newey, and J. Robins.
Double/debiased machine learning for treatment and structural parameters. The Econometrics
Journal, 2017.
[2] J. L. Powell, J. H. Stock, and T. M. Stoker. Semiparametric estimation of index coeﬃcients.
Econometrica: Journal of the Econometric Society, pages 1403–1430, 1989.