# Estimation for the Prediction of Point Processes with Many Covariates

## TL;DR

This paper develops a nonparametric estimation method for the intensity of point processes with many covariates, focusing on prediction accuracy and optimal convergence rates, demonstrated through financial data application.

## Contribution

It introduces a new additive modeling approach for point process intensity estimation with high-dimensional covariates, emphasizing prediction over variable screening.

## Key findings

- Optimal convergence rates achieved with many active covariates
- Application to New Zealand dollar futures trading data
- Simulation confirms theoretical properties

## Abstract

Estimation of the intensity of a point process is considered within a nonparametric framework. The intensity measure is unknown and depends on covariates, possibly many more than the observed number of jumps. Only a single trajectory of the counting process is observed. Interest lies in estimating the intensity conditional on the covariates. The impact of the covariates is modelled by an additive model where each component can be written as a linear combination of possibly unknown functions. The focus is on prediction as opposed to variable screening. Conditions are imposed on the coefficients of this linear combination in order to control the estimation error. The rates of convergence are optimal when the number of active covariates is large. As an application, the intensity of the buy and sell trades of the New Zealand dollar futures is estimated and a test for forecast evaluation is presented. A simulation is included to provide some finite sample intuition on the model and asymptotic properties.