Real-time estimation for functional stochastic regression models

ABSTRACT

In this paper, a heteroscedastic functional regression model with martingale difference errors is considered. We are interested in real-time estimation of the regression as well as the conditional variance operators when the response is a real-valued random variable and the covariate belongs to an infinite-dimensional space. A Robbins–Monro-type estimator of the conditional variance is introduced when a sample is collected from an underlying stationary and ergodic process. First, a local uniform $L^q$-consistency (for $q\ge 2$) rate of the recursive estimator of the regression operator is established. Then, a pointwise mean-square consistency rate of the conditional variance is given when the regression function is supposed to be known and when it is estimated recursively. Simulation studies are conducted to assess the proposed estimator's performance, in terms of reducing the computational time without affecting significantly the accuracy, compared to its natural competitor. An application to real environmental data is also carried out to illustrate the real-time on day ahead prediction of the maximum ozone concentration in Mexico city as well as its volatility.

Keywords:

• Conditional variance
• ergodic processes
• functional data
• Robbins–Monro algorithms
• nonparametric estimation
• kernel methods

2010 Mathematics Subject Classifications:

• 62G05
• 62L20
• 62G20
• 62G08

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Available on the website: www.lsp.ups-tlse.fr/staph/npfda

2 https://cran.r-project.org/web/packages/aire.zmvm/aire.zmvm.pdf

3 The characteristics of the computer we used to perform these forecasts were CPU: Duo E4700 2.60 GHz, HD: 149 Go, Memory: 3.23 Go.