Central Limit Theorem for Conditional Mode in the Single Functional Index Model with Data Missing at Random - Allal, Anis; Dib, Abdassamad; Rabhi, Abbes (Centralne twierdzenie graniczne dla trybu warunkowego w jednolitym funkcjonalnym modelu indeksowym z losowym brakiem danych, 2024)
This paper concentrates on nonparametrically estimating the conditional density function and conditional mode within the single functional index model for independent data, particularly when the variable of interest is affected by randomly missing data. This involves a semi-parametric single model structure and a censoring process on the variables. The estimator’s consistency (with rates) in a variety of situations, such as the framework of the single functional index model (SFIM) under the assumption of independent and identically distributed (i.i.d) data with randomly missing entries, as well as its performance under the assumption that the covariate is functional, are the main areas of focus. For this model, the nearly almost complete uniform convergence and rate of convergence are established. The rates of convergence highlight the critical part that the probability of concentration play in the law of the explanatory functional variable. Additionally, we establish the asymptotic normality of the derived estimators proposed under specific mild conditions, relying on standard assumptions in Functional Data Analysis (FDA) for the proofs. Finally, we explore the practical application of our findings in constructing confidence intervals for our estimators. The rates of convergence highlight the critical part that the probability of concentration play in the law of the explanatory functional variable.
Econometrics = Ekonometria, 2024, Vol. 28, No. 1, s. 39-60
Keywords: functional data analysis; functional single index process; kernel estimator; missing at random; nonparametric estimation; small ball probability; funkcjonalna analiza danych; funkcjonalny proces pojedynczego indeksu; estymator jądra; losowe braki; estymacja nieparametryczna; prawdopodobieństwo małej kuli