Smoothing estimation of distribution and density functions: A beta regression approach

Abstract

This study presents a new approach for smooth estimation of distribution and density functions by leveraging beta regression and generalized additive models (GAM). The approach estimates both functions by smoothing the first derivative of left mean absolute deviation (MAD) function under the condition of nondecreasing distribution function, and nonnegativity of the density. This is achieved by using beta regression and generalized additive models with various link functions (logit, probit, cloglog, and cauchit) that are applied to a polynomial function where the degree is selected based on minimum mean absolute regression error and positivity of the first derivative. Additionally, the confidence limits for the distribution function are derived based on normal approximation and the beta distribution to assess the precision of estimates. The proposed method is evaluated on simulated datasets featuring unimodal, multimodal and real datasets. The results suggest that the proposed method exhibits superior performance compared to the kernel-based estimators, particularly in terms of smoothness and accuracy for small sample sizes.

Journal of Statistical Research 2026, Vol. 60, No. 1, pp. 133-155.