  Smooth Estimation of probability density and distribution functions from its
sample is an attractive and an important problem that has applications in
several fields such as, business, medicine, and environment. This article
introduces a simple approach but novel for estimating both functions to have
smooth curves for both via left mean absolute deviation (MAD) function and beta
regression approach. Our approach explores estimation of both functions by
smoothing the first derivative of left MAD function to obtain the final optimal
smooth estimates. The final smooth estimates are obtained under the assumption
that nondecreasing distribution function and the density function remains
nonnegative. This is achieved by using beta regression with various link
functions (logit, probit, cloglog, and cauchit) applied to a polynomial whose
degree is determined by the first derivative of the left MAD function. The
chosen polynomial degree minimizes the mean absolute regression errors,
provided that the first derivative of the regression vector of expected values
is nonnegative. Additionally, pointwise confidence limits for the distribution
function are derived using the beta distribution. The method is utilized on
simulated datasets featuring unimodal, bimodal, and tri-modal, with a
demonstration on an actual dataset. The results suggest that this method
exhibits strong performance relative to the kerne-based method, especially
notable for its superior attributes in sample sizes and smoothness
