We consider the general modern notion of the so-called associated-kernels for smoothing density function on a given support. We first show that the unnormalized estimator is consistent and that the normalizing random variable converges in L^4 to 1. Then, we deduce the consistency of the considered normalized estimator. The comparison in favor of the normalized estimator is obtained by the mean squared error. We conclude by providing, for the first time, the general asymptotic normalities through some regularity assumptions for both unnormalized and normalized associated-kernel density estimators. The Gumbel, Weibull, lognormal, and other associated kernels are investigated for illustrating theoretically and numerically some of our results with an application to original data of automobile claim amounts from Covéa Affinity.