Remarks on Some Recursive Estimators of a Probability Density

The density estimator, f^ₙ(x) = n⁻¹∑ⁿj = 1h⁻¹ⱼK((x - Xⱼ)/hⱼ), as well as the closely related one f^ₙ(x) = n⁻¹hₙ-1/2∑ⁿj = 1hⱼ-1/2K((x - Xⱼ)/hⱼ) are considered. Expressions for asymptotic bias and variance are developed. Using the almost sure invariance principle, laws of the iterated logarithm are developed. Finally, illustration of these results with sequential estimation procedures are made.