Properties of Legendre Series Estimation of Probability Density
El Fahham Mostafa · The Egyptian Statistical Journal/The Egyptian Statistical Journal · 1985
An unknown density function f(x) and its derivatives have been estimated by means of Legendre functions pⱼ... The estimates use the partial sums of series of Legendre functions with coefficients âⱼₙ = n⁻¹ Σₖ₌₁ⁿ pⱼ(Xₖ) where X₁,...,Xₙ represent a sequence of independent identically distributed random variables (i.i.d.) with the unknown density function f. The integrated mean square rate of convergence of the sth derivative of the estimate is obtained. Similar results are obtained for the case of strictly mixing stationary processes.