ON THE PROBLEM OF BEST CONVERGENCE RATES OF DENSITY ESTIMATES
X Chen · 1984
Let X_1,…,X_n be iid samples drawn from an m-dimensional population with a probability density f,belonging to the family C_(ka),i.e.the family of all densities whose partial derivatives of order k are bounded by a.It is desired to estimate the value of f at some predetermined point a,for example a=0.Farrell obtained some results concerning the best possible convergence rates for all estimator sequence,from which it follows,for example,that there exists no estimator sequence{γ_n(0)=γ_n(X_1,…,X_n,0)}such that(?)E_f[γ_n(0)- f(0)]~2=o(n~(-2k/(2k+m))).This article pursues this problem further and proves that there exists no estimator sequence{γ_n(0)}such that n~(-k/(2k+m))(γ_n(0)-f(0))(?)0,for each f∈C_(ka), where(?)denotes convergence in probability.