STRONG CONSISTENCY OF THE PERIODIC SMOOTHING SPLINES  

作　　者：刘克俭 

机构地区：[1]Department of Mathematical Statistics and O.R.,Fudan University,Shanghai 200433,China

出　　处：《Systems Science and Mathematical Sciences》1991年第2期173-185,共13页

摘　　要：Consider the regression model Y_i=u（x_i）+ε_i,i=1,2,…,n,where u（x）∈W_2,per²[0,1],x_i=（i-1）/n,{ε_i}（?）are i.i.d,random variables.We use the periodic smoothing spline u_λp（x）toestimate u（x）.Under certain conditions,strong consistency results of u_λp（x）are obtained,i.e.,forall 0【a≤b【1,（?）((nλ^1/4)^1/2)/log n(u_λp（x）-u（x）)=0 a.s.and for all x∈（0, 1）,（?）((nλ^1/4)/（2loglogn）)^1/2(u_λp（x）-u（x）)=σ（integral from -∞ to ∞K²（x）dx）^1/2 a.s.,whereK（x）=(sin |x|/2^1/2+π/4)/2（?）Consider the regression model Y_i=u(x_i)+ε_i,i=1,2,…,n,where u(x)∈W_(2,per)~2[0,1],x_i=(i-1)/n,{ε_i}(?)are i.i.d,random variables.We use the periodic smoothing spline u_(λp)(x)toestimate u(x).Under certain conditions,strong consistency results of u_(λp)(x)are obtained,i.e.,forall 0<a≤b<1,(?)((nλ^(1/4))^(1/2))/log n(u_(λp)(x)-u(x))=0 a.s.and for all x∈(0, 1),(?)((nλ^(1/4))/(2loglogn))^(1/2)(u_(λp)(x)-u(x))=σ(integral from -∞ to ∞K^2(x)dx)^(1/2) a.s.,whereK(x)=(sin |x|/2^(1/2)+π/4)/2(?)

关 键 词：NONPARAMETRIC regression PERIODIC SMOOTHING SPLINE KERNEL estimate stron CONSISTENCY LIL 

分 类 号：N94[自然科学总论—系统科学]