Let (Y,X)={Y(t),X(t),-/spl infin/>t>/spl infin/} be real-valued continuous-time jointly stationary processes and let (t/sub j/) be a renewal point processes on (0,/spl infin/), with a finite mean rate, independent of (Y,X). We consider the estimation of regression function r(x/sub 0/, x/sub 1/,...,x/sub m-1/; /spl tau//sub 1/,...,/spl tau//sub m/) of /spl psi/(Y(/spl tau//sub m/)) given (X(0)=x/sub 0/, X(/spl tau//sub 1/)=x/sub 1/,...,X(/spl tau//sub m-1/)=/sub x-1/) for arbitrary lags 0>/spl tau//sub 1/>...> /spl tau//sub m/ on the basis of the discrete-time observations {Y(t/sub j/),X(t/sub j/),t/sub j/)/sub j=1//sup n/. We estimate the regression function and all its partial derivatives up to a total order p/spl ges/1 using high-order local polynomial fitting. We establish the weak consistency of such estimates along with rates of convergence. We also establish the joint asymptotic normality of the estimates for the regression function and all its partial derivatives up to a total order p/spl ges/1 and provide explicit expressions for the bias and covariance matrix (of the asymptotically normal distribution).
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机译:令(Y,X)= {Y(t),X(t),-/ spl infin /> t> / spl infin /}为实值连续时间联合平稳过程,并令(t / sub j /)是(0,/ spl infin /)上的更新点进程,其平均速率不受(Y,X)的限制。我们考虑回归函数r(x / sub 0 /,x / sub 1 /,...,x / sub m-1 /; / spl tau // sub 1 /,...,/ spl tau /给定(X(0)= x / sub 0 /,X(/ spl tau // sub 1 /)= x /的/ spl psi /(Y(/ spl tau // sub m /))的/ sub m /) sub 1 /,...,X(/ spl tau // sub m-1 /)= / sub x-1 /)任意滞后0> / spl tau // sub 1 /> ...> / spl tau // sub m /基于离散时间观测值{Y(t / sub j /),X(t / sub j /),t / sub j /)/ sub j = 1 // sup n /。我们使用高阶局部多项式拟合估计回归函数及其所有偏导数,直至总阶数p / spl ges / 1。我们建立了这种估计的弱一致性以及收敛速度。我们还建立了回归函数及其所有偏导数的估计的联合渐近正态性,直到总阶数p / spl ges / 1为止,并为(渐近正态分布的)偏差和协方差矩阵提供了明确的表达式。
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