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Let $\{X_i\}$ be a sequence of stationary $\a$-mixing random variables with probability density function $f(x)$. The recursive kernel estimators of $f(x)$ are defined by $$ \widehat{f}_n(x)=\frac{1}{n\sqrt{b_n}}\sum^n_{j=1}b_j^{-\frac{1}{2}}K\Big(\frac{x-X_j}{b_j}\Big)~~\mbox{and}~~ \widetilde{f}_n(x)=\frac{1}{n}\sum^n_{j=1}\frac{1}{b_j}K\Big(\frac{x-X_j}{b_j}\Big), $$ where $0<b_n\rightarrow 0$ is bandwith and $K$ is some kernel function. Under appropriate conditions, we establish the Berry-Esseen bounds for these estimators of $f(x)$, which show the convergence rates of asymptotic normality of the estimators.
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참고문헌 (23)
참고문헌 신청
B. W. Silverman / 1986 / Density Estimation for Statistics and Data Analysis / Chapman &Hall
D. W. Scott / 1992 / Multivariate Density Estimation: Theory, Practice, and Visualization / Wiley
E. J. Wegman / 1979 / Remarks on some recursive estimators of a probability density / Ann. Statist 7 (2) : 316 ~ 327
E. Liebscher / 1996 / Strong convergence of sums of -mixing random variables with applications to density estimation / Stochastic Process. Appl 65 (1) : 69 ~ 80
E. Liebscher / 1999 / Estimating the density of the residuals in autoregressive models / Stat. Inference Stoch. Process 2 (2) : 105 ~ 117
D. W. Scott / 1992 / Multivariate Density Estimation: Theory, Practice, and Visualization / Wiley
E. J. Wegman / 1979 / Remarks on some recursive estimators of a probability density / Ann. Statist 7 (2) : 316 ~ 327
E. Liebscher / 1996 / Strong convergence of sums of -mixing random variables with applications to density estimation / Stochastic Process. Appl 65 (1) : 69 ~ 80
E. Liebscher / 1999 / Estimating the density of the residuals in autoregressive models / Stat. Inference Stoch. Process 2 (2) : 105 ~ 117
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