Almost Sure $L_r$-Norm Convergence for Data-Based Histogram Density Estimates
Almost Sure $L_r$-Norm Convergence for Data-Based Histogram Density Estimates
复制标题
基于数据的直方图密度估计的几乎肯定 $L_r$-范数收敛
DOI：
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发表时间：
1991
期刊：
影响因子：
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通讯作者：
X. Chen
中科院分区：
文献类型：
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作者：
L. Zhao;P. R. Krishnaiah;X. Chen
Let $X_1 , cdots ,X_n $ be an independently and identically distributed sample drawn from a d-dimensional distribution with density f. Partition the space ${f R}^d$ into a union of disjoint intervals ${ I_j = I(j,X_1 , cdots ,X_n )} $ with the form $I_j = { x = (x^{(1)} , cdots ,x^{(d)} ): - infty < a_{mj} leqq x^{(m)} < b_{mj} < infty ,,m = 1, cdots ,d} $ Define the data-based histogram estimate of $f(x)$ based on this partition as: $f_n (x) = { ext{ the number of }}X_1 , cdots ,X_n { ext{ falling into }}I_j $
Let $X_1 , cdots ,X_n $ be an independently and identically distributed sample drawn from a d-dimensional distribution with density f. Partition the space ${f R}^d$ into a union of disjoint intervals ${ I_j = I(j,X_1 , cdots ,X_n )} $ with the form $I_j = { x = (x^{(1)} , cdots ,x^{(d)} ): - infty < a_{mj} leqq x^{(m)} < b_{mj} < infty ,,m = 1, cdots ,d} $ Define the data-based histogram estimate of $f(x)$ based on this partition as: $f_n (x) = { ext{ the number of }}X_1 , cdots ,X_n { ext{ falling into }}I_j $