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| Format: | Preprint |
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2015
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| Online Access: | https://arxiv.org/abs/1510.06940 |
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| _version_ | 1866910798596538368 |
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| author | Yatracos, Yannis G. |
| author_facet | Yatracos, Yannis G. |
| contents | A mixture density, $f_p,$ is estimable in $R^d, \ d \ge 1,$ but an estimate for the mixing density, $p,$ is usually obtained only when $d$ is unity; $h$ is the mixture's kernel. When $f_p$'s estimate has form $f_{\hat p_n}$ and $p$ is $\tilde q$-smooth, vanishing outside a compact in $R^d,$ plug-in upper bounds are obtained herein for the $L_u$-error (and risk)of $\hat p_n$ and its derivatives; $d \ge 1, 1 \le u \le \infty.$ The bounds depend on $f_{\hat p_n}$'s $L_u$-error (or risk), $h$'s Fourier transform, $\tilde h,$ and the bandwidth of kernel $K$ used in approximations. The choice of $\hat p_n,$ via $f_{\hat p_n},$ suggests that $\hat p_n$'s error rate could be only nearly optimal when $f_{\hat p_n}$ is optimal, but competing estimates and their error rates may not be available for $d>1.$ In examples with $d$ unity, the upper bound is optimal when $h$ is super smooth, misses the optimal rate by the factor $(\log n)^ξ, \ ξ>0,$ when $h$ is smooth, and is satisfactory when $\tilde h$ has periodic zeros. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1510_06940 |
| institution | arXiv |
| publishDate | 2015 |
| record_format | arxiv |
| spellingShingle | Plug-in error bounds for a mixing density estimate in $R^d,$ and for its derivatives Yatracos, Yannis G. Statistics Theory A mixture density, $f_p,$ is estimable in $R^d, \ d \ge 1,$ but an estimate for the mixing density, $p,$ is usually obtained only when $d$ is unity; $h$ is the mixture's kernel. When $f_p$'s estimate has form $f_{\hat p_n}$ and $p$ is $\tilde q$-smooth, vanishing outside a compact in $R^d,$ plug-in upper bounds are obtained herein for the $L_u$-error (and risk)of $\hat p_n$ and its derivatives; $d \ge 1, 1 \le u \le \infty.$ The bounds depend on $f_{\hat p_n}$'s $L_u$-error (or risk), $h$'s Fourier transform, $\tilde h,$ and the bandwidth of kernel $K$ used in approximations. The choice of $\hat p_n,$ via $f_{\hat p_n},$ suggests that $\hat p_n$'s error rate could be only nearly optimal when $f_{\hat p_n}$ is optimal, but competing estimates and their error rates may not be available for $d>1.$ In examples with $d$ unity, the upper bound is optimal when $h$ is super smooth, misses the optimal rate by the factor $(\log n)^ξ, \ ξ>0,$ when $h$ is smooth, and is satisfactory when $\tilde h$ has periodic zeros. |
| title | Plug-in error bounds for a mixing density estimate in $R^d,$ and for its derivatives |
| topic | Statistics Theory |
| url | https://arxiv.org/abs/1510.06940 |