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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2603.18490 |
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| _version_ | 1866912973793001472 |
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| author | Luo, Yiqi Luo, Xue |
| author_facet | Luo, Yiqi Luo, Xue |
| contents | We investigate Bayesian nonparametric density estimation via orthogonal polynomial expansions in weighted Sobolev spaces. A core challenge is establishing minimax optimal posterior convergence rates, especially for densities on unbounded domains without a strictly positive lower bound. For densities bounded away from zero, we give sufficient conditions under which the framework of \cite{shen2001} applies directly. For densities lacking a positive lower bound, the equivalence between Hellinger and weighted $L_2$-norm distance fails, invalidating the original theory. We propose a novel shifting method that lifts the true density $g_0$ to a sequence of proxy densities $g_{0,n}$. We prove a modified convergence theorem applicable to these shifted densities, preserving the optimal rate. We also construct a Gaussian sieve prior that achieves the minimax rate $\varepsilon_n=n^{-p/(2p+1)}$ for any integer $p\geq1$. Numerical results confirm that our estimator approximates the true density well and validates the theoretical convergence rate. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_18490 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The minimax optimal convergence rate of posterior density in the weighted orthogonal polynomials Luo, Yiqi Luo, Xue Statistics Theory 62G07, 62G20 We investigate Bayesian nonparametric density estimation via orthogonal polynomial expansions in weighted Sobolev spaces. A core challenge is establishing minimax optimal posterior convergence rates, especially for densities on unbounded domains without a strictly positive lower bound. For densities bounded away from zero, we give sufficient conditions under which the framework of \cite{shen2001} applies directly. For densities lacking a positive lower bound, the equivalence between Hellinger and weighted $L_2$-norm distance fails, invalidating the original theory. We propose a novel shifting method that lifts the true density $g_0$ to a sequence of proxy densities $g_{0,n}$. We prove a modified convergence theorem applicable to these shifted densities, preserving the optimal rate. We also construct a Gaussian sieve prior that achieves the minimax rate $\varepsilon_n=n^{-p/(2p+1)}$ for any integer $p\geq1$. Numerical results confirm that our estimator approximates the true density well and validates the theoretical convergence rate. |
| title | The minimax optimal convergence rate of posterior density in the weighted orthogonal polynomials |
| topic | Statistics Theory 62G07, 62G20 |
| url | https://arxiv.org/abs/2603.18490 |