Hu L, Zeng X, Wang J. Wavelet optimal estimations for a two-dimensional continuous-discrete density function over
L p risk.
JOURNAL OF INEQUALITIES AND APPLICATIONS 2018;
2018:279. [PMID:
30363759 PMCID:
PMC6182424 DOI:
10.1186/s13660-018-1868-7]
[Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 06/20/2018] [Accepted: 09/30/2018] [Indexed: 06/08/2023]
Abstract
The mixed continuous-discrete density model plays an important role in reliability, finance, biostatistics, and economics. Using wavelets methods, Chesneau, Dewan, and Doosti provide upper bounds of wavelet estimations on L 2 risk for a two-dimensional continuous-discrete density function over Besov spaces B r , q s . This paper deals with L p ( 1 ≤ p < ∞ ) risk estimations over Besov space, which generalizes Chesneau-Dewan-Doosti's theorems. In addition, we firstly provide a lower bound of L p risk. It turns out that the linear wavelet estimator attains the optimal convergence rate for r ≥ p , and the nonlinear one offers optimal estimation up to a logarithmic factor.
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