Abstract
This paper considers wavelet estimation for a multivariate density function based on mixing and size-biased data. We provide upper bounds for the mean integrated squared error (MISE) of wavelet estimators. It turns out that our results reduce to the corresponding theorem of Shirazi and Doosti (Stat. Methodol. 27:12–19, 2015), when the random sample is independent.
Highlights
Let {Yi, i ∈ Z} be a strictly stationary random process defined on a probability space (, F, P) with the common density function ω(y)f (y) g(y) =, y ∈ Rd, (1)μ where ω denotes a known positive function, f stands for an unknown density function of the unobserved random variable X and μ = Eω(X) = Rd ω(y)f (y) dy < +∞
An upper bound of wavelet estimation on Lp (1 ≤ p < +∞) risk in negatively associated case is given by Liu and Xu [9]
We provide two examples for strong mixing data
Summary
We want to estimate the unknown density function f from a sequence of strong mixing data Y1, Y2, . We give upper bounds for the mean integrated squared error (MISE) of wavelet estimators. Z denotes the integer set and N := {n ∈ Z, n ≥ 0}; (ii) j∈Z Vj = L2(Rd) This means the space j∈Z Vj being dense in L2(Rd); (iii) f (2·) ∈ Vj+1 if and only if f (·) ∈ Vj for each j ∈ Z; (iv) There exists a scaling function φ ∈ L2(Rd) such that {φ(· – k), k ∈ Zd} forms an orthonormal basis of V0 = span{φ(· – k)}. The independent and identically distributed (i.i.d.) data are strong mixing since P(A ∩ B) = P(A)P(B) and α(k) ≡ 0 in that case. When dealing with strong mixing data, it seems necessary to require the functions ω in (1) to be Borel measurable.
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
