Adaptive bandwidth choice

Abstract

In this paper, we consider the problem of bandwidth choice in the parallel settings of nonparametric kernel smoothed spectral density and probability density estimation. We propose a new class of ‘plug-in’ type bandwidth estimators, and show their favorable asymptotic properties. The new estimators automatically adapt to the degree of underlying smoothness which is unknown. The main idea behind the new estimators is the use of infinite-order ‘flat-top’ kernels for estimation of the constants implicit in the formulas giving the asymptotically optimal bandwidth choices. The proposed bandwidth choice rule for ‘flat-top’ kernels has a direct analogy with the notion of thresholding in wavelets. It is shown that the use of infinite-order kernels in the pilot estimator has a twofold advantage: (a) accurate estimation of the bandwidth constants, and (b) easy determination of the required ‘pilot’ kernel bandwidth.

Keywords:

• Bandwidth choice
• Density estimation
• Kernel smoothing
• Nonparametric function estimation
• Spectral estimation
• Time series

Acknowledgement

Many thanks are due to Professor Mary Ellen Bock of Purdue University for her suggestion in the early 90s that flat-top kernels behave like wavelets.

Notes

^*Research partially supported by NSF Grant DMS-01-04059

¹There exist different sets of conditions sufficient for Eq. (8); for example, we may assume E |X _t |^6+δ < ∞, and ∑ _k=1 ^∞ k ² (α _X (k))^δ/(6+δ) < ∞ for some δ > 0. The α-mixing coefficients are defined in the following way: let ℱ _n ^m be the σ-algebra generated by {X _t , n ≤ t ≤ m}, and define α _X (k) = sup _n sup _A,B |P(A ∩ B) − P(A)P(B)|, where A and B vary over the σ-fields ℱ_−∞ ⁿ and ℱ _n+k ^∞, respectively; see Rosenblatt (1956).

²There exist different sets of conditions sufficient for Eq. (9); see Brockwell and Davis (1991) or Romano and Thombs (1996). As a matter of fact, under further regularity conditions, the process √N(ρˆ(·) − ρ (·)) is asymptotically Gaussian with autocovariance tending to zero; consequently, Eq. (10) would follow from the theory of extremes of dependent sequences – see e.g., Leadbetter et al. (1983).

⁴There exist different sets of conditions sufficient for Eq. (18); for example, Hallin and Tran (1996) show Eq. (18) under an assumption of linearity for the time series {X _n }. Interestingly, as long as the linearity coefficients have a fast (polynomial) decay, the (weak) dependence present in the time series {X _n } does not seem to influence the large-sample variance of either fˆ or f¯ _p ; these variances can actually be calculated as if the sequence {X _n } were independent. On the other hand, such conditions on weak dependence are necessary since from recent work of Adams and Nobel (1998) it is known that ergodicity alone is not sufficient to guarantee consistent density estimation.