Family of the generalised gamma kernels: a generator of asymmetric kernels for nonnegative data

Abstract

Unlike symmetric kernels, so far exploring asymptotics on asymmetric kernels has relied on diversified approaches. This paper proposes a family of the generalised gamma (GG) kernels that is built on the probability density function of the GG distribution [Stacy, E.W. (1962), ‘A Generalization of the Gamma Distribution’, Annals of Mathematical Statistics, 33, 1187–1192] and a few common conditions. The family can generate asymmetric kernels that share appealing properties with the modified gamma kernel [Chen, S.X. (2000), ‘Probability Density Function Estimation Using Gamma Kernels’, Annals of the Institute of Statistical Mathematics, 52, 471–480]. Asymptotics on the kernels generated from the family can be delivered by manipulating the conditions directly, as with symmetric kernels.

Keywords:

• asymmetric kernel
• boundary correction
• density estimation
• kernel smoothing
• generalised gamma kernels

MSC 2010::

• 62G07
• 62G20

Conflict of interest disclosure statement

No potential conflict of interest was reported by the authors.

Conflict of interest disclosure statement

We would like to thank the editor Irène Gijbels, an associate editor and two anonymous referees for their comments that have substantially improved this paper. We are also grateful to Jin Seo Cho, Benedikt Funke, Bruce Hansen, Susumu Imai, Yoshihide Kakizawa, Yoshihiko Nishiyama, Katsumi Shimotsu, Nabil Zougab, and participants at Summer Workshop on Economic Theory 2012, New Zealand Econometric Study Group Meeting 2013, and seminars at the Development Bank of Japan, Kyoto University, the University of Sydney and the University of Tokyo for constructive comments and suggestions. The views expressed herein and those of the authors do not necessarily reflect the views of the Development Bank of Japan.

Funding

The first author gratefully acknowledges financial support from Japan Society of the Promotion of Science [grant number 23530259].

Notes

1. Conditions 1–5 even establish approximations to the bias and variance of nonparametric regression estimators (e.g. local constant and LL estimators) using the GG kernels.

2. Simulation results of GG density estimators with the rule-of-thumb smoothing parameters plugged in are available on the first author's webpage.

3. This definition differs from the one given in Kuruwita, Kulasekera, and Padgett (2010, Table 1), who construct their Weibull kernel from a different motivation. Apparently, their definition tends to be unbounded near the origin.

4. The pdf of is It is widely recognised that the Nakagami-m distribution was first proposed in Nakagami (1960). In reality, however, the distribution originates as early as in Nakagami (1943).

5. This kernel does not belong to the GG kernels, because it can be obtained by setting in Equation (9).

6. Because an error is found in the bias approximation for the Terrell and Scott (1980)-type MBC estimator in Hirukawa (2010) and Hirukawa and Sakudo (2014), we have corrected the error here. We thank Dr Nabil Zougab for pointing it out.

7. Because the third-order term in is O(b²), it would be possible in principle to improve the bias convergence of this estimator from O(b) to O(b²) by employing one of the bias correction techniques twice. However, it is doubtful whether there is much gain in practice from the bias correction via iteration, and thus we do not consider such inferior cases any further.

8. Carrasco and Chen (2002) provide the conditions that make GARCH processes stationary and β-mixing with exponential decay, whereas (2010) discuss the conditions that can establish β-mixing with exponential decay in scalar diffusion processes. In relation to the latter, Feller's square-root process and its inverse, which have been employed as models of short-term interest rates by (1985) and Ahn and Gao (1999), respectively, are examples of exponentially decaying β-mixing processes. Since β-mixing implies strong mixing, their assumption can cover many important applications in economics and finance.