New type of gamma kernel density estimator
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Online ISSN 2005-2863 Print ISSN 1226-3192
RESEARCH ARTICLE
New type of gamma kernel density estimator Rizky Reza Fauzi1 · Yoshihiko Maesono2 Received: 17 January 2019 / Accepted: 29 November 2019 © Korean Statistical Society 2020
Abstract We discuss a new kernel type estimator for density function f X (x) with nonnegative support. Here, we use a type of gamma density as a kernel function and modify it with expansions of exponential and logarithmic functions. Our modified gamma kernel density estimator is not only free of the boundary bias, but the variance is also in smaller orders, which are O(n −1 h −1/4 ) in the interior and O(n −1 h −3/4 ) in the boundary region. Furthermore, the optimal orders of its mean squared error are O(n −8/9 ) in the interior and O(n −8/11 ) in the boundary region. Simulation results that demonstrate the proposed method’s performances are also presented. Keywords Convergence rate · Density function · Exponential expansion · Gamma density · Kernel method · Logarithmic expansion · Nonparametric · Variance reduction Mathematics Subject Classification 62G07 · 62G20
1 Introduction Nonparametric methods are gradually becoming popular in statistical analysis for analyzing problems in many fields, such as economics, biology, and actuarial science. In most cases, this is because of a lack of information on the variables being analyzed. Smoothing concerning functions, such as density or cumulative distribution, plays a special role in nonparametric analysis. Knowledge on a density function, or its estimate, allows one to characterize the data more completely. We can derive other characteristics of a random variable from an estimate of its density function, such as the probability itself, hazard rate, mean, and variance value.
B
Yoshihiko Maesono [email protected]
1
Graduate School of Mathematics, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka-shi, Japan
2
Faculty of Science and Engineering, Chuo University, 1-13-27 Kasuga, Bunkyo-ku, Tokyo, Japan
123
Journal of the Korean Statistical Society
Let X 1 , X 2 , ..., X n be independently and identically distributed random variables with an absolutely continuous distribution function FX and a density f X . Parzen (1962) and Rosenblatt (1956) introduced the kernel density estimator (we will call it the standard one) as a smooth and continuous estimator for density functions. It is defined as n 1 x − Xi , K f X (x) = nh h
x ∈ R,
(1)
i=1
where K is a function called a “kernel”, and h > 0 is the bandwidth, which is a parameter that controls the smoothness of f X . It is usually ∞ assumed that K is a symmetric (about 0) continuous nonnegative function with −∞ K (v)dv = 1, as well as h → 0 and nh → ∞ when n → ∞. It is easy to prove that the standard kernel density estimator is continuous and satisfies all the properties of a density function. A typical general measure of the accuracy of f X (x) is the mean integrated squared error, defined as M I S E( fX ) = E
∞
−∞
{ f X (x) − f X (x)}2 w(x)dx ,
(2)
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