Comparing nonparametric density estimators using different canonical kernel functions

In this article we make a comparison between three different nonparametric density estimators .The first estimator is called fixed kernel which uses a fixed bandwidth. The second estimator is called variable kernel, which uses variable bandwidth, then the third estimator, refers to a new version of nonparametric density estimator (this estimator is a hybrid of the above estimators. This comparison depends on using different kernels, two of these functions are proposed in this article .To make this comparison suitable, we use standardization techniques, which are called canonical kernel. Simulation is used to find kernel function based on the standardization techniques (canonical kernel).


Introduction
The main property that describes the behavior of the random variables X can be seen from the probability density function.From the density function can be known the number of observations that lie in some groups.
The probability can be calculated as an integral on that group This value x lie in the interval that mentioned above, this will lead to prediction that the observations will appear frequently.Also, through using the graphical plots for probability density function.We will know that the observation may have long tails; also we can see that the observations have multimodality.Estimating the unknown probability density function will provide the researcher understanding for the behavior of the random variables, but in most studies the distribution is unknown and instead of that will depend on the given observation that regarded as identically and independent random variables have unknown probability density.
The goal is to estimate the probability density function on the basis of these observations.The methods are used in this article is of a nonparametric kind that does not have any restrictions on the form of the probability density function.These methods are estimating the density completely based on observations.Also in this article we want to compare different kernels, where two kernels are proposed here.This comparison rely on three nonparametric density estimators, one of these is a new version of estimators.

Kernel density estimators
The proposed kernel estimator is: Where ) ( ˆx f V refers to variable kernel estimator that has the form This estimator rely on using variable bandwidth at each point through using big window at low density and using small window at high density .The idea of this estimator come back to Breiman, Meisel and Purcell (1977).
The other nonparametric density estimator is , which refers to fixed kernel estimator that has the form: This estimator calls fixed because its rely on using fixed bandwidth on the real line of the density function.This estimator proposed from Rosenblatt (1956) and Parzen (1962).
We must note that the proposed estimator consists of two unknowns (K, h), that is kernel function and bandwidth.Therefore we will display the estimator of h and the possible kernel functions, and then we compare the performance of the proposed nonparametric estimator with the other estimators depending on the different kernel functions and using the standardization or rescale techniques to find the best estimator and kernel function.

Kernels
Kernel functions have different names such as window function or weight function.These functions are real, symmetric, continuous and these integrals equal to one.That is: Be sure that these conditions are satisfied when r=2.The following table display some kernels and there efficiencies: Refers to some kernels with two proposed kernels and their efficiencies relative to Epanchnikov The relative efficiency for the above kernels can be obtained from the following:

Canonical Kernels
We must note that if one used a kernel of the form , and rescale the bandwidth by the factorδ , one would obtain the estimate as with the original kernel estimator.
Kernel can therefore be seen as equivalence class of functions K with possible rescaling byδ .
A consequence of this scale dependence is that the bandwidth selection problem is not identifiable if the kernel is determined only up to scale.
Consider the situation in which two statisticians analyze the same data set but use different kernels for their estimators.Their bandwidths have been determined subjectively or automatically, but they have been computed for different kernels and therefore can't be compared directly.
In order to allow some comparison one needs a common scale for both bandwidths.To find such a scale, we must note that two kernel estimators with the same bandwidth should ascribe the same amount of smoothing to the data.Such a scale is given by canonical kernels in the class of kernels δ Where: To Scaling the bias and the variance equally we choose * δ such that: We call the parameter * δ the canonical bandwidth or canonical smoothing parameter for kernel * δ K and * δ can be achieved by defining Also we should be noting that: And the canonical bandwidth is:

Simulation
To know the performance of each of the kernels density estimators (Fixed, Variable and the proposed) with the kernel functions we do some simulation experiments.
In these experiments we use three different sizes (40,125,160), each with different distributions (Normal, Bimodal and contaminated distributions).
We repeat each experiment 100 once for each sample size and distribution.Also we use a plug-in estimator for estimating the bandwidth .The common distance measure that it used is mean averaged squared error (MASE) The following tables illustrate the values of   From the tables above we see the following: v For table (2) and for all the parameters are proposed we find that the proposed nonparametric estimator is the best for the probability density function, then variable kernel and lastly fixed kernel.

Table (3) Refers to MASE for different nonparametric estimators, kernels and distributions when using Bimodal and Contamination distributions
v The Gaussian kernel is the best kernel when we are using the proposed density estimator, then Epanchnikov and the two proposed kernels respectively.
v As a general case we see that the best kernel is Epanchnikov when we are using variable kernel, then the two proposed kernels except at one case when using the distribution N(-2,2) with sample size n=40.
v For the case N (0, 2.5) and n=40, the results refers to that the Gaussian kernel is the best, then Epanchnikov and the two proposed kernels respectively.
v The best kernels when we are using fixed kernel estimator are the two proposed kernels, then Epanchnikov.
v Refer to the results we see that the two kernels *Quartic and Triweight) are less efficient relative to the other kernels when we are using the proposed kernel estimator.
From the table (3) we obtain the following: v The best estimator for the probability density function is the proposed kernel estimator (for all the kernels).
v The Gaussian kernel is the best kernel when we are using the proposed density estimator, then Epanchnikov and the first proposed kernels equally.
v When we use the variable kernel, the results refer to that the best kernels are the first proposed kernel with Epanchnikov, then the second proposed kernels.
v When we use the fixed kernel, the results refer to that the best kernels are the proposed kernels, then Epanchnikov.