## 1 Introduction

Let , be independent realizations of

-dimensional random variable

having an unknown continuous-variate probability density function

. In this chapter, we concentrate on the problem of estimating by kernel density estimator, in which with support can be estimated by a variate classical kernel estimator (see, for example, Silverman, 1986; Wand and Jones, 1995). But this causes boundary bias in case of bounded or semi-bounded support. To solve this problem in univariate set-up, the associated kernels are proposed (see, for example, Chen, 1999, 2000; Libengué, 2013; Igarashi and Kakizawa, 2014), whereas, in multivariate set-up, the boundary bias can be omitted by using the product of univariate associated kernels (see, for example, Bouerzmarni and Rombouts, 2010). In the context of multivariate associated kernel, Kokonendji and Somé (2018) propose a bivariate beta kernel with a correlation structure. Now, when the support is a cartesian product of and bounded or semi-bounded sets, can be estimated using the product of univariate classical kernels and univariate associated kernels. Here, in particular, we consider the estimation of a bivariate density function with support .In this regard, Section 2 contains the properties of the estimators based on the product of a univariate classical kernel and a univariate gamma kernel. Section 3 provides bivariate density estimators based on normal-gamma () kernels. Section 4 discusses the relative performances of the estimators through simulation followed by data study in Section 5. Section 6 has the discussion, whereas some technical details are deferred to the Appendix.

## 2 Product of classical and gamma kernels

Consider a bivariate continuous density function satisfying (i) , (ii) is twice continuously partially differentiable on , (iii) and . To estimate , we consider the estimator as follows

(1) |

where is the classical kernel satisfying (a) , (b) , (c) and (d) , with bandwidth satisfying and as . is the first class of gamma kernels (Chen, ) defined as

where is the gamma function, with bandwidth satisfying and as . Bandwidths of the kernels are so chosen as to make the amount of smoothing in the same scale for both the kernels. In general, any associated kernel can be used here. However, we choose the gamma kernel due to its flexible properties (see, for example, Chen, ).

Now, using , we get

(2) |

where follows gamma.

Again, Taylor series expansion gives as

where and . Then, substituting the last expression in (2), we get

which implies is

(3) |

This shows estimator is free of boundary bias and the corresponding integrated squared bias is given by

(4) |

Now,

and

where follows gamma and . Lemma of Brown and Chen gives

if , | ||||

if (a non-negative constant), |

which implies

if , | |||||

if , | |||||

(5) |

where . Expressions (3) and (2) imply that for and as , the nonparametric density estimator is consistent for the true density function at each point x. Now, for with ,

(6) |

provided is finite.

Combining (2) and (6), the mean integrated squared error (MISE) is obtained as

(7) |

and the leading terms in (7) give the expression of the corresponding asymptotic mean integrated squared error (AMISE). AMISE is optimal for and , where and are constants, i.e. the optimal bandwidths for kernel density estimator are and which give as

For , the optimal is

which gives as

Another estimator of is considered as

(8) |

where is the second class of gamma kernels (Chen, ) defined as

if , | |||||

if . | (9) |

So, , given by

if , | |||||

if , | |||||

(10) |

which shows the boundary unbiasedness of estimator and for a non-negative constant ,

if , | ||||

if , |

imply

For , the optimal is

which corresponds to , given by

Observe that implies is expected to have a better asymptotic performance than .

## 3 Bivariate density estimation using kernel

Consider the density function

of a bivariate normal-gamma distribution defined as (Bernardo and Smith,

)(11) |

with , where and , respectively, stand for normal and gamma distributions. Using (3), we define the following estimator of as

(12) |

where is the kernel with such that the bandwidths and as Then,

where follows , which implies and . By Taylor series expansion we get (see, Appendix A.1),

Therefore, is given by

(13) |

which shows estimator is free of boundary bias, and the integrated squared bias is

(14) |

The variance of

isif , | ||||

if , | ||||

if , | ||||

if , , |

for non-negative constants (see, Appendix A.2), and

(15) |

assuming (see, Appendix A.3).