Point-Symmetric Multivariate Density Function and Its Decomposition

Abstract

For a -variate density function, the present paper defines the point-symmetry, quasi-point-symmetry of order (), and the marginal point-symmetry of order and gives the theorem that the density function is -variate point-symmetric if and only if it is quasi-point-symmetric and marginal point-symmetric of order . The theorem is illustrated for the multivariate normal density function.

1. Introduction

For square contingency tables, it is known that the symmetry model holds if and only if both the quasi-symmetry and the marginal homogeneity models hold (e.g., see Caussinus [1]; Tomizawa and Tahata [2]). For multiway contingency tables, Bhapkar and Darroch [3] defined the complete symmetry, quasi-symmetry, and marginal symmetry models and showed that the complete symmetry model holds if and only if both the quasi-symmetry and the marginal symmetry models hold.

Tomizawa et al. [4] gave a similar decomposition for the bivariate density function (instead of cell probabilities). Iki et al. [5] showed a similar decomposition for the multivariate density function.

On the other hand, for contingency tables, Wall and Lienert [6] defined the point-symmetry model for the cell probabilities, and Tomizawa [7] gave the theorem that the point-symmetry model holds for the cell probabilities if and only if both the quasi-point-symmetry and the marginal point-symmetry models hold (see also Tahata and Tomizawa [8]).

Tomizawa and Konuma [9] gave a similar decomposition for the bivariate point-symmetric density function. Now, we are interested in extending the decomposition of the point-symmetric density function to multivariate case.

In the present paper, we define the point-symmetry, quasi-point-symmetry, and marginal point-symmetry for the multivariate density function and decompose the point-symmetry into quasi-point-symmetry and marginal point-symmetry. Section 2 provides the decomposition for the trivariate density function. Section 3 extends the decomposition to multivariate density function. Section 4 illustrates our decomposition for the multivariate normal distribution.

2. Decomposition of Trivariate Density Function

Let , and be three continuous random variables with a density function , where with and where and , or and are finite. Let () denote a given point in domain , where if and are finite. Let when for . For example, when with , then . Note that, for , (i) is the symmetrical value of with respect to , (ii) , and (iii) .

We will define the point-symmetry (denoted by ) of density function with respect to the point by

Let , and be the marginal density functions of , and , respectively. For the density function , we will define the marginal point-symmetry of order 1 (denoted by ) by Let be the marginal density function of for . We define the marginal point-symmetry of order 2 (denoted by ) by Note that implies .

We can express the density function as where , and with similar properties of , and . The terms () correspond to main effects of the variable , () to interaction effects of and , and to interaction effect of , , and . We see with similar properties of , and . The term indicates the odds of density function with respect to -values with . Note that Thus, indicates the odds ratio of density function with respect to ()-values with . Also indicates the ratio of odds ratios of density function, that is, the ratio of odds ratio with respect to ()-values with to that with (or the ratio of odds ratio with respect to ()-values with to that with , where and ).

The density function is if and only if it is expressed as form (6) with

We will define the quasi-point-symmetry of order 1 (denoted by ) by (6) with The is equivalent to where with and so on. Therefore, indicates that the density function is point-symmetric with respect to the odds ratio.

Also, we will define the quasi-point-symmetry of order 2 (denoted by ) by (6) with The is equivalent to Therefore, indicates that the density function is point-symmetric with respect to the ratio of odds ratios. We note that implies . We obtain the following theorem.

Theorem 1. For fixed , the trivariate density function is if and only if it is both and .

Proof. Consider the case of . If a density function is , then it satisfies and . Assume that it is both and , and then we will show that it satisfies .
Let , and be three continuous random variables with a density function which satisfies both and . Therefore, we see where ,  , and .
Let where Note that . Then we have where .
Since satisfies , we see where and is the marginal density function of for . Denote (20) as () for .
Consider the arbitrary density function satisfying with From (19), (20), and (21), we have Using (22), we obtain where is the Kullback-Leibler information; that is, For fixed, we see and then uniquely minimizes .
Let for . Since satisfies , we see Since satisfies , we see where .
Consider the arbitrary density function satisfying with where . In a similar way, we see Thus, we obtain For fixed, we see and then uniquely minimizes . Therefore, we see . Thus, . Namely, satisfies . The case of can be proved in a similar way as the case of . So the proof is completed.

3. Decomposition of Multivariate Density Function

Let be continuous random variables with a density function , where for and is defined in a similar way to . Let denote a given point in , where if and are finite. Let when for . For the density function , we will define the point-symmetry (denoted by ) with respect to the point by Also, for , we will define the marginal point-symmetry of order (denoted by ) by where is the marginal density function of (). We note that implies ().

We can express the density function as where , and Then, the density function being is also expressed as (34) with For , we will define the quasi-point-symmetry of order (denoted by ) by (34) with We note that implies (). Then we obtain the following theorem.

Theorem 2. For fixed , the multivariate density function is if and only if it is both and .

The proof of Theorem 2 is omitted because it is obtained in a similar way to the proof of Theorem 1.

4. Point-Symmetry of Multivariate Normal Density Function

Consider a -dimensional random vector having a normal distribution with mean vector and covariance matrix . The density function is Denote by with . Then the density function can be expressed as where is positive constant and For an arbitrary given point , we set and (). Then noting that (), we see where is positive constant and Thus, and for , Since (), we see Therefore, the normal density function is for , without depending on the value of and on the values of parameters and . Thus, we see from Theorem 2 that, for fixed , the normal density function is if and only if is . Therefore, we see that the normal density function is not with respect to the point where , and it is only with respect to without depending on the value of . We see from Theorem 2 that when the normal density function is not , it is caused by the lack of the structure of .

5. Discussion

When a density function is not point-symmetric, Theorem 2 may be useful for knowing the reason, that is, for fixed, which structure of quasi-point-symmetry of order and marginal point-symmetry of order is lacking.

For symmetry of a multivariate distribution, there are various kinds of symmetry such as spherical symmetry, elliptical symmetry, and central symmetry (see, e.g., Kotz et al. [10, pages 5338–5341], Fang et al. [11, Chapter 2], Muirhead [12, pages 32–34], and Tong [13, Chapter 4]). The described in the present paper is equivalent to the central symmetry. Also, for the -variate spherical (elliptical) distribution, the probability density function is with respect to the mean vector, although when the density function is , the distribution is not always spherical (elliptical). Thus, for the -variate spherical (elliptical) distribution, the density function is and () with respect to the mean vector. We point out that, as described in Section 4, for -variate normal distribution, the density function is () with respect to the arbitrary point () (not only mean vector ()).

Testing spherical symmetry and elliptical symmetry is described in, for example, Fang and Zhang [14, Chapter 5], Muirhead [12, page 333], and Kotz et al. [10, pages 5341-5342]. Heathcote et al. [15] gave a procedure for testing a general multivariate distribution for symmetry about a point which indicates that the imaginary part of the characteristic function of centered variable vanishes identically. Although the readers may be interested in seeing the comparison of both approaches and the decomposition of into and , it seems difficult.

As (6), we have considered the multiplicative form of probability density function by the terms of the odds, the odds ratios, the ratios of odds ratios, and so on; as an analog to the log-linear model for the analysis of categorical data (Agresti [16, Chapter 9]). Although the readers also may be interested in the additive form of density function for point-symmetry, it seems difficult to consider it.

On discrete probability, the concept of odds ratio is important. Also it is important to use the odds ratio on probability density function (corresponding to a continuous random variable). For example, for bivariate probability density function , the odds ratio equals 1 for any , and fixed and , if and only if two variables are independent. So we are interested in how the structures of odds ratios, the ratios of odds ratios, and so on, of probability density function, are, for example, the point-symmetry. Note that Holland and Wang [17], Kotz et al. [18, page 74], and Tong [13, Chapter 4] discuss the properties of bivariate probability density function using the odds ratios, for example, as the local dependence function and the totally positive of density function, although the details are omitted.

In Section 4, we have shown that, for the -variate normal distribution, the density function is always but not (thus not ) with respect to the arbitrary point where is not equal to mean vector (). The readers may be interested in the probability density function such that it is not but it is . Consider the following density function: for () with satisfying . When is odd, the density function is with respect to the point because equals for (). Thus, from Theorem 2, when is odd, this density function is and (). However, when is even, the density function (47) is not . Also, for , the marginal density function of () is Namely, this is the uniform distribution. Therefore, the density function (47) is always () with respect to the point () without depending on whether is odd or even. In addition, when is even, the density function (47) is not (), because then and for .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors would like to thank two referees for their many helpful comments.