Estimation of the shape of the density contours of star-shaped distributions
Elliptically contoured distributions generalize the multivariate normal distributions in such a way that the density generators need not be exponential. However, as the name suggests, elliptically contoured distributions remain to be restricted in that the similar density contours ought to be elliptical. Kamiya, Takemura and Kuriki [Star-shaped distributions and their generalizations, Journal of Statistical Planning and Inference 138 (2008), 3429–3447] proposed star-shaped distributions, for which the density contours are allowed to be boundaries of arbitrary similar star-shaped sets. In the present paper, we propose a nonparametric estimator of the shape of the density contours of star-shaped distributions, and prove its strong consistency with respect to the Hausdorff distance. We illustrate our estimator by simulation.
Key words: density contour, direction, elliptically contoured distribution, Hausdorff distance, kernel density estimator, star-shaped distribution, strong consistency.
MSC2010: 62H12, 62H11, 62G07.
Elliptically contoured distributions generalize the multivariate normal distributions in such a way that the density generators need not be exponential (Fang and Zhang ). In this way, the class of elliptically contoured distributions includes, for example, distributions whose tails are heavier than those of the multivariate normal distributions. However, as the name suggests, elliptically contoured distributions remain to be restricted in that the similar density contours ought to be elliptical. Hence, in particular, no skewed distributions are members of this class. Skew-elliptical distributions (Genton ) allow skewness by introducing an extra parameter into elliptically contoured distributions.
Kamiya, Takemura and Kuriki  proposed a flexible class of distributions called star-shaped distributions, for which the density contours are allowed to be boundaries of arbitrary similar star-shaped sets (see also , ). Essentially the same idea can be found in -spherical distributions by Fernández, Osiewalski and Steel  and center-similar distributions by Yang and Kotz . Skewness as well as heavy-tailedness is allowed in star-shaped distributions. Thus, besides (centrally, reflectively or in some other ways) symmetric distributions such as elliptically contoured distributions and -spherical distributions, the class of star-shaped distributions also includes asymmetric distributions.
Kamiya, Takemura and Kuriki  studied distributional properties of star-shaped distributions, including independence of the “length” and the “direction,” and robustness of the distribution of the “direction.” However, they did not explore inferential aspects of star-shaped distributions. From the perspective of , the most important problem in the inference for star-shaped distributions is the estimation of the shape of the density contours.
In the present paper, we propose a nonparametric estimator of the shape of the density contours. The point is that the density of the usual direction under a star-shaped distribution is in one-to-one correspondence with a function which determines the shape of the density contours. Thus, by nonparametrically estimating the density of the direction, we can obtain a nonparametric estimator of the shape. We prove its strong consistency with respect to the Hausdorff distance.
In a recent paper, Liebscher and Richter  presented examples of parametric modeling and estimation concerning the shape of the density contours of two-dimensional star-shaped distributions (Section 2.2 as well as Sections 3.3 and 3.4 of ). They also investigated estimation about many other aspects of star-shaped distributions.
The organization of this paper is as follows. We describe a star-shaped distribution and define the shape of its density contours in Section 2. Next, we propose an estimator of the shape of the density contours of a star-shaped distribution in Section 3.1, and prove its strong consistency in Section 3.2. We illustrate our estimator by simulation in Section 4, and conclude with some remarks in Section 5.
2 Star-shaped distribution and the shape of its density contours
In this section, we describe a star-shaped distribution and define the shape of its density contours.
Suppose a random vector , is distributed as
where is continuous and equivariant under the action of the positive real numbers: for all . In (1), it is implicitly assumed that satisfies . In the particular case that for a positive definite matrix ( denotes the transpose of the column vector ) and that the density generator is exponential: , we obtain the multivariate normal distribution .
and write for . Then the density is constant on each of : for all . When is injective (e.g., strictly decreasing), each , is a contour of the density : , but in general, a contour of the density is a union of some ’s: .
Noticing that is a star-shaped set with respect to the origin, we say that in (1) has a star-shaped distribution. Also, we call the shape of the density contours of this star-shaped distribution, including cases where is not injective. When is strictly decreasing, is a density level set: .
3 Estimation of the shape
3.1 Proposed estimator
In this subsection, we propose an estimator of the shape .
Let denote the Euclidean norm. Under (1), the direction is distributed as
where stands for the volume element of and (Theorem 4.1 of ). Note the function in (3) is continuous and satisfies for all . Throughout this section (Section 3), we assume is taken so that and hence .
Now, we can write for . Thus, for , the condition that is equivalent to . Hence , and we can estimate by estimating the density of .
Suppose we are given an i.i.d. sample from (1), and consider estimating based on , where .
Let be an estimator of such that for all . Define the estimator of by
Then is also a star-shaped set with respect to the origin.
3.2 Strong consistency
In this subsection, we prove strong consistency of our estimator .
Let be the Hausdorff distance between and :
where , and denotes the Minkowski sum. Similarly, let be the Hausdorff distance between and . We note that and may not be compact. The purpose of this section is to show that, under some conditions, and converge to zero almost surely.
We begin by proving that and are bounded by :
Let be an arbitrary point of . Take . Then , and thus . This argument implies that . Similarly, holds true. Therefore, the second inequality in (4) is proved. The proof of the first inequality in (4) is similar.
Next we want to verify that almost surely for estimators having a certain property.
For each and each , we can write
for some between and .
Let . Then we have for all and all , and thus
for all . Since is continuous, is compact and for all , we have . Now, suppose the estimator satisfies
Then, with probability one, we have for all sufficiently large . Together with this fact, inequality (6) implies that, with probability one,
for all sufficiently large .
for all sufficiently large . Therefore, by (7) we obtain , as was to be verified.
Now, for estimating a general density on (i.e., not necessarily in (3)) based on an i.i.d. sample from , we can use the following kernel density estimator (Hall, Watson and Cabrera , Bai, Rao and Zhao ):
where , (), and satisfies . Notice that does not depend on and can be written as (equation (22) of , equation (1.6) of ). Recall, in passing, that the class of kernel estimators of the form (9) virtually “contains asymptotically” the class of kernel estimators of the form , , for a kernel and a smoothing parameter (see Hall, Watson and Cabrera ). The choice , is the von Mises kernel.
A sufficient condition for for a general density on , , and its kernel estimator in (9) was obtained by Bai, Rao and Zhao , Theorem 2: holds true if the following conditions are satisfied: 1. is continuous; 2. is bounded; 3. is Riemann integrable on any finite interval in with ; 4. as ; 5. as .
Note that under the fourth condition , we have (equation (1.7) of ).
The preceding arguments yield the following result:
Let , be an i.i.d. sample from a star-shaped distribution . Let be a kernel estimator of the density of , based on .
Assume the equivariant function under the action of the positive real numbers is continuous and normalized so that , and that is bounded and satisfies . Moreover, suppose is taken in such a way that and as .
Then, is a strongly consistent estimator of the shape of the density contours of the star-shaped distribution in the sense that the Hausdorff distance between and satisfies
In addition, is a strongly consistent estimator of :
It can easily be seen that and ( if and otherwise) satisfy and the other conditions of Theorem 3.1.
4 Illustrations by simulation
In this section, we illustrate our estimator by simulation.
In both cases, we use the von Mises kernel . We do not normalize , so is not equal to one in general and our estimator of is . We obtain the kernel estimator by making use of the R package circular222C. Agostinelli and U. Lund (2013). R package circular: Circular Statistics (version 0.4-7). URL https://r-forge.r-project.org/projects/circular/ . We select the bandwidth by simple trial and error. (If we did not know the true shape, we could use, e.g., cross-validation for minimizing the squared-error loss or the Kullback-Leibler loss in order to select the bandwidths ().)
Although we employ specific functions for below, these choices do not affect the estimation of (and hence of ) based on . This is because for , and the distribution of does not depend on (Theorem 4.1 of ).
4.1 Triangular shape
Essentially as in Example 3.1 of , we choose , which necessarily implies because of and . Hence, our star-shaped distribution is .
We can generate by , where is distributed as the Rayleigh distribution with scale parameter 1 (i.e., ), has density (with respect to the line element) on sides , respectively (Example 3.1 of ), and and are independently distributed.
Our estimator of is .
The true shape (blue, dashed line) and its estimator (red, solid line) for are shown in Figure 1.
4.2 -spherical shape
We take , . Then is the unit -sphere. We can calculate .
We choose , so , say. Then and hence . Thus, our star-shaped distribution is .
This star-shaped distribution is obtained as the distribution of with and being independently distributed according to the -generalized normal distribution with (this does not indicate the dimension of ): . We generate , by using the R package pgnorm333Steve Kalke (2015). pgnorm: The -Generalized Normal Distribution. R package version 2.0. https://CRAN.R-project.org/package=pgnorm .
Our estimator of is .
For visibility, we enlarge the shape and its estimator, and display (blue, dashed line) and (red, solid line) for in Figure 2.
5 Concluding remarks
In this paper, we proposed a nonparametric estimator of the shape of the density contours of star-shaped distributions, and proved its strong consistency with respect to the Hausdorff distance.
We can introduce the location parameter and consider a star-shaped distribution whose density contours are (unions of) boundaries of star-shaped sets with respect to the location. In that case, one possibility for estimating the shape is to plug in an estimator of the location and use our proposed nonparametric estimator of the shape. We might be able to estimate the location by characterizing it in some way. For example, if the star-shaped distribution may be assumed to be centrally symmetric about the location and have a finite first moment, the location can be characterized as the mean and may be estimated by, e.g., the sample mean. If, instead, in (1) is strictly decreasing, the location can be regarded as the mode and be estimated by means of various methods for estimating the multivariate mode.
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