Multivariate Confidence Region (MCR) cannot be used to obtain the confidence region of the mean vector of multivariate data when the normality assumption is not satisfied; however, the Quantile Confidence Region (QCR) could be used with a Multivariate Quantile Vector in these cases. The coverage rate of the QCR is better than MCR; however, it has a disadvantage because the QCR has a wide shape when the probability density function follows a bimodal form. In this study, we propose a Quantile Confidence Region using the Highest density (QCRHD) method with the Highest Density Region (HDR). The coverage rate of QCRHD was superior to MCR, but is found to be similar to QCR. The QCRHD is constructed as one region similar to QCR when the distance of the mean vector is close. When the distance of the mean vector is far, the QCR has one wide region, but the QCRHD has two smaller regions. Based on these features, it is found that the QCRHD can overcome the disadvantages of the QCR, which may have a wide shape.
Under the assumption of a multivariate normal distribution, the multivariate confidence region (MCR) of the mean vector is expressed in the form of an ellipse and sphere in two and three dimensions, respectively (Chew, 1966; Fan and Zhang, 2000; Frank, 1996; Johnson and Whichern, 2002; Sun and Loader, 1994). However, constructing a confidence region using the MCR of the population mean vector is not easy when a large amount of real data does not satisfy the normality assumption (Asgharzadeh and Abdi, 2011). For these non-normal real data, Hong and Kim (2017) proposed the quantile confidence region (QCR) using the multivariate quantile vector suggested by Hong
This study proposed a method of QCR using HDR called a quantile confidence region using highest density (QCRHD) to overcome the disadvantage of QCR. The QCRHD is a method to construct QCR using information obtained from the probability density function. We discuss the characteristics and advantages of the QCRHD by comparing the shapes and coverage rates of the QCR and MCR with various probability density functions. In particular, the shapes and coverage rates of the QCRHD are compared with both QCR and MCR for asymmetric probability density functions; in addition, we also include bimodal probability density functions.
Section 2 introduces a method to obtain the QCRHD, which is also an alternative QCR using HDR. In Section 3, the QCRHDs for various probability density functions that do not follow the normal distribution function are obtained and compared with QCR and MCR. In Section 4, the performance and coverage rate of QCRHD is evaluated and compared with the QCR and MCR. Finally, Section 5 discusses future studies and derives some conclusions.
We introduce an alternative QCR method using the HDR, which is called the QCRHD. Before explaining the multivariate QCRHD generally, we discuss only the bivariate QCRHD that can be visually explained. First, in order to use the HDR method, the (1 –
where
Whereas the QCR of Hong and Kim (2017) is derived using upper
For a given
For an initial
With the set of the vector
For a given (
where
Repeat Step 1 through 3 until
Finally, (1–
If there exist two sets of the vector
The shapes of the 95% QCRHD, QCR, and MCR are obtained and visualized, and their characteristics are discussed under the following mixture distributions:
where
In the case of
The common features of the QCRHD, QCR, and MCR can be seen in Figure 3. First, the QCRHD, QCR, and MCR narrow as the
Next, a bimodal probability density function where the distance between the mean vector is sufficiently long, i.e.,
The common characteristics on the shapes of the QCRHD, QCR, and MCR with respect to
It is remarkable in Figures 3 and 4 that the shapes of the QCRHD are different, whereas those of the QCR are similar. First, the QCRHD in Figure 4 consists of two regions. Note that the QCRHD in Figure 3 only has one region. Among the two regions of the QCRHD, the region located at the lower left has a similar shape to the QCR. However, the upper right region has a more angled lower bound than the QCR. These characteristics stand out when
Another comparison of shapes between the QCRHD, QCR, and MCR can be explored in Figure 6 and 8. In both cases, the lower bound of the lower left QCRHD almost overlap with the QCR. It also can be found that the QCR is always composed of one region, even when the probability density function has a bimodal form. While the QCRHD and QCR have similar regions in Figure 6 where the mean vectors are close, the QCRHD has a smaller region than the QCR in Figure 8 where mean vectors are distant. When
In the case of MCR, it is in the form of an ellipse regardless of
As mentioned at the end of Section 2, when there exist two sets of the vector
In Section 3.3, the normal mixture distribution with different variance-covariance matrix,
It could be observed that the QCR also has similar characteristic with the QCRHD, and its lower bound in Figure 5 has a longer length than in Figure 4. The MCR still has the form of a single ellipse, but the MCR in Figure 4 has a slimmer shape than Figure 5.
Figure 5 provides only the case of
We have found that the QCRHD can overcome the disadvantage of the QCR which has wide region. In this section, we compare the performance of the QCRHD with the QCR and MCR. After obtaining the 95% QCRHD, QCR, and MCR, their coverage rates are calculated and compared. In order to calculate coverage rates, 100 sample means are generated from (
When
In case of
Whereas the coverage rates of the QCRHD, QCR, and MCR under the normal mixture distribution are considered with equal variance-covariance matrix in Sections 4.1 and 4.2, one compares their coverage rates under (
The blue, red, green lines in Figure 10 are represented by shapes of the QCRHD, QCR, and MCR, respectively according to the change of
The coverage rates of the 95% QCRHD, QCR, and MCR are provided in Table 3. Coverage rates of the QCRHD and QCR are close to 0.95, whereas the MCR has smaller value than 0.95. Therefore, one might conclude that the QCRHD has better performance than the MCR in this Section.
Table 3 provides only the case of
The well-known MCR cannot be used when the multivariate data does not satisfy the normality assumption. Hong and Kim (2017) proposed a QCR using the multivariate quantile vector and showed that the coverage rate of the QCR is better than MCR when the assumption of normality is not satisfied.
However, the QCR has a disadvantage of a wide region when the probability density function has the bimodal form. In order to overcome the disadvantage of the QCR, an alternative QCR, called QCRHD, is proposed using the HDR of Hyndman (1996) that can utilize information obtained from the probability density function.
The QCRHD has either one big region or two small regions whose shapes are dependent on the type of distribution. The QCRHD has one single region for mixture distributions with a short distance between mean vectors or heavy tails of the variance-covariance matrix. However, the QCRHD is represented by two small regions when the distance between the mean vector of mixture distribution is large or the variance-covariance matrix has a light tail.
The results of this study show that the QCRHD has better coverage rates than MCR with various probability density functions and shows a similar performance with the QCR. In addition, the QCRHD has an advantage that it has smaller region than the QCR. Therefore, one could conclude that QCRHD are better than QCR in terms of coverage rates and shapes.
We consider only the cases of one region of the QCRHD and two non-overlapped regions of the QCRHD. When there exist two regions of the QCRHD and their two regions overlap, it is found that the regions of the QCRHD become too narrow, so that the coverage rate of that case becomes worse than those of other cases. Consequently, we consider two cases in this work. One is that there exists only one set of the vector
The discussion on the QCRHD is restricted to the bivariate probability density function in this study. However, the QCRHD study might be extended to multivariate probability density functions. Even though the multivariate QCRHD cannot be visualized, it is left to a future study to discuss the characteristics of the multivariate QCRHD. In addition, research on cases greater than a trimodal will be an interesting problem.
Comparison of coverage rates between QCRHD, QCR, and MCR when
95% QCRHD | 95% QCR | 95% MCR | |||||
---|---|---|---|---|---|---|---|
Coverage | SE | Coverage | SE | Coverage | SE | ||
0.3 | −0.5 | 94.57 | 2.22 | 95.25 | 2.06 | 96.87 | 1.71 |
0.0 | 94.78 | 2.18 | 95.05 | 2.18 | 96.61 | 1.79 | |
0.5 | 94.78 | 2.14 | 95.02 | 2.24 | 96.38 | 1.84 | |
0.5 | −0.5 | 95.10 | 2.09 | 94.92 | 2.12 | 97.35 | 1.60 |
0.0 | 94.50 | 2.25 | 95.10 | 2.18 | 97.18 | 1.69 | |
0.5 | 95.07 | 2.10 | 95.10 | 2.12 | 97.02 | 1.74 | |
0.7 | −0.5 | 94.95 | 2.24 | 95.13 | 2.13 | 96.92 | 1.70 |
0.0 | 94.91 | 2.21 | 94.96 | 2.14 | 96.56 | 1.83 | |
0.5 | 94.86 | 2.06 | 95.05 | 2.19 | 96.33 | 1.78 |
QCRHD = QCR using highest density; QCR = quantile confidence region; MCR = multivariate confidence region.
Comparison of coverage rates between QCRHD, QCR, and MCR when
95% QCRHD | 95% QCR | 95% MCR | |||||
---|---|---|---|---|---|---|---|
Coverage | SE | Coverage | SE | Coverage | SE | ||
0.3 | −0.5 | 95.09 | 2.27 | 95.15 | 2.13 | 96.91 | 1.70 |
0.0 | 95.19 | 2.14 | 94.98 | 2.21 | 96.55 | 1.79 | |
0.5 | 95.07 | 2.24 | 95.11 | 2.10 | 96.89 | 1.74 | |
0.5 | −0.5 | 94.93 | 2.14 | 95.21 | 2.11 | 97.29 | 1.58 |
0.0 | 95.18 | 2.13 | 95.00 | 2.13 | 97.24 | 1.61 | |
0.5 | 94.98 | 2.20 | 94.97 | 2.19 | 97.36 | 1.57 | |
0.7 | −0.5 | 95.07 | 2.11 | 95.10 | 2.14 | 96.79 | 1.73 |
0.0 | 94.92 | 2.15 | 95.11 | 2.07 | 96.76 | 1.75 | |
0.5 | 94.91 | 2.26 | 95.01 | 2.19 | 96.79 | 1.75 |
QCRHD = QCR using highest density; QCR = quantile confidence region; MCR = multivariate confidence region.
Comparison of coverage rates between QCRHD, QCR, and MCR when
95% QCRHD | 95% QCR | 95% MCR | |||||
---|---|---|---|---|---|---|---|
Coverage | SE | Coverage | SE | Coverage | SE | ||
−0.5 | 0.3 | 94.90 | 2.05 | 95.05 | 2.11 | 90.49 | 2.52 |
0.5 | 95.02 | 2.05 | 95.12 | 2.31 | 92.46 | 2.61 | |
0.7 | 95.11 | 2.15 | 94.95 | 2.17 | 90.95 | 2.80 |
QCRHD = QCR using highest density; QCR = quantile confidence region; MCR = multivariate confidence region.