The purpose of the multidimension uniformity test is to check whether the underlying probability distribution of a multidimensional population differs from the multidimensional uniform distribution. The multidimensional uniformity test has applications in various fields such as biology, astronomy, and computer science. Such a test, however, has received less attention in the literature compared with the univariate case. A new test statistic for checking multidimensional uniformity is proposed in this paper. Some important properties of the proposed test statistic are discussed. As a special case, the bivariate statistic test is discussed in detail in this paper. The Monte Carlo simulation is used to compare the power of the newly proposed test with the distance-to-boundary test, which is a recently published statistical test for multidimensional uniformity. It has been shown that the test proposed in this paper is more powerful than the distance-to-boundary test in some cases.
Testing uniformity in the univariate case has been studied by many researchers, whereas the multidimensional uniformity test seems to have received less attention in the literature. Testing whether a pattern of points in the multidimensional space is distributed uniformly has applications in many fields such as biology, astronomy, and computer science. A commonly used goodness-of-fit test for uniformity is the chi-square test [
The main purpose of this paper is to propose a new test statistics for testing multidimensional uniformity. It is expected that the newly proposed test may improve the power of the multidimensional uniformity tests. Since the distance-to-boundary test is a recently published test in multivariate case, the power of test proposed in this paper will be compared with the power of the distance-to-boundary test. While the statistical test can be used for any multidimensional case, the discussion will be mainly based on the bivariate case. Some techniques used in nonparametric statistics are adopted to modify the test statistic for the purpose of raising the power of the test for the bivariate case.
Suppose
The test statistic proposed in this paper is defined as
Here it is assumed that
In order to raise the power of the test, the test statistic defined in (
No comparison is made if
Let
Define
It can be seen that if the underlying distribution is a uniform distribution on
It can be shown that
To show
It should be mentioned that the lower and upper bounds of the above inequality cannot be improved. It fact, one may construct bivariate data sets easily such that the values of
Monte Carlo simulation is used to find the critical values of the test statistic described in (
Critical values of
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|
|
|
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5 | 0.05194 | 0.06722 | 0.08300 | 0.10584 |
6 | 0.03355 | 0.04323 | 0.05385 | 0.06894 |
7 | 0.02321 | 0.02997 | 0.03695 | 0.04726 |
8 | 0.01676 | 0.02154 | 0.02678 | 0.03430 |
9 | 0.01256 | 0.01610 | 0.01990 | 0.02562 |
10 | 0.00973 | 0.01250 | 0.01525 | 0.01955 |
11 | 0.00773 | 0.00984 | 0.01210 | 0.01509 |
12 | 0.00618 | 0.00790 | 0.00972 | 0.01212 |
13 | 0.00515 | 0.00657 | 0.00802 | 0.01002 |
14 | 0.00426 | 0.00540 | 0.00657 | 0.00829 |
15 | 0.00359 | 0.00455 | 0.00552 | 0.00698 |
16 | 0.00308 | 0.00388 | 0.00468 | 0.00585 |
17 | 0.00265 | 0.00334 | 0.00404 | 0.00502 |
18 | 0.00231 | 0.00292 | 0.00356 | 0.00439 |
19 | 0.00201 | 0.00254 | 0.00308 | 0.00379 |
20 | 0.00177 | 0.00222 | 0.00270 | 0.00333 |
21 | 0.00158 | 0.00200 | 0.00241 | 0.00298 |
22 | 0.00140 | 0.00176 | 0.00212 | 0.00262 |
23 | 0.00126 | 0.00158 | 0.00191 | 0.00236 |
24 | 0.00113 | 0.00141 | 0.00169 | 0.00211 |
25 | 0.00102 | 0.00127 | 0.00153 | 0.00188 |
26 | 0.00093 | 0.00116 | 0.00139 | 0.00171 |
27 | 0.00085 | 0.00106 | 0.00127 | 0.00156 |
28 | 0.00077 | 0.00096 | 0.00114 | 0.00139 |
29 | 0.00071 | 0.00088 | 0.00106 | 0.00129 |
30 | 0.00065 | 0.00081 | 0.00096 | 0.00117 |
31 | 0.00060 | 0.00074 | 0.00089 | 0.00108 |
32 | 0.00055 | 0.00069 | 0.00082 | 0.00099 |
33 | 0.00052 | 0.00064 | 0.00077 | 0.00094 |
34 | 0.00048 | 0.00059 | 0.00070 | 0.00085 |
35 | 0.00044 | 0.00055 | 0.00065 | 0.00079 |
36 | 0.00041 | 0.00051 | 0.00061 | 0.00075 |
37 | 0.00039 | 0.00048 | 0.00057 | 0.00070 |
38 | 0.00036 | 0.00045 | 0.00053 | 0.00064 |
39 | 0.00034 | 0.00042 | 0.00050 | 0.00060 |
40 | 0.00032 | 0.00040 | 0.00047 | 0.00057 |
41 | 0.00030 | 0.00037 | 0.00044 | 0.00053 |
42 | 0.00028 | 0.00035 | 0.00041 | 0.00050 |
43 | 0.00027 | 0.00033 | 0.00039 | 0.00047 |
44 | 0.00025 | 0.00031 | 0.00037 | 0.00044 |
45 | 0.00024 | 0.00029 | 0.00035 | 0.00041 |
46 | 0.00023 | 0.00028 | 0.00033 | 0.00040 |
47 | 0.00021 | 0.00026 | 0.00031 | 0.00037 |
48 | 0.00020 | 0.00025 | 0.00029 | 0.00035 |
49 | 0.00019 | 0.00024 | 0.00028 | 0.00034 |
50 | 0.00018 | 0.00022 | 0.00026 | 0.00032 |
The power of the test statistic proposed in this paper is compared with the recently published distance-to-boundary test by Berrendero et al. [
The probability density function of the univariate Beta distribution is
The bivariate Beta distribution is formed by two independent
Power comparison. Alternative distribution 1. Marginal distribution is Beta
The bivariate Beta distribution is formed by two independent
Power comparison. Alternative distribution 2. Marginal distribution is Beta
The bivariate Beta distribution is formed by two independent
Power comparison. Alternative distribution 3. Marginal distribution is Beta
The Metatype uniform distribution was mentioned in the papers of Liang et al. [
The basic idea for creating metatype multivariate distribution is as follows. Let the random vector
MNU is obtained from bivariate normal distribution with
Power comparison. Alternative distribution 4. MNU: metatype uniform distribution.
MTU is obtained from bivariate Student’s-
Power comparison. Alternative distribution 5. MTU: metatype uniform distribution.
In this paper, the new multidimensional uniformity test is proposed. The basic idea is from univariate uniform distribution test in the paper of Chen and Ye [
The distance-to-boundary is a recently published multivariate uniformity test by Berrendero et al. [
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors sincerely thank two anonymous referees for their comments and suggestions that greatly improved the quality of the paper.