TEXT SIZE

search for



CrossRef (0)
Nonparametric analysis of income distributions among different regions based on energy distance with applications to China Health and Nutrition Survey data
Communications for Statistical Applications and Methods 2019;26:57-67
Published online January 31, 2019
© 2019 Korean Statistical Society.

Zhihua Maa, Yishu Xuea, Guanyu Hu1,a

aDepartment of Statistics, University of Connecticut, USA
Correspondence to: 1Corresponding author: Department of Statistics, University of Connecticut, Room 323, Philip E. Austin Building, 215 Glenbrook Rd. U-4120, Storrs, CT 06269, USA. E-mail: guanyu.hu@uconn.edu
Received October 23, 2018; Revised December 4, 2018; Accepted December 5, 2018.
 Abstract

Income distribution is a major concern in economic theory. In regional economics, it is often of interest to compare income distributions in different regions. Traditional methods often compare the income inequality of different regions by assuming parametric forms of the income distributions, or using summary statistics like the Gini coefficient. In this paper, we propose a nonparametric procedure to test for heterogeneity in income distributions among different regions, and a K-means clustering procedure for clustering income distributions based on energy distance. In simulation studies, it is shown that the energy distance based method has competitive results with other common methods in hypothesis testing, and the energy distance based clustering method performs well in the clustering problem. The proposed approaches are applied in analyzing data from China Health and Nutrition Survey 2011. The results indicate that there are significant differences among income distributions of the 12 provinces in the dataset. After applying a 4-means clustering algorithm, we obtained the clustering results of the income distributions in the 12 provinces.

Keywords : income distribution, nonparametric test, energy distance, K-means clustering
1. Introduction

The income distribution in economic theory describes how a region’s total wealth is distributed amongst its population (Sullivan, 2003). Back to classical economists time, income distribution was a key concern of economic theory, as it plays an important role in measuring the health of a region’s economy. For modern economics, more attention is paid to inequalities in certain measurements of income distribution, an example of which is the Gini coefficient (Yitzhaki, 1979), now universally used by many international organizations such as the United Nations and the World Bank. The Gini coefficient, however, summarizes information about a distribution in one number, and loses some information such as the intrinsic structure of the distribution. Other measurements such as the Lorenz curve (Lorenz, 1905), also have such kind of disadvantage.

The probability density of the income distribution can often be estimated when the sample size is large enough. From the 1890s, many parametric distributions were introduced to model income distribution. In 1895, Pareto (1964) (originally published in 1895) first proposed the Pareto density function to model the income distribution. Gibrat (1931) suggested the usage of a two-parameter log-normal distribution. Salem and Mount (1974) proposed a two-parameter Gamma density to approximate the distribution of personal income. Bartels and Van Metelen (1975) suggested another two-parameter density, the Weibull distribution, to model personal income. McDonald (1984) considered two generalized Beta distributions as models for the distribution of income. Furthermore, McDonald and Xu (1995) introduced a five-parameter beta distribution which nests the generalized Beta and Gamma distributions to model income distribution. In summary, these methods used heavy tail distributions to model personal income. When testing the equality of income distributions, people often compared parameters in the parametric models. This method leans on the assumption that the parametric models sufficiently describe both income distributions. Still, some information loss would occur in such comparisons. In addition, it is often not reasonable to assume the same parametric model form for different regions, and such comparison would be impossible when the regional models are parameterized differently.

The energy distance (Székely and Rizzo, 2004) is a measure that is often used to test for equality of distributions. Rizzo and Székely (2010) provided a nonparametric version of the analysis of variance (ANOVA) called the distance components (DISCO), which partitions the total dispersion in the dataset into components that are analogies of ANOVA’s variance components. In addition, the energy distance can also be used in feature selection and generalizations of clustering algorithms (Rizzo and Székely, 2015). Li and Rizzo (2017) extended the usage of the energy distance and proposed k-groups, a generalization of the K-means clustering algorithm. K-groups aims to put similar samples in the same cluster so that the dispersion between the k groups are maximized. The maximization is achieved upon computation of energy distances, instead of Euclidean distances.

In order to address the disadvantages of the Gini coefficient and parametric models, we use the energy distance between different regions to test the equality of income distributions, which is a parametric-free method. Based on the energy distance, we propose a clustering method which clusters not individual observations, but income distributions, in different regions. The main contribution of this work is that all proposed methods are nonparametric - we do not require any parametric assumptions for income distribution.

The remainder of this paper is organized as follows. In Section 2, we give a brief introduction to the energy distance, and propose our testing and clustering procedures. In Section 3, we present the performance of the proposed illustrate our proposed procedures using extensive simulation results. In Section 4, we apply our methods on China Health and Nutrition Survey data. We conclude this paper with a brief discussion in Section 5.

2. Methodology

2.1. Energy distance

The energy distance is a statistical distance defined for two probability distributions. Following Székely and Rizzo (2004), we consider two distributions denoted by FX and FY. The energy distribution between these two distributions is defined as:


De(FX,FY)=2EXYEXXEYY,

where X, X′, Y, Y′ are independent random vectors in 꽋d with finite expectations, and || · || denotes the Euclidean norm. The distance De(FX, FY) equals 0 if and only if FXFY, while a large energy distance corresponds to very different distributions. Hence it provides a parametric free similarity measure of two distributions.

Now consider two independent samples from FX and FY: X = {x1, …, xn} and Y = {y1, …, ym}. From Székely and Rizzo (2004), the energy distance associated with the independent random samples can be written as:


dn,m=nmn+m(2nmi=1nj=1mxiyj1n2i=1nj=1nxixj1m2i=1mj=1myiyj),

where || · || is the Euclidean distance. Based on (2.2), we can calculate the distance between two income distributions.

2.2. Hypothesis testing

For the two-sample problem, suppose we have two samples X1, …, Xn1 and Y1, …, Yn2. The null hypothesis of two-sample is:


H0:F1=F2,

where F1 and F2 are the respective distributions of X1, …, Xn1 and Y1, …, Yn2. The two-sample test statistic is defined using (2.2):

T=n1n2n1+n2(2n1n2i=1n1j=1n2XiYj1n12i=1n1j=1n1XiXj1n22i=1n2j=1n2YiYj).

Let n = n1 + n2 be the total sample size of the sample. Under the null hypothesis, X1, …, Xn1 and Y1, …, Yn2 follow the same distribution. Therefore, the random permutation without replacement from X1, …, Xn1, Y1, …, Yn2 is W1π,,Wn1π,,Wnπ. Then we can calculate the test statistics Tπ based on W1π,,Wn1π,,Wnπ. For a fixed number of permutations, under the null hypothesis, the empirical distribution of the test statistic can be obtained. Based on the empirical distribution of the test statistic, the critical value for the statistic, cα, can be chosen to satisfy:


limnP(Tn>cα)=α,

where α ∈ (0, 1), and Tn is the full permutation test statistic. Proofs of the limiting distribution of test statistics T, the existence of cα and consistency results are given in Székely and Rizzo (2004). From this limiting distribution, we can reject null hypothesis if T > cα.

It is easy to extend the two-sample testing problem to k-sample problem. Suppose X1, …, Xk are k independent samples of random vectors with respective distributions F1, …, Fk. The null hypothesis is:


H0:F1==Fk,

and the alternative is Fi, Fj for some ij.

For Xi and Xj, where ij, we can calculate the energy distance Ti j between Xi and Xj using (2.4). The k-sample test statistic is:


Tk=1i<jkTij.

Similar to in the two-sample case, we can get the critical value from the random permutations from X1, …, Xk.


Multidimensional scaling
  • Assign points to arbitrary coordinates in p-dimensional space

  • Compute the Euclidean distances among all pairs of points, to form the D^ matrix

  • Compare the D^ matrix with the input D matrix by evaluating the stress function. The smaller the value, the greater the correspondence between the two

  • Adjust coordinates of each point in the direction that best minimize the stress

  • Repeat step 2 through 4 until stress won’t get any lower


K-means clustering
  • Set K-means {m(k)} to random values

  • Each data point x(n) is assigned to the nearest mean. We denote our guess of the cluster k(n) that the point x(n) belongs to by k^(n)=argmink(d(mk,x(n))).

    An alternative, equivalent representation of this assignment of points to cluster i given by “responsibilities”, which are indicator variables:Rkn=1 if k^(n)=k,Rkn=0 otherwise.

  • The model parameters, the means, are adjusted to mach the sample means of the data points that they are responsible for m(k)=nrk(n)x(n)R(k),

    where R(k) is the total responsibility of mean k, R(k)=nrk(n).

  • Repeat the step 2 and step 3 until the assignments do not change

2.3. K-means clustering

According to the energy distance calculated by (2.2), we can perform clustering for different samples. Before clustering, we need to describe the relationship between the different samples based on the energy distance. The technique we use is the multidimensional scaling (MDS) (Kruskal, 1964).

The purpose of the MDS is to provide a visual representation for the pattern of proximities among a set of objects, when only a table of the distances between them is given. When the objects are income distributions, the MDS is used to reduce the matrix of distance to two dimensions, which can be plotted on a two-dimensional coordinate system. Similar functions are placed near each other on the map, while very different functions are placed far from each other. See Algorithm 1 for a brief description of the MDS. Based on the MDS, following the steps of K-means clustering in Algorithm 2, we are able to cluster income distributions from different regions using their energy distance matrix.

3. Simulation study

In this section, we set up different scenarios to assess the accuracy of hypothesis testing and K-means clustering using the methods described in Section 2. For hypothesis testing, we test cases of two-sample and multi-sample under the null and alternative hypotheses respectively. Different sample sizes are considered to compare the error rates under different circumstances. For K-means clustering, we consider 2-means clustering procedure within different sample sizes.

3.1. Simulation under the null hypothesis

3.1.1. Two-sample test

We first consider the two-sample tests under two different circumstances. One case is that two groups of data X1 = (x11, …, x1k) and X2 = (x21, …, x2k) are generated from Gamma(2, 0.2), and another case is that X1 and X2 are generated from Log-Normal(0, 1). The test statistic based on energy distance (2.4) is then calculated along with other commonly used test statistics. We considered k ∈ {20, 40, …, 250} to illustrate the performances of hypothesis testing of different statistics under different sample sizes with a significance level of 0.05. For each k, a total of 5,000 replicates are performed. We report the error rates, i.e., empirical probabilities of falsely rejecting the null hypothesis, in Table 1. Generally, under these two circumstances, at small sample sizes, the error rates of the proposed test statistics are slightly above 0.05, and as the sample size increases, the error rates decrease and fluctuate around the nominal value 0.05. These results indicate that the proposed test holds its size under H0 in the two-sample case. By comparing the results with those of the other traditional testing procedures, we can see that the Energy test outperforms the K-S test under both circumstances, and behaves similarly to the other four testing procedures. the results in Table 1, we can find that the Energy test has similar performance with other traditional testing procedure.


3.1.2. Multi-sample test

For the multi-sample scenario, the same simulation setting is used as the two-sample test in Section 3.1.1 except for the number of groups and the values of k. Here we generated five groups of data X1, …, X5 and calculated the test statistics with k ∈ {10, 20, …, 55}. The error rates under different testing procedures are presented in Table 2. Under both circumstances, the error rates of the Energy test fluctuate around 0.05 slightly. When the samples are generated from Gamma(2, 0.2), the performance of the Energy test is better than the other tests when the sample sizes are larger than 40. Under the case of Log-Normal(0, 1), the Energy test behaves similarly to the other testing procedures.

3.2. Simulation under alternative hypothesis

3.2.1. Two-sample test

In this section, we consider two-sample tests under the alternative hypothesis. We generated two groups of data, X1 and X2, with different sample sizes k ∈ {20, 40, …, 250}. Two different circumstances are considered. The first one is that X1 ~ Gamma(2, 0.2) and X2 ~ Gamma(2, 0.3), and second case is that X1 ~ Log-Normal(0, 1) and X2 ~ Log-Normal(0.5, 1). The other simulation settings remain the same as Section 3.1.1. The Energy test and the other test procedures are used for hypothesis testing. Under the alternative hypothesis, the error rate is the empirical probability of failing to reject a false null hypothesis, which is equal to 1 minus the power of a test statistics. The error rates under different sample sizes are reported in Table 3. Under both cases, for all the testing procedures, the error rates are larger with smaller sample sizes, and decrease sharply when the sample sizes increase.

Under the first case, the Energy test behaves the best compared to the other testing procedures since it has the greatest power with different values of k. And under the second case, the Energy test still behaves better than the K-S test. Under these two circumstances, the power of the Energy test is greater than 0.95 when k reaches 100 and 150, respectively.


3.2.2. Multi-sample test

For multiple-sample testing, we generated five groups of data X1, …, X5 with k ∈ {10, 20, …, 55} and kept the other simulation settings same as Section 3.2.1. For the first case, X1 ~ Gamma(2, 0.2), X2 ~ Gamma(2, 0.25), X3 ~ Gamma(2, 0.3), X4 ~ Gamma(2, 0.35), and X5 ~ Gamma(2, 0.4). For the second case, X1 ~ Log-Normal(0, 1), X2 ~ Log-Normal(0.1, 1), X3 ~ Log-Normal(0.3, 1), X4 ~ Log-Normal(0.5, 1), and X5 ~ Log-Normal(0.7, 1). Hypothesis testings are performed with α = 0.05 and the corresponding error rates are reported in Table 4. Similarly, the error rates decrease as the sample sizes increase. Under the first case, the Energy test behaves the best under different values of the sample size, and reaches a power larger than 0.95 when the sample size in each group exceeds 40. Under the second case, even though the Energy test cannot outperform the other testing procedures, it still has the power larger than 0.88 when the sample size reaches 55.

3.3. Simulation: K-means clustering

In this case, we test the accuracy of K-means clustering using energy distance through simulation. We considered the case where in each replicate, 60% of the data are generated from Gamma(2, 0.2), and the other 40% are generated from another Gamma distribution with different rates as displayed in Table 5. For each sample size k ∈ {500, 600, 700, 800, 900, 1000, 1500}, 1,000 replicates of simulations are performed. The error rate in clustering procedure is the proportion of observations that are incorrectly clustered. They are reported in Table 5. It can be seen that the accuracy is related to the sample size as well as the extent how different the two distributions are. With the same alternative distribution, the accuracy of clustering increases with larger sample size. With the same sample size, as the difference between the distributions gets larger, the accuracy of clustering increases.

4. Illustration: China Health and Nutrition Survey data

We apply the proposed methods to the analysis of a China Health and Nutrition Survey (CHNS) dataset for year 2011. The dataset contains information of 4,346 households in 12 provinces. Based on our simulation results, this sample size is big enough to have powerful testing results and clustering results. A brief description of the dataset by province is given in Table 6.

From the Gini coefficient calculated in Table 6, Beijing has the lowest Gini coefficient, which means that Beijing has the most balanced income distribution. On the contrary, Henan has the highest Gini coefficient, which means the income distribution in Henan is the least balanced. Most provinces have similar Gini coefficients, and it is difficult to set a threshold to say whether there are statistically significant differences among these provinces.

4.1. Hypothesis testing results

We calculated the pairwise test statistic in (2.4) for the 12 provinces, giving rise to 66 statistic values. The resulted p-values are visualized in Figure 1. While most p-values are small, indicating a significant difference of the two distributions they correspond to, some p-values suggest that the two income distributions are not significantly different, such as Liaoning and Hubei, Liaoning with Hunan, and Hunan with Hubei.

Also, we do the multi-sample test for all the provinces using (2.6). The energy statistic of this data is calculated to be 87,944,000. Based on the permutation test procedure, the p-value is 0.000999. We therefore reject the null hypothesis, and conclude that not all income distributions are the same in these 12 provinces.

4.2. Clustering results

From the test results of previous analysis, we learned that the income distributions among 12 provinces are different, but smaller subsets of provinces have similar poverty pattern. Therefore, it is of interest to further cluster the 12 provinces. Some exploratory data analysis is performed before clustering. We used the hierarchical clustering methods (Bibby et al., 1979; Johnson and Wichern, 2007) based on the energy distances calculated using (2.1) to see the clustering pattern of different provinces. The result is plotted in Figure 2.

From Figure 2, we can easily find that there are four major clusters based on the energy distance. Therefore, we chose k = 4 for the clustering method described in Section 2.3, and the result is visualized in Figure 3.

From Figure 3, it is found that Beijing and Shanghai are in cluster 1. Jiangsu province is the only one province in cluster 2. Hunan, Hubei, Chongqing, Guizhou, Heilongjiang, and Liaoning are in cluster 3, and Henan, Guangxi, and Shandong are in cluster 4. These clustering results are consistent with explanatory analysis based on the Gini coefficient calculation and testing results. Also, the clustering results reflects actual development of economy in these provinces. Beijing, Shanghai, and Jiangsu have relatively better economic development, while Henan, Guangxi, and Shandong have worse economic development.

5. Discussion

We developed nonparametric test and clustering procedure for income distribution in different geographical regions. The test was inspired by the energy distance between distributions (Székely and Rizzo, 2004), and the MDS (Kruskal, 1964) for reducing the dimension of data based on a distance matrix so that an appropriate clustering algorithm can be implemented. In simulation studies, the nonparametric test has size close to the nominal 0.05 level. When the samples are generated from different distributions, or the same distribution with different parameters, the proposed test has strong power in detecting the violation of the null hypothesis, and the power is even stronger than other frequently used testing procedures. In the application to CHNS data, the proposed methods rejected the null hypothesis, and identified four clusters of income distribution, which is robust against choice of initial centroids and consistent with the benchmark hierarchical clustering results. In this work, there are no significant differences between the results of hierarchical clustering and K-means clustering because of the relative small number of provinces we analyzed. If we compare the county level data, however, it is very difficult for us to interpret results form the dendrograms. K-means clustering method is more robust than the HC.

In this work, we are only concerned with using household incomes to calculate the energy distance between different regions. There could be other information sources, such as the value of property, amount of mortgage, and consumptions, that can be used to provide more accurate description of the distribution of income, as the energy distribution provides good measurement of high-dimensional cumulative distribution functions. Geographical information, in addition, can be taken into account in comparing income distributions.

Acknowledgement

Dr. Hu’s research was supported by Dean’s office of College of Liberal Arts and Sciences in University of Connecticut.

Figures
Fig. 1. A visualization of pairwise p-values.
Fig. 2. Hierarchical clustering results.
Fig. 3. 4-means clustering results.
TABLES

Algorithm 1

Multidimensional scaling

Assign points to arbitrary coordinates in p-dimensional space

Compute the Euclidean distances among all pairs of points, to form the D^ matrix

Compare the D^ matrix with the input D matrix by evaluating the stress function. The smaller the value, the greater the correspondence between the two

Adjust coordinates of each point in the direction that best minimize the stress

Repeat step 2 through 4 until stress won’t get any lower


Algorithm 2

K-means clustering

Set K-means {m(k)} to random values

Each data point x(n) is assigned to the nearest mean. We denote our guess of the cluster k(n) that the point x(n) belongs to by k^(n)=argmink(d(mk,x(n))).

An alternative, equivalent representation of this assignment of points to cluster i given by “responsibilities”, which are indicator variables:Rkn=1 if k^(n)=k,Rkn=0 otherwise.

The model parameters, the means, are adjusted to mach the sample means of the data points that they are responsible for m(k)=nrk(n)x(n)R(k),

where R(k) is the total responsibility of mean k, R(k)=nrk(n).

Repeat the step 2 and step 3 until the assignments do not change


Table 1

Error rates under null hypothesis: two-sample (α = 0.05)

Sample size k 20 40 60 80 100 150 200 250
Energy test 0.0524 0.0532 0.0518 0.0516 0.0494 0.0490 0.0486 0.0466
X1 ~ Gamma(2, 0.2)
X2 ~ Gamma(2, 0.2)
Wilcoxon rank sum test 0.0528 0.0520 0.0518 0.0499 0.0516 0.0484 0.0502 0.0486
Kolmogorov-Smirnov test 0.0464 0.0371 0.0354 0.0330 0.0244 0.0436 0.0402 0.0394
Anderson-Darling test 0.0589 0.0531 0.0513 0.0502 0.0514 0.0494 0.0494 0.0462
Jonckheere-Terpstra test 0.0522 0.0504 0.0488 0.0499 0.0536 0.0542 0.0518 0.0484
Rank Score test 0.0560 0.0545 0.0494 0.0502 0.0482 0.0494 0.0470 0.0484

X1 ~ Log-Normal(0, 1)
X2 ~ Log-Normal(0, 1)
Energy test 0.0502 0.0488 0.0510 0.0506 0.0504 0.0492 0.0472 0.0474
Wilcoxon rank sum test 0.0554 0.0472 0.0508 0.0500 0.0520 0.0482 0.0502 0.0478
Kolmogorov-Smirnov test 0.0380 0.0334 0.0490 0.0330 0.0400 0.0422 0.0410 0.0386
Anderson-Darling test 0.0546 0.0498 0.0530 0.0494 0.0550 0.0506 0.0504 0.0458
Jonckheere-Terpstra test 0.0516 0.0430 0.0498 0.0454 0.0516 0.0514 0.0466 0.0452
Rank Score test 0.0532 0.0444 0.0490 0.0510 0.0526 0.0482 0.0482 0.0492

Table 2

Error rates under null hypothesis: multi-sample

Sample size k 10 20 30 40 45 50 55
Xi ~ Gamma(2, 0.2) (i = 1, …, 5) Energy test 0.0502 0.0497 0.0510 0.0476 0.0478 0.0488 0.0484
Wilcoxon rank sum test 0.0466 0.0523 0.0492 0.0570 0.0468 0.0480 0.0464
Anderson-Darling test 0.0510 0.0486 0.0504 0.0460 0.0476 0.0440 0.0440
Jonckheere-Terpstra test 0.0492 0.0542 0.0502 0.0514 0.0466 0.0460 0.0476
Rank Score test 0.0498 0.0484 0.0514 0.0470 0.0464 0.0424 0.0448

Xi ~ Log-Normal(0, 1) (i = 1, …, 5) Energy test 0.0488 0.0548 0.0462 0.0482 0.0492 0.0494 0.0484
Wilcoxon rank sum test 0.0468 0.0540 0.0436 0.0506 0.0500 0.0560 0.0430
Anderson-Darling test 0.0488 0.0566 0.0440 0.0524 0.0500 0.0536 0.0476
Jonckheere-Terpstra test 0.0538 0.0494 0.0462 0.0498 0.0494 0.0534 0.0452
Rank Score test 0.0508 0.0546 0.0436 0.0536 0.0514 0.0532 0.0504

Table 3

Error rates under alternative hypothesis: 2 groups

Sample size k 20 40 60 80 100 150 200 250
X1 ~ Gamma(2, 0.2)
X2 ~ Gamma(2, 0.3)
Energy test 0.6308 0.3428 0.1746 0.0798 0.0354 0.0054 0.0004 0.0000
Wilcoxon rank sum test 0.6460 0.3778 0.2038 0.0994 0.0474 0.0064 0.0010 0.0000
Kolmogorov-Smirnov test 0.7612 0.5598 0.2944 0.2060 0.1212 0.0202 0.0046 0.0004
Anderson-Darling test 0.6528 0.3786 0.1940 0.0936 0.0438 0.0070 0.0008 0.0001
Jonckheere-Terpstra test 0.9992 0.9942 0.9814 0.9781 0.9583 0.9016 0.8875 0.8666
Rank Score test 0.6360 0.3638 0.1860 0.0886 0.0384 0.0048 0.0006 0.0000

X1 ~ Log-Normal(0, 1)
X2 ~ Log-Normal(0.5, 1)
Energy test 0.7088 0.4920 0.3040 0.1870 0.1086 0.0240 0.0048 0.0010
Wilcoxon rank sum test 0.6800 0.4476 0.2552 0.1590 0.0802 0.0184 0.0038 0.0006
Kolmogorov-Smirnov test 0.7868 0.6398 0.3750 0.3004 0.1944 0.0482 0.0146 0.0036
Anderson-Darling test 0.6836 0.4622 0.2654 0.1700 0.0876 0.0174 0.0040 0.0010
Jonckheere-Terpstra test 0.5650 0.3214 0.1670 0.0906 0.0428 0.0060 0.0014 0.0000
Rank Score test 0.6744 0.4324 0.2374 0.1418 0.0700 0.0126 0.0028 0.0004

Table 4

Error rates for the proposed nonparametric test under the alternative hypothesis: multi-sample

Sample size k 10 20 30 40 45 50 55
Xi ~ Gamma(2, ai)
a = (0.2, 0.25, 0.3, 0.35, 0.4)
Energy test 0.6144 0.2646 0.0946 0.0220 0.0154 0.0074 0.0026
Wilcoxon rank sum test 0.9968 0.9978 0.9992 0.9994 0.9998 0.9998 0.9998
Anderson-Darling test 0.6420 0.3054 0.1194 0.0306 0.0202 0.0116 0.0036
Jonckheere-Terpstra test 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
Rank Score test 0.6216 0.2896 0.1048 0.0282 0.0176 0.0086 0.0038

Xi ~ Lognormal(bi, 1)
b = (0, 0.1, 0.3, 0.5, 0.7)
Energy test 0.8118 0.6004 0.3972 0.2692 0.1940 0.1534 0.1138
Wilcoxon rank sum test 0.6994 0.4516 0.2612 0.1564 0.1122 0.0794 0.0622
Anderson-Darling test 0.7888 0.5540 0.3430 0.2034 0.1548 0.1076 0.0790
Jonckheere-Terpstra test 0.4784 0.2128 0.0818 0.0332 0.0208 0.0118 0.0060
Rank Score test 0.7780 0.5254 0.3068 0.1734 0.1272 0.0826 0.0616

Table 5

Error rate for the proposed nonparametric clustering procedure under the alternative hypothesis: multi-sample

Density 1
Density 2
Gamma(2, 0.2)
Gamma(2, 0.25)
Gamma(2, 0.2)
Gamma(2, 0.3)
Gamma(2, 0.2)
Gamma(2, 0.35)
Gamma(2, 0.2)
Gamma(2, 0.4)
Gamma(2, 0.2)
Gamma(2, 0.45)
k = 500 0.854 0.361 0.090 0.035 0.008
k = 600 0.823 0.303 0.073 0.012 0.001
k = 700 0.767 0.159 0.020 0.003 0.001
k = 800 0.721 0.133 0.015 0.001 0.001
k = 900 0.635 0.066 0.011 0.001 0.000
k = 1000 0.624 0.064 0.006 0.001 0.000
k = 1500 0.408 0.015 0.000 0.000 0.000

Table 6

Description of China Health and Nutrition Survey data

Province Sample size Average household income in thousands (SD) Gini coefficient
Beijing 416 74.98 (49.9) 0.3236
Liaoling 395 49.43 (47.9) 0.3953
Heilongjiang 398 46.11 (44.4) 0.4307
Shanghai 424 87.46 (68.7) 0.3700
Jiangsu 414 60.93 (43.5) 0.3712
Shandong 399 41.00 (40.9) 0.4031
Henan 299 36.57 (42.7) 0.5027
Hubei 339 50.04 (57.3) 0.4142
Hunan 245 47.95 (43.5) 0.3986
Guangxi 362 36.81 (33.4) 0.3800
Guizhou 339 45.39 (52.7) 0.4172
Chongqing 329 40.98 (39.9) 0.4361

References
  1. Bartels, CP, and Van Metelen, H (1975). Alternative probability density functions of income: A comparison of the lognormal-, Gamma-and Weibull-distribution with Dutch data: Vrije Universiteit, Economische Faculteit
  2. Bibby, J, Kent, J, and Mardia, K (1979). Multivariate Analysis. London: Academic Press
  3. Gibrat, R (1931). Les inégalités économiques: Recueil Sirey
  4. Johnson, R, and Wichern, D (2007). Discrimination and classification. Applied Multivariate Statistical Analysis. 4.
  5. Kruskal, JB (1964). Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis. Psychometrika. 29, 1-27.
    CrossRef
  6. Li, S, and Rizzo, ML (2017). K-groups: a generalization of K-means clustering. ArXiv e-prints.
  7. Lorenz, MO (1905). Methods of measuring the concentration of wealth. Publications of the American Statistical Association. 9, 209-219.
    CrossRef
  8. McDonald, JB (1984). Some generalized functions for the size distribution of income. Econometrica: Journal of the Econometric Society. 52, 647-665.
    CrossRef
  9. McDonald, JB, and Xu, YJ (1995). A generalization of the beta distribution with applications. Journal of Econometrics. 66, 133-152.
    CrossRef
  10. Pareto, V (1964). Cours d’économie politique: Librairie Droz
    CrossRef
  11. Rizzo, ML, and Székely, GJ (2010). Disco analysis: a nonparametric extension of analysis of variance. The Annals of Applied Statistics. 4, 1034-1055.
    CrossRef
  12. Rizzo, ML, and Székely, GJ (2015). Energy distance. Wiley Interdisciplinary Reviews: Computational Statistics. 8, 27-38.
    CrossRef
  13. Salem, ABZ, and Mount, TD (1974). A convenient descriptive model of income distribution: the gamma density. Econometrica: Journal of the Econometric Society. 42, 1115-1127.
    CrossRef
  14. Sullivan, A (2003). Economics: Principles in action
  15. Székely, GJ, and Rizzo, ML (2004). Testing for equal distributions in high dimension. InterStat. 5, 2004.
  16. Yitzhaki, S (1979). Relative deprivation and the Gini coefficient. The Quarterly Journal of Economics. 93, 321-324.
    CrossRef