### Abstract

A consistent test for difference in locations between two bivariate populations is proposed. The test is similar as the Mann-Whitney test and depends on the exceedances of slopes of the two samples where slope for each sample observation is computed by taking the ratios of the observed values. In terms of the slopes, it reduces to a univariate problem. The power of the test has been compared with those of various existing tests by simulation. The proposed test statistic is compared with Mardia's(1967) test statistics, Peters-Randies (1991) test statistic, Wilcoxon's rank sum test statistic and Hotelling's T^{2} test statistic using Monte Carlo technique. It performs better than other statistics compared for small differences in locations between two populations when underlying population is population 7(light tailed population) and sample sizes are 15 and 18 respectively. When underlying population is population 6(heavy tailed population) and sample sizes are 15 and 18 it performs better than other statistics compared except Wilcoxon's rank sum test statistic for small differences in location between two populations. It performs better than Mardia's(1967) test statistic for large differences in location between two populations when underlying population is bivariate normal mixture with probability P=0.5, population 6, Pearson type II population and Pearson type VII population for sample sizes 15 and 18. Under bivariate normal population it performs as good as Mardia's (1967) test statistic for small differences in locations between two populations and sample sizes 15 and 18. For sample sizes 25 and 28 respectively it performs better than Mardia's (1967) test statistic when underlying population is population 6, Pearson type II population and Pearson type VII population.

Original language | English (US) |
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Pages (from-to) | 417-436 |

Number of pages | 20 |

Journal | Communications in Statistics - Theory and Methods |

Volume | 29 |

Issue number | 2 |

State | Published - Feb 1 2000 |

### Fingerprint

### Keywords

- Bivariate population
- Consistent
- Power

### ASJC Scopus subject areas

- Statistics and Probability
- Safety, Risk, Reliability and Quality

### Cite this

*Communications in Statistics - Theory and Methods*,

*29*(2), 417-436.

**A test for bivariate two sample location problem.** / Sen, Kanwar; Mathur, Sunil Kumar.

Research output: Contribution to journal › Article

*Communications in Statistics - Theory and Methods*, vol. 29, no. 2, pp. 417-436.

}

TY - JOUR

T1 - A test for bivariate two sample location problem

AU - Sen, Kanwar

AU - Mathur, Sunil Kumar

PY - 2000/2/1

Y1 - 2000/2/1

N2 - A consistent test for difference in locations between two bivariate populations is proposed. The test is similar as the Mann-Whitney test and depends on the exceedances of slopes of the two samples where slope for each sample observation is computed by taking the ratios of the observed values. In terms of the slopes, it reduces to a univariate problem. The power of the test has been compared with those of various existing tests by simulation. The proposed test statistic is compared with Mardia's(1967) test statistics, Peters-Randies (1991) test statistic, Wilcoxon's rank sum test statistic and Hotelling's T2 test statistic using Monte Carlo technique. It performs better than other statistics compared for small differences in locations between two populations when underlying population is population 7(light tailed population) and sample sizes are 15 and 18 respectively. When underlying population is population 6(heavy tailed population) and sample sizes are 15 and 18 it performs better than other statistics compared except Wilcoxon's rank sum test statistic for small differences in location between two populations. It performs better than Mardia's(1967) test statistic for large differences in location between two populations when underlying population is bivariate normal mixture with probability P=0.5, population 6, Pearson type II population and Pearson type VII population for sample sizes 15 and 18. Under bivariate normal population it performs as good as Mardia's (1967) test statistic for small differences in locations between two populations and sample sizes 15 and 18. For sample sizes 25 and 28 respectively it performs better than Mardia's (1967) test statistic when underlying population is population 6, Pearson type II population and Pearson type VII population.

AB - A consistent test for difference in locations between two bivariate populations is proposed. The test is similar as the Mann-Whitney test and depends on the exceedances of slopes of the two samples where slope for each sample observation is computed by taking the ratios of the observed values. In terms of the slopes, it reduces to a univariate problem. The power of the test has been compared with those of various existing tests by simulation. The proposed test statistic is compared with Mardia's(1967) test statistics, Peters-Randies (1991) test statistic, Wilcoxon's rank sum test statistic and Hotelling's T2 test statistic using Monte Carlo technique. It performs better than other statistics compared for small differences in locations between two populations when underlying population is population 7(light tailed population) and sample sizes are 15 and 18 respectively. When underlying population is population 6(heavy tailed population) and sample sizes are 15 and 18 it performs better than other statistics compared except Wilcoxon's rank sum test statistic for small differences in location between two populations. It performs better than Mardia's(1967) test statistic for large differences in location between two populations when underlying population is bivariate normal mixture with probability P=0.5, population 6, Pearson type II population and Pearson type VII population for sample sizes 15 and 18. Under bivariate normal population it performs as good as Mardia's (1967) test statistic for small differences in locations between two populations and sample sizes 15 and 18. For sample sizes 25 and 28 respectively it performs better than Mardia's (1967) test statistic when underlying population is population 6, Pearson type II population and Pearson type VII population.

KW - Bivariate population

KW - Consistent

KW - Power

UR - http://www.scopus.com/inward/record.url?scp=26944476335&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=26944476335&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:26944476335

VL - 29

SP - 417

EP - 436

JO - Communications in Statistics - Theory and Methods

JF - Communications in Statistics - Theory and Methods

SN - 0361-0926

IS - 2

ER -