# An Exact and Near-Exact Distribution Approach to the Behrens–Fisher Problem

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## Abstract

**:**

## 1. Introduction

## 2. The Exact Distribution of the Behrens–Fisher Statistic for Odd-Numbered Sample Sizes

## 3. The Exact and Near-Exact Distribution of the Behrens–Fisher Statistic for Even-Numbered Sample Sizes

#### 3.1. The Exact Distribution

#### 3.2. Near-Exact Distribution

## 4. One of the Sample Sizes Is Even and the Other Is Odd

## 5. Comparison of the Exact or Near-Exact Distribution and Welch’s t Test

#### 5.1. Odd n_{1} and n_{2}

#### 5.2. Even ${n}_{1}$ and ${n}_{2}$

#### 5.3. ${n}_{1}$ Is Even and ${n}_{2}$ Is Odd

#### 5.4. Brief Study of Power Evolution for Increasing Sample Sizes

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The left panel is based on ${\sigma}_{1}^{2}=5$, ${\sigma}_{2}^{2}=45$. $({n}_{1},{n}_{2})=(5+2k,5+2k)$ for $k=0,1,\dots ,12$ and ${\delta}_{k}=({\mu}_{1}-{\mu}_{2})/\sqrt{\frac{{\sigma}_{1}^{2}}{{n}_{1}}+\frac{{\sigma}_{2}^{2}}{{n}_{2}}}=\sqrt{40}/\sqrt{\frac{5}{5+2k}+\frac{45}{5+2k}}$. The right panel is based on ${\sigma}_{1}^{2}=15$, ${\sigma}_{2}^{2}=27$ and $({n}_{1},{n}_{2})=(15+2k,3+2k)$ for $k=0,1,\dots ,9$ and ${\delta}_{k}=({\mu}_{1}-{\mu}_{2})/\sqrt{\frac{{\sigma}_{1}^{2}}{{n}_{1}}+\frac{{\sigma}_{2}^{2}}{{n}_{2}}}=\sqrt{40}/\sqrt{\frac{15}{15+2k}+\frac{27}{3+2k}}$. Red and blue lines represent the power curves from exact distribution and Welch’s test, respectively.

**Figure 2.**The left panel is based on ${\sigma}_{1}^{2}=8$, ${\sigma}_{2}^{2}=72$. $({n}_{1},{n}_{2})=(8+2k,8+2k)$ for $k=0,1,\dots ,17$ and ${\delta}_{k}=({\mu}_{1}-{\mu}_{2})/\sqrt{\frac{{\sigma}_{1}^{2}}{{n}_{1}}+\frac{{\sigma}_{2}^{2}}{{n}_{2}}}=\sqrt{40}/\sqrt{\frac{8}{8+2k}+\frac{72}{8+2k}}$. The right panel is based on ${\sigma}_{1}^{2}=12$, ${\sigma}_{2}^{2}=36$ and $({n}_{1},{n}_{2})=(12+2k,4+2k)$ for $k=0,1,\dots ,11$ and ${\delta}_{k}=({\mu}_{1}-{\mu}_{2})/\sqrt{\frac{{\sigma}_{1}^{2}}{{n}_{1}}+\frac{{\sigma}_{2}^{2}}{{n}_{2}}}=\sqrt{40}/\sqrt{\frac{12}{12+2k}+\frac{36}{4+2k}}$ Red and blue lines represent the power curves from the nearly-exact distribution and Welch’s test, respectively.

**Figure 3.**The left panel is based on ${\sigma}_{1}^{2}=8$, ${\sigma}_{2}^{2}=63$. $({n}_{1},{n}_{2})=(8+2k,7+2k)$ for $k=0,1,\dots ,14$ and ${\delta}_{k}=({\mu}_{1}-{\mu}_{2})/\sqrt{\frac{{\sigma}_{1}^{2}}{{n}_{1}}+\frac{{\sigma}_{2}^{2}}{{n}_{2}}}=\sqrt{40}/\sqrt{\frac{8}{8+2k}+\frac{63}{7+2k}}$. The right panel is based on ${\sigma}_{1}^{2}=12$, ${\sigma}_{2}^{2}=27$ and $({n}_{1},{n}_{2})=(12+2k,3+2k)$ for $k=0,1,\dots ,9$ and ${\delta}_{k}=({\mu}_{1}-{\mu}_{2})/\sqrt{\frac{{\sigma}_{1}^{2}}{{n}_{1}}+\frac{{\sigma}_{2}^{2}}{{n}_{2}}}=\sqrt{40}/\sqrt{\frac{12}{12+2k}+\frac{27}{3+2k}}$. Red and blue lines represent the power curves from the nearly exact distribution and Welch’s test, respectively.

$\mathsf{\theta}$ | $\mathsf{\alpha}$ | $\mathsf{\delta}=0$ | $\mathsf{\delta}=1$ | $\mathsf{\delta}=2$ | |||
---|---|---|---|---|---|---|---|

E | W | E | W | E | W | ||

$0.1$ | $0.1$ | $0.1009$ | $0.1000$ | $0.3578$ | $0.3558$ | $0.7059$ | $0.7036$ |

${\sigma}_{1}^{2}=5$ | $0.05$ | $0.0518$ | $0.0506$ | $0.2234$ | $0.2199$ | $0.5368$ | $0.5313$ |

${\sigma}_{2}^{2}=45$ | $0.01$ | $0.0123$ | $0.0114$ | $0.0731$ | $0.0676$ | $0.2291$ | $0.2139$ |

$0.3$ | $0.1$ | $0.0971$ | $0.0965$ | $0.3664$ | $0.3648$ | $0.7205$ | $0.7187$ |

${\sigma}_{1}^{2}=15$ | $0.05$ | $0.0480$ | $0.0473$ | $0.2271$ | $0.2241$ | $0.5528$ | $0.5482$ |

${\sigma}_{2}^{2}=35$ | $0.01$ | $0.0100$ | $0.0092$ | $0.0636$ | $0.0601$ | $0.2376$ | $0.2263$ |

$0.5$ | $0.1$ | $0.0958$ | $0.0953$ | $0.3628$ | $0.3614$ | $0.7221$ | $0.7205$ |

${\sigma}_{1}^{2}=5$ | $0.05$ | $0.0462$ | $0.0456$ | $0.2243$ | $0.2218$ | $0.5582$ | $0.5545$ |

${\sigma}_{2}^{2}=5$ | $0.01$ | $0.0094$ | $0.0005$ | $0.0641$ | $0.0606$ | $0.2375$ | $0.2288$ |

$\mathsf{\theta}$ | $\mathsf{\alpha}$ | $\mathsf{\delta}=0$ | $\mathsf{\delta}=1$ | $\mathsf{\delta}=2$ | |||
---|---|---|---|---|---|---|---|

E | W | E | W | E | W | ||

$0.1$ | $0.1$ | $0.1100$ | $0.1048$ | $0.3542$ | $0.3402$ | $0.6614$ | $0.6441$ |

${\sigma}_{1}^{2}=15$ | $0.05$ | $0.0660$ | $0.0595$ | $0.2304$ | $0.2090$ | $0.4893$ | $0.4522$ |

${\sigma}_{2}^{2}=27$ | $0.01$ | $0.0273$ | $0.0053$ | $0.1076$ | $0.0843$ | $0.2609$ | $0.1987$ |

$0.3$ | $0.1$ | $0.1052$ | $0.1013$ | $0.3636$ | $0.3538$ | $0.7033$ | $0.6900$ |

${\sigma}_{1}^{2}=45$ | $0.05$ | $0.0593$ | $0.0557$ | $0.2434$ | $0.2287$ | $0.5414$ | $0.5133$ |

${\sigma}_{2}^{2}=21$ | $0.01$ | $0.0180$ | $0.0161$ | $0.0990$ | $0.0872$ | $0.2940$ | $0.2533$ |

$0.5$ | $0.1$ | $0.1001$ | $0.0976$ | $0.3743$ | $0.3679$ | $0.7263$ | $0.7185$ |

${\sigma}_{1}^{2}=15$ | $0.05$ | $0.0532$ | $0.0507$ | $0.2390$ | $0.2301$ | $0.5657$ | $0.5488$ |

${\sigma}_{2}^{2}=3$ | $0.01$ | $0.0133$ | $0.0119$ | $0.0866$ | $0.0788$ | $0.2949$ | $0.2687$ |

$\mathsf{\theta}$ | $\mathsf{\alpha}$ | $\mathsf{\delta}=0$ | $\mathsf{\delta}=1$ | $\mathsf{\delta}=2$ | |||
---|---|---|---|---|---|---|---|

NE | W | NE | W | NE | W | ||

$0.1$ | $0.1$ | $0.1002$ | $0.0998$ | $0.3731$ | $0.3722$ | $0.7319$ | $0.7311$ |

${\sigma}_{1}^{2}=8$ | $0.05$ | $0.0525$ | $0.0519$ | $0.2384$ | $0.2371$ | $0.5775$ | $0.5750$ |

${\sigma}_{2}^{2}=72$ | $0.01$ | $0.0111$ | $0.0106$ | $0.0755$ | $0.0726$ | $0.2747$ | $0.2672$ |

$0.3$ | $0.1$ | $0.1001$ | $0.0992$ | $0.3746$ | $0.3741$ | $0.7392$ | $0.7395$ |

${\sigma}_{1}^{2}=24$ | $0.05$ | $0.0497$ | $0.0488$ | $0.2373$ | $0.2360$ | $0.5927$ | $0.5922$ |

${\sigma}_{2}^{2}=56$ | $0.01$ | $0.0105$ | $0.0094$ | $0.0760$ | $0.0744$ | $0.2929$ | $0.2879$ |

$0.5$ | $0.1$ | $0.0981$ | $0.0968$ | $0.3775$ | $0.3772$ | $0.7370$ | $0.7385$ |

${\sigma}_{1}^{2}=8$ | $0.05$ | $0.0511$ | $0.0495$ | $0.2403$ | $0.2393$ | $0.5942$ | $0.5947$ |

${\sigma}_{2}^{2}=8$ | $0.01$ | $0.0103$ | $0.0087$ | $0.0757$ | $0.0739$ | $0.2911$ | $0.2890$ |

$\mathsf{\theta}$ | $\mathsf{\alpha}$ | $\mathsf{\delta}=0$ | $\mathsf{\delta}=1$ | $\mathsf{\delta}=2$ | |||
---|---|---|---|---|---|---|---|

NE | W | NE | W | NE | W | ||

$0.1$ | $0.1$ | $0.1061$ | $0.1020$ | $0.3554$ | $0.3458$ | $0.6925$ | $0.6823$ |

${\sigma}_{1}^{2}=12$ | $0.05$ | $0.0578$ | $0.0539$ | $0.2293$ | $0.2129$ | $0.5233$ | $0.4974$ |

${\sigma}_{2}^{2}=36$ | $0.01$ | $0.0175$ | $0.0148$ | $0.0894$ | $0.0725$ | $0.2568$ | $0.2082$ |

$0.3$ | $0.1$ | $0.1031$ | $0.1009$ | $0.3666$ | $0.3612$ | $0.7186$ | $0.7120$ |

${\sigma}_{1}^{2}=36$ | $0.05$ | $0.0545$ | $0.0523$ | $0.2361$ | $0.2266$ | $0.5657$ | $0.5492$ |

${\sigma}_{2}^{2}=28$ | $0.01$ | $0.0131$ | $0.0117$ | $0.0864$ | $0.0769$ | $0.2781$ | $0.2498$ |

$0.5$ | $0.1$ | $0.0991$ | $0.0976$ | $0.3730$ | $0.3701$ | $0.7332$ | $0.7308$ |

${\sigma}_{1}^{2}=12$ | $0.05$ | $0.0511$ | $0.0492$ | $0.2388$ | $0.2340$ | $0.5802$ | $0.5718$ |

${\sigma}_{2}^{2}=4$ | $0.01$ | $0.0117$ | $0.0103$ | $0.0785$ | $0.0732$ | $0.2841$ | $0.2683$ |

$\mathsf{\theta}$ | $\mathsf{\alpha}$ | $\mathsf{\delta}=0$ | $\mathsf{\delta}=1$ | $\mathsf{\delta}=2$ | |||
---|---|---|---|---|---|---|---|

NE | W | NE | W | NE | W | ||

$0.1$ | $0.1$ | $0.1015$ | $0.1012$ | $0.3684$ | $0.3674$ | $0.7279$ | $0.7269$ |

${\sigma}_{1}^{2}=8$ | $0.05$ | $0.0504$ | $0.0497$ | $0.2377$ | $0.2354$ | $0.5669$ | $0.5638$ |

${\sigma}_{2}^{2}=63$ | $0.01$ | $0.0117$ | $0.0111$ | $0.0744$ | $0.0712$ | $0.2642$ | $0.2549$ |

$0.3$ | $0.1$ | $0.0980$ | $0.0971$ | $0.3736$ | $0.3730$ | $0.7366$ | $0.7367$ |

${\sigma}_{1}^{2}=24$ | $0.05$ | $0.0496$ | $0.0486$ | $0.2354$ | $0.2341$ | $0.5874$ | $0.5857$ |

${\sigma}_{2}^{2}=49$ | $0.01$ | $0.0115$ | $0.0104$ | $0.0747$ | $0.0722$ | $0.2876$ | $0.2820$ |

$0.5$ | $0.1$ | $0.1015$ | $0.0995$ | $0.3711$ | $0.3706$ | $0.7389$ | $0.7405$ |

${\sigma}_{1}^{2}=8$ | $0.05$ | $0.0494$ | $0.0475$ | $0.2410$ | $0.2393$ | $0.5897$ | $0.5896$ |

${\sigma}_{2}^{2}=7$ | $0.01$ | $0.0116$ | $0.0089$ | $0.0737$ | $0.0701$ | $0.2844$ | $0.2806$ |

$\mathsf{\theta}$ | $\mathsf{\alpha}$ | $\mathsf{\delta}=0$ | $\mathsf{\delta}=1$ | $\mathsf{\delta}=2$ | |||
---|---|---|---|---|---|---|---|

NE | W | NE | W | NE | W | ||

$0.1$ | $0.1$ | $0.1099$ | $0.1045$ | $0.3495$ | $0.3355$ | $0.6590$ | $0.6418$ |

${\sigma}_{1}^{2}=12$ | $0.05$ | $0.0632$ | $0.0574$ | $0.2311$ | $0.2099$ | $0.4807$ | $0.4455$ |

${\sigma}_{2}^{2}=27$ | $0.01$ | $0.0254$ | $0.0206$ | $0.1066$ | $0.0849$ | $0.2586$ | $0.1994$ |

$0.3$ | $0.1$ | $0.1074$ | $0.1035$ | $0.3632$ | $0.3523$ | $0.6966$ | $0.6817$ |

${\sigma}_{1}^{2}=36$ | $0.05$ | $0.0582$ | $0.0546$ | $0.2417$ | $0.2261$ | $0.5417$ | $0.5120$ |

${\sigma}_{2}^{2}=21$ | $0.01$ | $0.0157$ | $0.0137$ | $0.0929$ | $0.0808$ | $0.2825$ | $0.2428$ |

$0.5$ | $0.1$ | $0.1006$ | $0.0986$ | $0.3671$ | $0.3611$ | $0.7185$ | $0.7102$ |

${\sigma}_{1}^{2}=12$ | $0.05$ | $0.0503$ | $0.0486$ | $0.2351$ | $0.2263$ | $0.5664$ | $0.5493$ |

${\sigma}_{2}^{2}=3$ | $0.01$ | $0.0115$ | $0.01058$ | $0.0817$ | $0.0755$ | $0.2826$ | $0.2581$ |

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**MDPI and ACS Style**

Hong, S.; Coelho, C.A.; Park, J. An Exact and Near-Exact Distribution Approach to the Behrens–Fisher Problem. *Mathematics* **2022**, *10*, 2953.
https://doi.org/10.3390/math10162953

**AMA Style**

Hong S, Coelho CA, Park J. An Exact and Near-Exact Distribution Approach to the Behrens–Fisher Problem. *Mathematics*. 2022; 10(16):2953.
https://doi.org/10.3390/math10162953

**Chicago/Turabian Style**

Hong, Serim, Carlos A. Coelho, and Junyong Park. 2022. "An Exact and Near-Exact Distribution Approach to the Behrens–Fisher Problem" *Mathematics* 10, no. 16: 2953.
https://doi.org/10.3390/math10162953