^{1}

^{1}

Based on the Confidence Distribution method to the Behrens-Fisher problem, we consider two approaches of combining Confidence Distributions: P Combination and AN Combination to solve the Behrens-Fisher problem. Firstly, we provide some Confidence Distributions to the Behrens- Fisher problem, and then we give the Confidence Distribution method to the Behrens-Fisher problem. Finally, we compare the “combination” and the “single” through the numerical simulation.

Let

In the case of 1)

In Bayesian inference, researchers typically rely on a posterior distribution to make inference on a parameter of interest, where the posterior is often viewed as a “distribution estimator” [

Confidence Distribution is one such a “distribution estimator” that can be defined and interpreted in a frequentist framework, in which the parameter is a fixed and non-random quantity. The concept of confidence distribution has a long history. The following definition is proposed and utilized in Schweder & Hjort (2002) [

Definition 1.1: Given:

1) For each given

2) At the true parameter value

Also, the function

And, we got a theorem in the author’s another paper [

Theorem 1.1: If for each given

According to Theorem 1.1, we can see that if the

To put it simply, Confidence Distribution is a distribution of the parameter, we can know almost all of the information of the parameter. But methods to the construction of the Confidence Distribution are not unique, so we can get different Confidence Distributions and then find the optimal one.

According to the conclusion of the author’s another paper: in both small sample size and big sample size, the effectiveness of WS and CA are relatively close, but CA is a little better then WS in the optimality; And then we consider the forms of WS and CA are relatively simple.

So we choose WS and CA to combine Confidence Distribution. The following are the conclusions of WS and CA.

1) WS Distribution’s density function:

where

2) CA Distribution’s density function:

where parameters in

The notion of a Confidence Distribution is attractive for the purpose of combining information. The main reasons are that there is a wealth of information on θ inside a Confidence Distribution, the concept of Confidence Distribution is quite broad, and the Confidence Distributions are relatively easy to construct and interpret.

Multiplying likelihood functions from independent sources constitutes a standard method for combining parametric information. Naturally, this suggests multiplying CD densities and normalizing to possibly derive combined CDs as follows [

where

Now, we use the P Combination to combine WS and CA:

Theorem 2.1:

Proof: Obviously, the

Then we consider an asymptotic normality method based on asymptotic Confidence Distributions [

where

Now, we use the AN Combination to combine WS and CA:

where

Theorem 2.2:

Proof: Let

First of all, we need to consider the effectiveness of the Confidence Distribution in Behrens-Fisher problem. Here, we define the effectiveness of the Confidence Distribution:

In this problem, we have a very small sample. In the numerical simulation, we define:

where, I is a indicative function. The more

After the text edit has been completed, the paper is ready for the template. Duplicate the template file by using the Save As command, and use the naming convention prescribed by your journal for the name of your paper. In this newly created file, highlight all of the contents and import your prepared text file. You are now ready to style your paper.

Both

In the case of similar effectiveness, we consider the length of the confidence interval, the shorter length of the confidence interval corresponding Confidence Distribution is optimum.

According to the result of numerical simulation (

Confidence Distribution | the effectiveness | ||||||
---|---|---|---|---|---|---|---|

0.1 | 0.5 | 1.0 | 2.0 | 5.0 | 10.0 | ||

(10, 5) | WS | 0.05013 | 0.05176 | 0.06225 | 0.07228 | 0.07980 | 0.08321 |

CA | 0.04623 | 0.04372 | 0.04188 | 0.03812 | 0.03595 | 0.02665 | |

PC | 0.04760 | 0.04613 | 0.05138 | 0.05333 | 0.05874 | 0.06078 | |

AN | 0.04829 | 0.04712 | 0.04936 | 0.04781 | 0.04525 | 0.04357 | |

(10, 10) | WS | 0.05188 | 0.04839 | 0.04753 | 0.04914 | 0.05024 | 0.05241 |

CA | 0.04256 | 0.04820 | 0.04873 | 0.04724 | 0.04582 | 0.04413 | |

PC | 0.04691 | 0.04835 | 0.04872 | 0.04860 | 0.04602 | 0.04608 | |

ANC | 0.04722 | 0.04845 | 0.04860 | 0.04724 | 0.04682 | 0.04613 | |

(10, 25) | WS | 0.05027 | 0.04988 | 0.04931 | 0.04601 | 0.04079 | 0.03856 |

CA | 0.04182 | 0.04590 | 0.04836 | 0.04993 | 0.04918 | 0.04825 | |

PC | 0.04658 | 0.04781 | 0.04944 | 0.04894 | 0.04825 | 0.04829 | |

ANC | 0.04711 | 0.04861 | 0.04920 | 0.04820 | 0.04861 | 0.04853 | |

(25, 25) | WS | 0.04943 | 0.04893 | 0.04977 | 0.04971 | 0.05103 | 0.05054 |

CA | 0.04927 | 0.04982 | 0.04979 | 0.04930 | 0.04791 | 0.04632 | |

PC | 0.04951 | 0.04915 | 0.04957 | 0.04970 | 0.04884 | 0.04807 | |

ANC | 0.04911 | 0.04900 | 0.04949 | 0.04931 | 0.04916 | 0.04835 | |

(25, 50) | WS | 0.05004 | 0.05028 | 0.04868 | 0.04860 | 0.04694 | 0.04606 |

CA | 0.04837 | 0.04962 | 0.04938 | 0.04987 | 0.04880 | 0.04973 | |

PC | 0.04922 | 0.04929 | 0.04938 | 0.04919 | 0.04872 | 0.04817 | |

ANC | 0.04902 | 0.04983 | 0.04912 | 0.04909 | 0.04885 | 0.04859 | |

(50, 50) | WS | 0.04986 | 0.05027 | 0.04988 | 0.04900 | 0.04970 | 0.04990 |

CA | 0.04928 | 0.04904 | 0.04978 | 0.04885 | 0.04903 | 0.04820 | |

PC | 0.04926 | 0.04921 | 0.04942 | 0.04966 | 0.04948 | 0.04822 | |

ANC | 0.04987 | 0.04912 | 0.04960 | 0.04873 | 0.04915 | 0.04884 |

Where PC is for P Combination; ANC is for AN Combination.

1) With the increase of sample size, the effectiveness of each Confidence Distribution and combining Confidence Distribution increase.

2) The effectiveness of PC and ANC is better than WS and CA.

We use two different methods to combine Confidence Distributions. Thought the numerical simulation we can find the optimal Confidence Distribution. In small sample size, the effectiveness of PC and ANC is better than WS and CA (PNC is a little better than PC); in the relatively big sample size, WS and CA are also effective. So, this paper argues that the PNC Combination is the optimal combining Confidence Distribution to solve the Behrens-Fisher Problem.

Wenyong Tao,Wanzhou Ye, (2016) The Combining Confidence Distribution Method to the Behrens-Fisher Problem. Advances in Pure Mathematics,06,532-536. doi: 10.4236/apm.2016.68041