The Combining Confidence Distribution Method to the Behrens-Fisher Problem

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 BehrensFisher 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.


Introduction 1.Behrens-Fisher Problem
n n = , we can use frequentist approach to solve Behrens-Fisher problem; In general case, we usually use of large sample theory to find the approximate confidence interval [1].

Confidence Distribution
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" [2] for the parameter.
Confidence Distribution is one such a "distribution estimator" that can be defined and interpreted in a fre-quentist 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) [3] and Singh et al. (2005Singh et al. ( , 2007) ) [4] [5].Definition 1.1: Given: Θ is the parameter space of the unknown parameter of interest θ ; X is the sample space corresponding to sample data 1 , , n x x x =  .We called the function ⋅ a confidence distribution (CD) for a parameter θ , if 1) For each given 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.

Some Confidence Distributions to the Behrens-Fisher problem
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.

Combination
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.

P Combination
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 [3]: θ µ µ = − .

AN Combination
Then we consider an asymptotic normality method based on asymptotic Confidence Distributions [7]: , here we let i T are means of normal samples, ( ) ( ) Now, we use the AN Combination to combine WS and CA: ( ) , The normality based asymptotic Confidence Distribution is ( ) ( ) with asymptotic Confidence Distribution density ( ) ( )

Effectiveness
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 η is close to α , the more Confidence Distribution is efficient.
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Numerical Simulation
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 (Table 1), we can see: , the effectiveness η of the different confidence distribution.
( ) 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.
σ are assumed to be unknown and not necessarily equal.Behrens-Fisher problem is to give the interval estimation of the parameter Confidence Distribution densities from L independent studies.Now, we use the P Combination to combine WS and CA: is a distribution function of θ , meet the condition 1) in definition 1.1; according to theorem 1.1, meet the condition 2) in definition 1.1.So ( ) P H θ is a Confidence Distribution of 1 2

Table 1 .
Under the condition of 0.05