The Confidence Distribution Method to the Behrens-Fisher Problem

We use the methods of “The Welch-Satterthwaite test”, “The Cochran-Cox test”, “The Generalized p-value test”, “Computational Approach test” to structure different Confidence Distributions, and use the Confidence Distributions to give an new solution the confidence interval of the difference between two population means where the populations are assumed to be normal with unknown and unequal variances. Finally, we find the most effective solution 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 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.A nice aspect of using a distribution estimator is that it contains a wealth of information for almost all types of inference.In frequentist inference, however, we often use a single point or an interval to estimate a parameter of interest.A simple question is: Can we also use a distribution function, or a "distribution estimator", to estimate a parameter of interest in frequentist inference in the style of a Bayesian posterior?
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, especially with its early interpretation associated with fiducial reasoning.Historically, it has been long misconstrued as a fiducial concept, and has not been fully developed under the frequentist framework.One nice aspect of treating a confidence distribution as a purely frequentist concept is that the confidence distribution is now a clean and coherent frequentist concept.Recently as the development of Confidence Distribution's applications (such as Monte Carlo simulation based on Confidence Distribution), this approach has again aroused general concern.
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.
The main work of this article is using the Confidence Distribution method to solve the Behrens-Fisher problem.Firstly, we define the Confidence Distribution; then, we construct Confidence Distribution through some test method and prove these distributions to meet the definition; finally we find the optimal solution Confidence Distribution through numerical simulation.

Definition
The concept of "confidence" was first introduced by Neyman (1934Neyman ( , 1937) ) in his seminal papers on confidence intervals, where frequentist repetition properties for confidence were clarified.The earliest use of the terminology "confidence distribution" that we can find so far in a formal publication is Cox (1958) Thus, ( ) ( )

Structure and Proof
There is no fixed method to structure Confidence Distribution, we only need to construct to meet the definition of confidence distribution.Here we use some test method of Behrens-Fisher problem to structure Confidence Distribution.

WS Distribution
Firstly, we use one of the most widely used method-"Welch-Satterthwaite test" (WST) [6] [7] to structure Confidence Distribution.The known conclusion is: where is the 2 α quantile of a Student's t distribution with k degrees of freedom.
On the basis of the conclusion above, we can easily get a probability distribution function (PDF) of θ : θ µ µ = − .Proof: According to the probability distribution function (PDF) of the Student's t distribution: where k is the degree of freedom, µ is the non-central parameter.

CC Distribution
The most prominent test after the WST is the one proposed by Cochran and Cox (1950) [8].The known conclusion is: is the 2 α quantile of a Student's t distribution with i n degrees of freedom.
On the basis of the conclusion above, we can easily get a probability distribution function (PDF) of θ :

GP Distribution
Taking advantage of the computational resources that are available today, the generalized p-value test (GPT) [9] uses a suitable pivot to come up with a simple test procedure.The known conclusion is: On the basis of the conclusion above, we can easily get a probability distribution function (PDF) of θ : 1) The probability distribution function (PDF) of 2) The probability distribution function (PDF) of 2 2 3) The probability distribution function (PDF) of x z = : The probability distribution function (PDF) of θ µ µ = − .

CA Distribution
According to the classical Likelihood Ratio Tests and parameters bootstrap method, Chang and Pal (2008) [10] proposed Computational Approach Test (CAT).The known conclusion is: the solutions of the following equations: On the basis of the conclusion above, we can easily get a probability distribution function (PDF) of θ :

Proof:
( ) ca f θ is a normal distribution with a mean value of ( ) X X − and a variance of

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 (Table 1).
According to the result of numerical simulation, we can see: 1) With the increase of sample size, the effectiveness of each Confidence Distribution increase; 2) In the small sample size, the effectiveness of ws and ca is relatively high; 3) In the relatively big sample size, cc, ws, ca are relatively stable and highly effective.

Conclusion
We construct four different Confidence Distributions.Through the numerical simulation we can find the optimal Confidence Distribution.In small sample size and relatively big sample size, the effectiveness of ws care rela- tively close, so we can compare the length of the confidence interval.Due tows come from the Student's t distribution, ca comes from the standard normal distribution and the Student's t distribution is a fat tail distribution, so ca is better than cc and ws.Therefore, this paper argues that the ca distribution is the optimal Confidence Distribution to solve the Behrens-Fisher problem.
1, 2, be i.i.d.samples from two normal populations σ are assumed to be unknown and not necessarily equal.Behrens-Fisher problem is to give the interval estimation of the parameter

F
θ is a cumulative distribution function, meet the condition 1) in Definition 2.1; according to Theorem 2.1, meet the condition 2) in Definition 2.1.So ( ) ws f θ is a Confidence Distribu- tion of 1 2 composed of two t distribution obtained through convolution formula, so the ( ) cc F θ is a cumulative distribution function, meet the condition 1) in Definition 2.1; according to Theorem 2.1, meet the condition 2) in Definition 2 composed of two chi-square distribution and a normal distribution obtained through con- volution formula, so the ( ) gp F θ is a cumulative distribution function, meet the condition 1) in Definition 2.1; according to Theorem 2.1, meet the condition 2) in Definition 2.1.So ( ) gp f θ is a Confidence Distribution of 1 2

Fnition 2 . 1 ;
θ is a cumulative distribution function, meet the condition 1) in Defi- according to Theorem 2.1, meet the condition 2) in Definition 2this section, we construct four different Confidence Distributions to solve the Behrens-Fisher problem.Through this method, we the Confidence Distribution method, we can get almost all of the information of the parameter

Table 1 .
Under the condition of