Fisher’s Fiducial Inference for Parameters of Uniform Distribution

Fisher’s Fiducial Inference for the parameters of a totality uniformly distributed on   ,   is discussed. The corresponding fiducial distributions are derived. The maximum fiducial estimators, fiducial median estimators and fiducial expect estimators of  and  are got. The problems about the fiducial interval, fiducial region and hypothesis testing are discussed. An example which showed that Neyman-Pearson’s confidence interval has some place to be improved is illustrated. An idea about deriving fiducial distribution is proposed.


Introduction
In 1930 Fisher proposed an inference method based on the idea of fiducial probability [1,2].Fisher's fiducial inference has been much applied in practice.The fiducial argument stands out somewhat of an enigma in classical statistics.The enigma mentioned above need statistical scholar to solve.
Fisher's fiducial inference for the parameters of a totality   ,   is discussed.The corresponding fiducial distributions are derived.The maximum fiducial estimators, the fiducial median estimators and the fiducial expect estimators of U  and  are got.The problems about the fiducial interval, fiducial region and hypothesis testing are discussed.
The example below shows that Neyman-Pearson's confidence interval has some place to be improved.Let .By [3] p. 16 Corollary 3.2 the density function of And using its density 1 2 1 2 2 ,if 0.5 0.5 0, otherwise the 95% confidence interval of  can be got as where  is the solution of The length of interval (*) is independent of the sample value!Assam that Is got in a certain sample (Note that   0.58 1.39, 1, 2, , 0.85 0.08 the above data can illustrate the common problems).The probability that

 
Fisher's fiducial inference offered a selection in solving the problems similar with above.

Fiducial Distribution
. As well known, their sufficient statistics of least dimension is It is not difficult to show that Y and Z are the minimum and maximum order statistics of the sample from respectively, and by [3] p. 16 Corollary 3.2, the density function of is See parameters  and  as r.v.'s, see Applying the relative results about the transformation of r.v.'s, it can be show that: Theorem 1.The fiducial density function of vector if min and max 0, otherwise If only one parameter need to be considered, the another parameter is then so-called nuisance parameter.We insist that the marginal distribution should be used in this situation.Hence find the two marginal density functions of Corollary 1.The fiducial density functions of only one parameters  or  as r.v.'s are given by (2.5) and (2.6).

Estimation
It is easy to see that fiducial density Theorem 2. The maximum fiducial estimators of  and  are given by (3.1) and (3.

 and ˆm
 are coincided with the maximum likelihood estimators of  and  .
To find the median of Found the median of

  f b
 by using the same method, and have Theorem 3. The fiducial median estimators of  and  are given by (3.4) and (3.5).
The maximum fiducial estimators ˆm  and ˆm  are extreme a little, Equation (3.1) can be written as E  can be calculated by using the same method.□ ˆe  is a better modify to ˆm  , and ˆe  is a better modify to ˆm  as well.We suggest using ˆe  and ˆe  .The fiducial probability that  belongs to a certain interval estimator   In the same way Give a fiducial probability let us consider the In order to set the length of the interval as shorter as possible, we choice Min X as the right end point of the fiducial And the bellow equation can be got by using (3.8) In order to set the area of the region as smaller as possible, we choice the region as the following rectangular triangle: choice the same value when b a  equals to a constant, and 12) is used here.□

The Case That One Parameter Is in Variation
Let us consider the case that only one parameter is in variation.
is a distribution with single-parameter when one end point of U The maximum fiducial estimators, the fiducial median estimators and the fiducial expect estimators of  can be got easily by using (4.1).
The fiducial probability of one interval estimator   for  can be calculated as The 1   fiducial interval of  can be got as follows by using (4.3).
The similar results for  can be got easily as well.
If there is a relation between the parameters, such as the example in Section 1, this situation may be thought as missing parameter(s).We insist that the conditional distribution should be used in this situation.Under the condition that , is a constant in the interval on which its value isn't zero, because . This is the fiducial density of the parameter  of a totality It can be seen that for distribution , Using the above results to the example in Section 1 it can be got that any subinterval of [0.89, 1.08] with the length 0.95 × 0.19 is the 95% fiducial interval of  .Its length 0.1805 is much smaller than , the length of interval (*).

Hypothesis Testing
Let us consider the hypothesis testing problem.Equation (3.7) and (3.8) can be used to calculate the fiducial probability when the parameter would belong to the range that a certain hypothesis is true.Theorem 7.For hypothesis And should rejected rion is that reject H 0 when Note that the left hand of (5.2) is the quantile of order  : , The fiducial probability Proof.The result can be got just like theorem 7. □ The parallel results for  can be got by using the same method as well.
Theorem 9. Hypothesis This theorem can be got by calculating the above integral.
□ The fiducial probability in the situation that the parameters would belong to the range that a certain hypothesis in Theorem 7 or 8 is true can be easily got by using (4.3) in the case that one parameter is in variation.
Example.For the example in Section 1, consider the hypothesis So the criterion is that reject H 0 when     0 max min 0.5 0.5 (5.4) Please note that the left hand of (5.4) is the quantile of

Discussion
Up to now, the discussion on Fisher's fiducial inference has still remained intuitive and imprecise.There are two problems: 1) Just what a fiducial probability means?2) How can one derive the only fiducial distribution of the 12) Proof.At first Equation (3.12) has a positive solution Copyright © 2013 SciRes.AM d because its left side equal to 1 when and tends to 0 when d tends to .Hence the fiducial probability that 0 d    ,    belongs to the region given by (3.11) is

. 1 )
It should noted that using (2.4) and (2.6) the conditional density of  under b   can be got as 1) and (4.2) is to say that (4.1) is coincided with the conditional density of  under 0 !b   in (3.7).□If for a certain 0   , the decision is made by com- and Me  is a modify to ˆm 