Statistical Inferences on Uniform Distributions: The Cases of Boundary Values Being Parameters

If a continuous random variable X is uniformly distributed over the interval and if any of the two boundary values is unknown, it is necessary to make inferences related to the unknown parameter. In this work, for the unknown boundary values of X, some unbiased estimators based on certain order statistics and sample mean are suggested. These estimators are compared in terms their efficiencies. The most efficient unbiased estimator is used to provide confidence intervals and tests of hypotheses procedures for the unknown parameter (the unknown boundary value).


Background
The books and articles, listed in the reference list of this study with the reference numbers from [1] to [18], and many more not in the list, studied order statistics in such a way that every aspects of the topic has already been well explored.
However, there is no inferential study, as far as I am aware of, on the boundary values of the uniform distributions. This work aims at the determination of good estimators for the boundary values of uniform distributions. Based on the determined good estimator, construction of confidence intervals and procedures for the test of hypotheses are established.
For illustration, a simulation study is conducted and summaries of the simulation study are provided. The raw data and computations are provided in the appendix of this paper.

Introduction
A uniformly distributed continuous random variable, assuming real values in the interval ) , ( 2 1 θ θ , has the following probability density function (pdf) The pdf of ith order statistic is obtained by the following general formula, as given in almost all mathematical statistics textbooks, like the ones with the reference numbers [1], [2], [3], [4], [6], and [7].
Where, n= size of the random sample, ) ( y F is the cumulative distribution function (cdf) of the distribution, and ) ( y f i Y is the pdf of the random variable ( ith order statistic) i Y .
Specifically, if the pdf given in (1) is used we obtain the following cdf.
Using the equations (3) and (4), we obtain the following pdfs for 1 Y and n Y .  (6) Use of the above pdfs will enable us to obtain the expected values and the variances of 1 Y , and n Y .
To take advantage of the computational simplicity lets introduce the following transformations.
Here we established the distributions, expected values, and variances of the smallest (the first) and the largest (the last) ordered statistics that are to be used in subsequent sections of this study.

Estimation of the Parameterθ of the Distribution
For any distribution, if the sample mean is X , for any random sample of size n, the followings hold true. is to be obtained, by the use of (3), as follows.
We need to show that for any m and for n=2m+3 the statements of the Theorem 3.4.1 are true. For n=(2m+3) If we let k m = + ) 1 ( then in accordance with (3.4.2) and (3.4.3) we conclude the following.
For the case of n being an odd number, the proof is completed. For any random sample taken from the distribution of ) ( y f , joint pdf of the ordered statistics r U , and t U , (r < t) can be obtained by the use of the following general formulation as given in [6].
For n=4: The joint pdf of 2 U and 3 U is obtained as given below.

Tests of Hypotheses Related to the Parameter
Don't reject H0 otherwise Don't reject H0 otherwise Where, If the level of significance is chosen to be α , then the decision rules, as given in the following table, are applicable.
It is concluded that the best unbiased estimator, among the ones suggested, of the parameter 1 θ for the uniform distribution over Since T 1 is a linear function of the first order statistic Y 1 , construction of confidence interval and tests of hypotheses procedures are related to and dependent upon the observed value of the first order statistic Y 1 and the chosen level of significance α .

Statistical Inferences Related to the Parameter of
and its pdf is as given below.
If a random sample of size is taken from the distribution of Y, then the ordered statistics will be denoted by The parameter of this distribution, η , can be estimated by 1 U . The pdf of 1 U is given below.
Hence, an estimator of 2 By using (6.1.3) and (6.1.5) we obtained the variance of 1 [ ] By the utilization of (6.1.3) we can obtain an unbiased estimator for θ as a function of 1

Estimation of the Parameter θ of the Distribution
The Maximum Likelihood estimator for the parameter 2 θ is n Y .
Since, a X Y − = , then the following will be true.
n a n a n a n a n From (6.2.4) we obtain an unbiased estimator for θ as a function of n n n a n X Var n W Var n θ (6.2.7)

Estimation of the Parameter
For any distribution, if the sample mean is X , for any random sample of size n, the following hold true.
From (4.3.1) we can obtain an unbiased estimator for θ , as function of the sample mean X .
and its pdf is as given below.

Comparisons of Unbiased Estimators in Terms of Their Efficiencies
The unbiased estimators and their comparisons are given in Table 3.
We see that, for n >1, the most efficient unbiased estimator among the ones given above, is

Confidence Interval for the Parameter
The most efficient unbiased estimator of θ is seen to be n a Y n T n − + = ) 1 ( 2 . By the use of the pdf of n Y we can construct a )% 1 ( 100 α − confidence interval forθ . As it is shown before By the use of following probability statement we can obtain a confidence interval forθ .
n n y a n n a a y dw w a n dy a y a n a nL nL n nL a a y If the results in (7.2) and (7.3) are substituted in (7.1) Solving the above inequalities for θ , the following Confidence Interval is obtained.

Tests of Hypotheses Related to the
is to be tested against to any proper alternative hypothesis, a plausible test statistic is to be If the level of significance is chosen to be α , then the decision rules, as given in the following table, are applicable.

Simulation Study
To see the match between the established theoretical findings and the empirical results, a simulation study on a uniform distribution over the interval  Table 5.