1. 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.

2. Introduction

A uniformly distributed continuous random variable, assuming real values in the interval, has the following probability density function (pdf)

(1)

For this distribution, the Maximum Likelihood Estimators

(MLE) of and will be and, the smallest and the largest order statistics, respectfully.

and need to be estimated for that each moment of the random variable X is , as shown below, a function of these parameters.

(2)

Let be a random sample of size n from a uniform distribution over the interval, and let ith order statistic, i=1, 2,..., n.

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].

(3)

Where, n= size of the random sample, is the cumulative distribution function (cdf) of the distribution, and is the pdf of the random variable ( ith order statistic) .

Specifically, if the pdf given in (1) is used we obtain the following cdf.

(4)

Using the equations (3) and (4), we obtain the following pdfs for and.

(5)

(6)

Use of the above pdfs will enable us to obtain the expected values and the variances of, and.

To take advantage of the computational simplicity lets introduce the following transformations.

(7)

(8)

(9)

(10)

From (7) we get . It then follows that

(11)

From (7) we observe that the pdfs of are exactly the same.

Hence,

From (7) we get. It then follows that

(12)

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.

3. Statistical Inferences Related to the Parameter of Distribution

3.1. Estimation of the Parameter of the Distribution by the First Order Statistic

A uniformly distributed continuous random variable X, over the interval, where b is given constant, has the following pdf

The MLE of of will be and its expected value and variance are from Equations given in (11),

(3.1.1)

An unbiased estimator of , based on the first order statistic is

(3.1.2)

3.2. Estimation of the Parameter of the Distributionby the Last Order Statistic

Another estimator of of will be and its expected value and variance are from Equations given in (12),

(3.2.1)

By the utilization of the first equation of (3.2.1) we can obtain an unbiased estimator for as a function of the last order statistic

(3.2.2)

3.3. Estimation of the Parameter of the Distribution by

If then its pdf is. Then by equation (1), as given below:

For k=1 and

(3.3.1)

For k=2 and ,

(3.3.2)

From (1.3.1) and (1.3.2) and we obtain

.

For any distribution, if the sample mean is , for any random sample of size n, the followings hold true.

,

and

Hence, for,

(3.3.3)

(3.3.4)

From (3.3.3) an unbiased estimator of will be

(3.3.5)

3.4. Estimation of the Parameter of the Distributionby the Sample Median M

If then is uniformly distributed over the interval

To take advantage of computational simplicity, the parameter (and in turn of X), of the distribution, is to be estimated by the sample median.

Sample Median (M):

Let is a random sample of size n from

The sample median M, depending on if n is odd or even, is obtained as follows.

If are ordered, in the order of their magnitude, we obtain the following order statistics. .

If n is odd:

If n is even:

3.4.1. Estimation of the Parameter θ₁ of the Distribution X∼Uniform(θ₁, b) by the Sample Median n is Odd

Theorem 3.4.1. If and if an odd sized random sample is taken from this distribution then for the sample median M,

Proof:

Proof will be given by induction.

for n=1: and .

For n=3: the sample median is .The pdf of is to be obtained, by the use of (3), as follows.

Since, and , then

(3.4.1)

Similarly, for n=5, and from (3)

For n=2m+1, then .

Now, let’s assume that the following hold true for any n=2m+1.

(3.4.2)

(3.4.3)

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 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.

3.4.2. Estimation of the Parameter θ of the Distribution X∼Uniform(θ₁, b) by the Sample Median n is even

Theorem 3.4.2 If and if an even sized random sample is taken from this distribution then for the sample median M,

Proof:

Note:

If n is even, then the sample median is .

To compute the expected value and the variance of M we need to have the joint pdf of .

For any random sample taken from the distribution of, joint pdf of the ordered statistics, and, (r < t) can be obtained by the use of the following general formulation as given in[6].

(3.4.4)

For n=4: . The joint pdf of and is obtained as given below.

Since, and according to (3.4.5) the joint pdf is

(3.4.5)

For n=6: , the joint pdf of and is obtained as given below.:

(3.4.6)

Now, let’s assume that the following, given in (3.4.7) and (3.48), hold true.

(3.4.7)

(3.4.8)

We need to show that the statements of the Theorem 3.4.2 hold true for n=2m+4.

For n=2m+4,

The joint pdf of is obtained as follows.

(3.4.9)

In (3.4.9 take, observing the identity between (3.4.7) and (3.4.10) we conclude the following.

(3.4.10)

Similarly,

(3.4.11)

In (3.4.11), if we let, observing the identity between (3.4.8) and (3.4.12) we conclude the following.

(3.4.12)

For the case of n being an even number, the proof is completed.

3.4.3. Unbiased Estimators of θ₁ for X∼Uniform(θ₁, b) by M

n is odd: , , and

Where , hence an unbiased estimator of as function of is M

(3.4.13)

n is even: , , and .

Table 1. Unbiased estimators of       for the Distribution      and their variances

Where, hence an unbiased estimator of as function of is M ,

(3.4.14)

If we give the above comparisons in a tabulated form, we obtain the following:

For n>1;

We see that, for n >1, the most efficient unbiased estimator, among the ones as given above, is .

4. Confidence Interval for the Parameter of the Distribution

The most efficient unbiased estimator of is seen to be. By the use of the pdf of we can construct a confidence interval for. As it is shown before

.

By the use of following probability statement we can obtain a confidence interval for.

(4.1)

If we let then the following are true:

(4.2)

Similarly,

(4.3)

If the results in (4.2) and (4.3) are substituted in (4.1)

Solving the above inequalities for we obtain the following Confidence Interval.

(4.4)

A confidence interval for :

(4.5)

5. Tests of Hypotheses Related to the Parameter of the Distribution

Table 2. Tests of Hypotheses Related to the Parameter       for the Distribution

If is to be tested against to any proper alternative hypothesis, a plausible test statistic is to be for that the most efficient unbiased estimator is , which is a linear function of .

If the level of significance is chosen to be, 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 for the uniform distribution over is.

Since T₁ is a linear function of the first order statistic Y₁, construction of confidence interval and tests of hypotheses procedures are related to and dependent upon the observed value of the first order statistic Y₁ and the chosen level of significance .

6. Statistical Inferences Related to the Parameter of Distribution

6.1. Estimation of the Parameter θ₂ of the Distributionby the First Order Statistic

A uniformly distributed continuous random variable X, over the interval, has the following pdf.

If then, 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, , can be estimated by . The pdf of is given below.

(6.1.1)

(6.1.2)

If we let then the following will hold true.

(6.1.3)

Hence, an estimator of of will be and its expected value, from the transformation, is obtained as given below.

(6.1.4)

Similarly,

(6.1.5)

By using (6.1.3) and (6.1.5) we obtained the variance of and .

(6.1.6)

Since, , then .

(6.1.7)

By the utilization of (6.1.3) we can obtain an unbiased estimator for as a function of.

(6.1.8)

6.2. Estimation of the Parameter of the Distribution by the Last Order Statistic

If we want to estimate the parameter of by,

(6.2.1)

(6.2.2)

(6.2.3)

The Maximum Likelihood estimator for the parameter is.

Since, , then the following will be true.

(6.2.4)

(6.2.5)

(6.2.6)

.

From (6.2.4) we obtain an unbiased estimator for as a function of.

(6.2.7)

6.3. Estimation of the Parameter of the Distribution by the Sample Mean

If then the pdf is , and , , .

For any distribution, if the sample mean is , for any random sample of size n, the following hold true.

(6.3.1)

(6.3.2)

From (4.3.1) we can obtain an unbiased estimator for , as function of the sample mean .

(6.3.3)

6.4. Estimation of the Parameter of the Distribution by the Median M

A uniformly distributed continuous random variable X , over the interval , has the following pdf

If , then as it is stated in Note 1, and its pdf is as given below.

.

Estimation procedures and the findings, related to the parameter of , will be exactly the same as the one given in sections 1.4.1 and 1.4.2, with the exception that

In other words, the statements of Theorem 1.4.1 and Theorem 1.4.2 hold true.

That is: if n is odd then, , , and ,

If n is even, then

The unbiased estimators of as functions of the sample median M are as follows:

(when n is odd), (when n is even).

The variances of these unbiased estimators are:

6.5. 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 .

7. Confidence Interval for the Parameter of the Distribution

The most efficient unbiased estimator of is seen to be. By the use of the pdf of we can construct a confidence interval for. As it is shown before

.

By the use of following probability statement we can obtain a confidence interval for.

(7.1)

If we let then the following are true:

(7.2)

Similarly,

(7.3)

Table 3. Unbiased estimators of       for the Distribution      and their variances

Table 4. Tests of Hypotheses Related to the Parameter       of

If the results in (7.2) and (7.3) are substituted in (7.1)

.

Solving the above inequalities for , the following Confidence Interval is obtained.

(7.4)

A confidence interval for is given as follows.

(7.5)

8. Tests of Hypotheses Related to the Parameter of the Distribution

If is to be tested against to any proper alternative hypothesis, a plausible test statistic is to be for that the most efficient unbiased estimator is

, which is a linear function of.

If the level of significance is chosen to be, then the decision rules, as given in the following table, are applicable.

9. Conclusions

For uniform distributions, of the types , and , some unbiased estimators for the unknown boundary values and are suggested. Among the suggest estimators, the most efficient estimator is selected. By the use the most efficient estimators, confidence interval and tests of hypotheses procedures are established.

10. 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 is carried out. 100 independent samples of size 100 are drawn from this uniform distribution. From each sample the first order statistic, the last order statistic, sample mean, sample medians (for n=100), and for n=99) are computed. Unbiased estimator values are computed from each sample. Summary of the simulation results are given in the following Table 5.

Table 5. Simulation summaries

Based on a simulation of 100 independent samples of size n=100 from a uniform distribution over the interval, summary statistics, of 100 observations are obtained.

Values of five different estimators for, namely T1, T2, T3, T4, and T5 are computed from each sample by the use of proper summary statistics. The second and third columns of Table 5 contain the computed means and the variances of each summary statistics and of estimate values for.

From Table 5, we observe that T1, T2, T3, and T5 are unbiased estimators of. Among these unbiased estimators T1 is the best (the most efficient) estimator, for that it has the smallest variance.

In the same line, values of five different estimators for, namely W1, W2, W3, W4, and W5 are computed from each sample by the use of proper summary statistics.

From the second and the third columns of Table 5, we observe that W1, W2, W3, W4 and W5 are unbiased estimators of. Among these unbiased estimators W2 is the best estimator, for that it has the smallest variance.

By using the formula (2.5), a 95% confidence interval is computed for from each sample. Exactly 95 out of 100 confidence intervals contained the parameter value 5 of .

Similar procedures are followed for. By the use of the formula (5.5), 95% confidence intervals for are computed. 94 out of 100 confidence intervals contained the parameter value 10 of.

Simulation study results are seen to be in accordance with the theoretical findings.