A New Class of Rectangular Distribution Properties and Application

In this article, we generalize the Rectangular distribution using the quadratic rank transmutation map studied by Shaw and Buckley (2007) to develop a transmuted Rectangular distribution. The expectation, variance, moments, reliability function, hazard rate function, Cumulative Hazard function, the moment generating function


1-Introduction and Motivation
In probability theory and statistics, the continuous uniform distribution or Rectangular distribution is a family of symmetric probability distributions, such that for each member of the family all intervals of the same length on the distribution's support are equally probable. A uniform random variable is equally likely to take any value between its lower limit (a) and its upper limit (b). The distribution is often abbreviated U(a,b), [Park and Anil, 2009] . For a uniform random variable between a and b, the function is constant between a and b.
Since the probability that the random variable falls between a and b must be 1, the function is given by Notice that, if has a standard uniform distribution, then by the inverse transform sampling method, Y = − λ −1 has an exponential distribution with (rate) parameter λ. If has a standard uniform distribution, then has a beta distribution with parameters .
As such, If has a standard uniform distribution, then is also a special case of the beta distribution with parameters (1,1 Fx is the cumulative distribution function (cdf) of the base distribution, which on differentiation yields, () fx, such that

2-Transmuted Rectangular Distribution
By using Eq.(1) and Eq.(2), the cdf of transmuted Rectangular distribution (TRD) has the following form Where  is the transmuted parameter. The corresponding probability density function of Eq.
(3) is given as follows It is observed that, the transmuted Rectangular distribution is an extended model to analyze data from complex situations, as shown in the following plots which are the pdf and cdf of TRD distribution for selected values of the parameters.

3-Statistical Properties
In this section, some statistical properties of the new generalization are provided.

3-1 The Reliability or the Survivor function:
There is a relation between the cdf and the reliability function, i.e., ( ) 1 RF F x  . Therefore, the reliability function of the transmuted Rectangular distribution (RF TRD ) also known as the survivor function and is defined as:

3-2 The Hazard function :
There is a relation between the pdf, reliability and hazard . Therefore, the hazard function of the transmuted Rectangular distribution (HF TRD ) is defined as:

3-3 The Cumulative Hazard function
It is observed that, the transmuted Rectangular distribution has increasing patterns reliability and cumulative hazard function, as shown in the following plots.

3-4 Random number generation and parameter estimation
To generate random numbers when the parameters a and b are known, we can use the method of inversion from the transmuted Rectangular distribution as . (1 ) , t have a closed form solution, so u will be generated as uniform random variables from U(0,1), and then solve for x r.v in order to generate random numbers from TRD distribution.

3-5 Moments
In applications, it is necessary and important in any statistical analysis to derive moments, so in this subsection, the r th moment for (TRD) distribution are derived.
11 12 (2), The r th ordinary moment of the (TRD) distribution is given by

3-6 Moment Generating Function
The moment generating function is important especially if it is existing. Then in this section, the moment generating function of (TRD) distribution is derived. From Eq. (2), The moment generating function of the (TRD) distribution is given by In addition, we can obtain the characteristic function of the TRD distribution.

Theorem 3:
If has TRD with the moment generating function MGF(11), then the characteristic function QF of the TRD distribution is From Eq.(2), and MGF(11) of the TRD the characteristic function QF of the TRD is given which is the characteristic function of the Rectangular distribution.

4-Maximum Likelihood Estimates
The maximum likelihood estimates, MLEs, of the parameters that are inherent within the transmuted Rectangular probability distribution function is given by the following: Let be a sample of size n from a transmuted Rectangular distribution, then the likelihood function is given by

5-Application of Transmuted Rectangular distribution
The estimators and the corresponding summary statistics are obtained by the proposed model using MathCAD program. For a given samples with different choices of , b, and a  we obtain the maximum likelihood estimators (MLEs), the mean squared error (MSE), relative bias (RAB) and the confidence interval for λ, Table 1 summarizes the results. Estimate the true parameter λ , well with relatively small MSEs and RAB. We also notice that the coverage probabilities of the asymptotic confidence interval are close to the nominal level. These results indicate that the proposed model and the asymptotic approximation work well under the situation.

6-Conclusion
In this article, a new model called the transmuted Rectangular distribution was proposed, which extends the Rectangular, or the uniform distribution in the analysis of data with real support. Expansions for the expectation, variance, moments, reliability function, hazard rate function, the moment generating function, the characteristic function, the MLE of the unknown parameter with its variance were derived. An application of the transmuted Rectangular distribution to generated data show that the new distribution can be used quite effectively to provide better fits than the Rectangular distribution, and that an obvious reason for generalizing a standard distribution of the Rectangular distribution, so it provides greater flexibility in modeling real data.