E-Bayesian estimation and e-posterior risk for Erlang distribution for known shape parameter

I. Introduction

Erlang Distribution is a queuing distribution. It was first proposed by Agner Krarup Erlang. Erlang distribution has following probability density function: -𝑓(𝑥; 𝑘, λ) = λ 𝑘 𝑥 𝑘-1 𝑒 -λ𝑥 (𝑘-1)! for 𝑥, λ ≥ 0

Where λa rate parameter setting and k is is shape parameter setting.

Erlang distribution main usage comes in queueing theory, reliability analysis, mathematical biology, etc. Erlang distribution has found its way into statistical research and has been extensively used as well. Kindly refer to Therrar Kadri et al. [1] , Erik Levén et al. [2] , Aslam et al. [3] , M. M. E. Abd El-Monsef [4] , etc. The foundation for Bayesian Statistics was first laid by Thomas Bayes in 18 th century.

III. Bayesian Estimation

Let X1, X2, X3… Xn be observed random variables for the distribution (1). The likelihood function (LF) can be written as

Where 𝑥̅ = {𝑥 1 , 𝑥 2 , … , 𝑥 𝑛 }. Let 𝑝(𝑥̅ | 𝜆) be the prior distribution. On simplifying -

For prior distribution we will use gamma distribution. Gamma distribution has two parameter which we will assume to be known parameter. It has the following pdf

Joint distribution of λ and 𝑥̅ is -

The marginal distribution is -

Posterior distribution has the following mathematical structure -

Using the squared loss error function, and differentiating for reducing the value of loss function we get -

According to Ali et al [10] and Han [7] , the posterior risk of bayesian estimate

IV. E-Bayesian Estimation and E-Posterior Risk

The E-Bayesian Estimation of λ becomes

Where p is domain or precinct of the prior parameter𝛼 𝑎𝑛𝑑 𝛽. The E-Posterior Estimation of λ becomes -

For simplification, this paper assumes the prior parameters to be independent with bivariate density function -

The above assumption has also been used for the calculation of E-Bayesian for record statistics in the paper Okasha & Wang [5] from the same paper, we take the following distribution for 𝛼 𝑎𝑛𝑑 𝛽 (this will further strengthens their work)

I. E-Bayesian Estimation of 𝜆

The following distribution, as mentioned earlier, has been chosen -

Both of these distributions follow our assumption of independent bivariate density as presented in the equation (10). The calculation for E-bayesian for the respective distributions have been done below a) 𝜋 1 (𝛼, 𝛽) Distribution -

II. E-Posterior Risk Estimation of 𝜆 a) 𝜋 1 (𝛼, 𝛽) Distribution -

V. Monte Carlo Simulation

Erlang distribution can be generated using the following transformation -

For our simulation the following steps are used a) We take fixed values for our prior (u, v) and (0, s). With these values we generate 𝛼 𝑎𝑛𝑑 𝛽 from ( 14) and (15). b) We fix our shape parameter k = 3. c) For generating 𝜆 ̂(𝛼, 𝛽) we take mean ofπ

VI. Conclusion

As you can see from table caption the parameter study has been done on rate parameter, and two parameters of prior distribution. Following can be observed a) On inverting (u, v) from (2, 3) to (3, 2), the bayesian and e-bayesian converges to rate parameter slowly. b) For increasing rate parameter, a larger sample is required to reduce the error for converging to the rate parameter c) In all study e-bayesian study has better perform than traditional likelihood estimation method. (Kindly refer to last row of each study) d) As the sample size increases, the posterior risk decreases in all. Thus, this is in good agreement with the theory. Also, the eposterior risk has been far lower than the posterior risk.

VII. Further Reading

E-Bayesian Estimation has seen a keen interest by other research and author in the field of mathematics and statistics. The analysis presented in this paper has been done for exponential distribution Han [7] . For lomax distribution kindly refer Liu & Zhang [8] . Njomen et al. [9] has done an excellent work by relating competing risks and e-bayesian. The author has further referenced the work of Panda and Bhoi [6] .

VIII. Acknowledgements

The author thanks the editor and the reviewers for their time and attention this paper. Their criticisms have played a significant role in improving the quality of the research.

References

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