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11
BAYESIAN INTERVAL ESTIMATION FOR THE PARAMETERS OF POISSON TYPE RAYLEIGH
CLASS MODEL
Rajesh Singh
Department of Statistics, S. G. B. Amravati University, Amravati, India rsinghamt@hotmail.com
Preeti A. Badge
Department of Statistics, S. G. B. Amravati University, Amravati, India preetibadge10@gmail.com
Pritee Singh
Department of Statistics, Institute of Science, Nagpur, India. priteesingh25@gmail.com
Abstract
In this article, two sided Bayesian interval is proposed for the parameters of Poisson type Rayleigh class software reliability growth model. In this work, the failure intensity function, mean time to failure function and likelihood function of this model have been derived by considering parameter total number of failures i.e. ya and scale parameter yi. The mathematical expressions of Bayesian interval for the parameters have been obtained by considering non informative priors. The performance of proposed Bayesian interval is studied on the basis of average length and coverage probability. Average length and coverage probability is obtained by using Monte Carlo simulation technique after generating 1000 random samples. From the obtained results, it is concluded that Bayesian interval of parameters perform better for appropriate choice of execution time and certain values of parameters.
Keywords: Rayleigh distribution, Non informative prior, Software reliability growth model, Bayesian interval, average length, coverage probability.
1. Introduction
Software reliability is the quality characteristic of operation system which can measure, predict and estimate quality of software system. In last several decades various model have been proposed to assess software reliability. Most of them are probabilistic models. Software modeling techniques can be divided into two categories: Prediction and estimation models. Estimation models determines the current software reliability by applying statistical inference techniques to failure data while the prediction models determines future software reliability based upon available software metrics and measures. The parameters included in software reliability models can be estimated by using some basic procedures like maximum likelihood, least square estimation and Bayesian point estimation, etc.
The research in this area of software reliability has been started since 1950's and a large number of researchers have done work in this field. Most of the past research work in software reliability modeling has concentrated on the point estimation of the parameters. The uncertainty of the estimates by using interval estimation has not been fully discussed. The most commonly applied interval estimation technique is based on the central limit theorem assuming large sample size. However, in real world testing the number of software failures observed is usually not large enough. Whereas Bayesian approaches produce interval estimates even in the case of small sample size, by utilizing prior knowledge.
This paper considers Poisson type Rayleigh model as per classification scheme of Musa and Okumoto [10] Rayleigh distribution has wide application in life time data especially in reliability theory and survival analysis. A specific case of Weibull distribution exhibiting aging effect with an integer valued shape parameter is known as "Rayleigh distribution." Dey and Dey [3] presented Bayes estimators for the parameter of Rayleigh model on the basis of loss function. Also provided Highest Posterior Density (HPD) for the unknown parameter of Rayleigh model. Lee et al [8] proposed software reliability growth model (SRGM) and obtained confidence interval using Obha's inflection S shaped model that can assess software developers optimal release time of software testing tasks. Rabie and Li [12] has studied Bayesian and E-Bayesian approaches under squared error loss function LINEX loss functions and constructed confidence interval for maximum likelihood and credible interval. Xie et al [18] estimated software reliability using Goel-Okumoto model and obtained confidence interval for failure intensity. Wu et al [17] obtained Bayes estimates and credible interval for Rayleigh distribution for the parameters and reliability function. Shrestha and Kumar [13] have obtained Bayes parameter estimates such as reliability function, hazard function under loss function for lomax distribution. Also provides Bayesian credible interval and Highest Posterior Density (HPD) interval for the corresponding parameters. Ogura and Yanagimoto [11] proposed a novel credible interval of the binomial proportion by improving the Highest Posterior Density (HPD) interval using logit transformation. The 100(1-a)% confidence interval through MLE is compared with corresponding level of credible interval. The reason for this is that MLE is preferred by researchers and Bayesian inference is effective for small sample size. Cunha and Rao [2] estimated credible interval and confidence interval through MLE for lognormal distribution also compared average length and coverage probability of the calculated interval. Fang and Yeh [4] proposed a software reliability estimation process that uses stochastic differential equations (SDEs) with fault detection function to construct confidence interval of mean value function m(t) of SRGMs. Song et al [16] proposed a new NHHP software reliability model and estimated confidence interval.
The association of the paper is such that section 2 presents derivation of failure intensity and expected number of failures using Rayleigh distribution. Section 3 presents selection of priors and posterior distribution of model. Section 4 presents two sided Bayes interval for the parameters yo and yi. Results are discussed in the section 5 while concluding remarks are provided in section 6.
Considering that software failure time of a system following Rayleigh distribution with scale parameter yi and software failures occurred in Poisson manner. Let t be the positive random variable having Rayleigh distribution then its probability density function is given by
2. Model Formulation
—i—1"
/(t) = (t y-2e "W , t>0,y1 > 0
Assuming that the total number of failures remaining in the program at time t = 0 is a Poisson random variable with mean yo then the failure intensity A (t) = yo f (t) can be obtained as follows (cf Musa et al [9])
JILI2
A(t) = y0yJ2te 2W , t>0,Vl >0,y0 >0 (2)
The mean time to failure function i.e. expected number of failures at time te using equation (2) comes to be
c ii-l2 KO = Y0Yl2 J0te 2W dx
On simplification it can be written as
tite) = ^$ ] (3)
Now assuming that me software failures for a system are experienced at times ti , i = 1,2,......,m
up to execution time te (> tme) and the likelihood function of parameters yo and y 1 can be obtained as L(0,t) = [nj/!^(tj)] exp exp [—u(te)] (cf. Singh et al [14]). Using failure intensity function given in (2) and mean time to failure function given in (3) the likelihood function is obtained as
e-frYi2e -T0exp >'Y0 e ? (4)
¿(To/YO = T.T"2m*[n.! ti] e~"rT-2e exp {'Y0 e""3^"?
3. Choice of priors and Posterior distribution
In Bayesian estimation appropriate choice of the prior(s) for the parameter is necessary. However Bayesian analyst pointed out that there is no perfect technique from which one can conclude that one prior is better than the other. Very often, priors are chosen according to one's subjective knowledge and beliefs. However, in case of adequate information about the parameter is available one can use informative prior(s) otherwise it is preferable to use non informative prior(s). Bayesian estimation is a method that combines prior information with information obtained from sample data. While testing the software, the experimenter have very little knowledge relative to the total number of failures present in the software i. e. yo and yi. Here insufficient prior information is available about parameters yo and yi, hence noninformative priors are considered.
Jeffrey's [6] has suggested the use of non-informative priors. Jeffrey's prior is widely used because it is proper under slight conditions. It requires likelihood function from which the prior is then derived using Jeffrey's rule. More discussion properties of Jeffrey's prior has been studied by Chen et al [1].The following non informative prior distributions g (yo) and g (yi) are considered for parameters yo and yi which are as follows:
and
9(Yo) K [y0 1 ,Y0e
g(Yi) K {/1 1 ,Yi£ [0,ra)
(5)
(6)
The joint posterior of T0 and T1 given t (= ti, i=1, 2,..........me) is obtained by using equations
(4),(5) and (6) is as follows:
n
(t) = D-1Y™e-1Y-2me-1e-2Ty- e —^exp \yn e
me <Y0 < ,0 < Y1 <
(7)
Where D is normalizing constant
D
= G
9=1
r( m* +j, m* )r(2m*) j!
(2/S)(#.
Where, S = (T + jt2e), T = £j=1 t# The marginal posterior distribution of y0 given t is obtained by integrating equation (8) over the whole range of 71i.e.
n(t) = D-1 z;=n [-i2^^] [c+—vn, 3 , m*
<Yo <
(8)
Similarly, the marginal posterior distribution of y1 given t is as
n(t) = D—1 S;=n [
r(me+j,me)
J'
—2m„ — 1 —Svi'
e rri I ,0<Y1 <™
(9)
4. Bayesian interval estimation of parameters yo and yi The equal tailed 100(1- a)% Bayes probability interval is given as: /8 n[t)dt = a/2 and /^nfydt = a/2
Where (t) is the marginal posterior distribution and yt lower limit and y*upper limit of the Bayesian interval respectively. For details see Martz and Waller [9], B.K. Kale [7], S. K. Sinha [15].
Now by integrating equations (8) and (9) w.r.t. yo and yi respectively 100(1- a)% two sided Bayesian interval for the parameter yo and yi can be obtained as follows:
CO
4 = D-! £7=1
(2/S)(2meV(2 mP)
9
— T( m* +;,Kot )
^ r( m* +i,Y0 )
4 = D-1 Zf=i
(2/J)(2me)r(2me)
9
4 = D-1 Zf=i
4 = £-1 zr=i
K( meH
K( meH
Where, 4 and 4 is the Bayes lower limit and upper limit of parameter yo i.e. total number of failures, y4i and 4 is the Bayes lower limit and upper limit of parameter yi. And T( me + j, y*0 ) and r(2me, S/2 yl) are incomplete gamma functions.
The details about the incomplete gamma function can be seen from Gradshteyn and Ryzhik [5].
5. Discussion
Table (1) to (8) represents average length and coverage probability of the Bayesian two sided interval is obtained for the parameter yo i.e. total number of failures and parameter^. The Bayesian interval depends upon the values of execution time i.e. te and me failures experienced
at times ti, i = 1,2...........,me. Bayesian interval is studied by calculating the average length and
coverage probability of the simulated interval. To study the performance, a sample size was generated from the Rayleigh distribution and it is repeated 1000 times. Average length and coverage probability is calculated for Bayes two sided interval for different execution time te for different values of parameters. Monte Carlo simulation is used to study the performance of Bayesian interval. Average length and coverage probability have been calculated by assuming parameter yo (=1(1)5) and y1(= 0.25(0.25)1.25) using 1000simulations.
From tables (1) to (4) it is observed that Bayesian interval's average length decreases as y0increases and it is increases as y1 increases for different execution time i.e. te. It can be seen that as execution time increases average length also increases. Here assumes that Bayesian interval maintains the credible level if the estimated coverage probability is in between the range of 0.940 to 0.960 i.e. (1-a)±0.01 where, a = 0.05. It was found that the coverage probabilities of the interval decreases as y0increases and coverage probability increases as y1 increases.
Here, table (5) to (8) represents average length and coverage probability for Bayesian two sided interval for the parameter yi.It can be seen that average length computed for Bayesian interval is increases as total number of failures i.e. y0 increases. And as y1 increases average length also increases. It also observes that average length decreases as execution time i.e. te increases. From tables it can be seen that coverage probability increases as y0and y1 increases. When average length will be shorter coverage probability will decreases. As execution time te increases coverage probability decreases.
Table 1: Average length and coverage probability of Bayesian interval &0 calculated for different values of parameters y0 and y^when execution time te = 5
Y° 1 2 3 4 5
Yi
2.14713 2.023915 1.574669 1.02334 0.295229
0.25 (0.996) (0.996) (0.995) (0.995) (0.994)
2.14714 2.058673 1.604102 1.04157 0.375861
0.50 (0.996) (0.996) (0.995) (0.995) (0.994)
2.14715 2.066402 1.621196 1.30856 0.624627
0.75 (0.996) (0.996) (0.995) (0.995) (0.994)
2.14723 2.078305 1.683277 1.40131 0.674976
1 (0.996) (0.996) (0.995) (0.995) (0.994)
2.14768 2.081441 1.751818 1.41164 0.718004
1.25 (0.997) (0.996) (0.995) (0.995) (0.994)
*The values in the parenthesis are coverage probability.
Table 2: Average length and coverage probability of Bayesian interval &0 calculated for different values of parameters y0 and y^when execution time te = 10
\Y» 1 2 3 4 5
Y1
2. 42186 2.14698 2.13883 2.051196 1.664166
0.25 (0.998) (0.996) (0.996) (0.995) (0.994)
2.46526 2.22720 2.147932 2.06754 1.673916
0.50 (0.998) (0.996) (0.996) (0.995) (0.994)
2.48655 2.247208 2.147905 2.07153 1.67429
0.75 (0.998) (0.996) (0.996) (0.995) (0.994)
2.48658 2.327472 2.14932 2.07210 1.67455
1 (0.998) (0.997) (0.997) (0.996) (0.994)
2.49144 2.348076 2.15390 2.074465 1.68610
1.25 (0.999) (0.998) (0.997) (0.996) (0.994)
Table 3: Average Length and Coverage Probability of Bayesian interval y0calculated for different values of parameters y0 and y± when execution time te = 15
\V0 1 2 3 4 5
Y1
2.43761 2.24616 2.18678 2.151995 2.08162
0.25 (0.998) (0.998) (0.997) (0.997) (0.996)
2.43786 2.26656 2.18701 2.152415 2.082083
0.50 (0.998) (0.998) (0.997) (0.997) (0.996)
2.43791 2.271643 2.20720 2.15321 2.083737
0.75 (0.998) (0.998) (0.997) (0.997) (0.996)
2.44656 2.33112 2.21472 2.153597 2.084265
1 (0.998) (0.998) (0.998) (0.997) (0.996)
2.48656 2.35643 2.22476 2.15383 2.102458
1.25 (0.999) (0.998) (0.998) (0.998) (0.997)
*The values in the parenthesis are coverage probability.
Table 4: Average Length and Coverage Probability of Bayesian interval y0 calculated for different values of parameters y0 and y± when execution time te = 20
1 2 3 4 5
Yi
2.46155 2.25882 2.14701 2.14309 2.10662
0.25 (0.998) (0.998) (0.996) (0.995) (0.994)
2.46237 2.26956 2.14701 2.14382 2.10883
0.50 (0.998) (0.998) (0.996) (0.995) (0.994)
2.46256 2.27436 2.14720 2.14445 2.11537
0.75 (0.998) (0.998) (0.996) (0.995) (0.995)
2.46506 2.34452 2.147682 2.14894 2.12780
1 (0.998) (0.998) (0.996) (0.995) (0.995)
2.49164 2.36436 2.14807 2.14979 2.12858
1.25 (0.999) (0.998) (0.996) (0.995) (0.995)
Table 5: Average length and coverage probability of Bayesian interval yt calculated for different values of parameters y0 and y^when execution time te = 5
1 2 3 4 5
Yi
0.001098 0.003211 0.006615 0.009148 0.011176
0.25 (0.995) (0.996) (0.997) (0.998) (0.998)
0.001114 0.003322 0.007081 0.009530 0.012966
0.50 (0.996) (0.996) (0.997) (0.998) (0.998)
0.001116 0.003406 0.007276 0.010573 0.013338
0.75 (0.996) (0.996) (0.997) (0.998) (0.998)
0.001116 0.003485 0.007954 0.013072 0.014761
1 (0.996) (0.996) (0.998) (0.998) (0.998)
0.001125 0.003821 0.008098 0.013388 0.015302
1.25 (0.996) (0.997) (0.998) (0.998) (0.998)
*The values in the parenthesis are coverage probability.
Table 6: Average length and coverage probability of Bayesian interval yt calculated for different values of parameters y0 and y^when execution time te = 10
\Yo 1 2 3 4 5
Yi \
0.25 0.001071 0.001090 0.001723 0.002980 0.002112
(0.995) (0.995) (0.996) (0.996) (0.997)
0.50 0.001074 0.001102 0.002021 0.00302 0.002406
(0.995) (0.995) (0.996) (0.996) (0.997)
0.75 0.001074 0.001102 0.002023 0.003133 0.002988
(0.995) (0.995) (0.996) (0.996) (0.997)
1 0.001078 0.001103 0.002036 0.003106 0.003079
(0.995) (0.996) (0.996) (0.996) (0.997)
1.25 0.001079 0.001103 0.002221 0.003374 0.003684
(0.995) (0.996) (0.996) (0.996) (0.997)
Table 7: Average Length and Coverage Probability of Bayesian interval yt calculated for different values of parameters y0 and y± when execution time te = 15
\ Y» 1 2 3 4 5
Yi
0.001069 0.001082 0.001091 0.001114 0.001116
0.25 (0.995) (0.995) (0.996) (0.996) (0.997)
0.001071 0.001085 0.001113 0.001115 0.001119
0.50 (0.995) (0.995) (0.996) (0.996) (0.997)
0.001072 0.001086 0.001115 0.001115 0.001120
0.75 (0.995) (0.995) (0.996) (0.996) (0.997)
0.001075 0.001086 0.001116 0.001116 0.001122
1 (0.995) (0.995) (0.996) (0.996) (0.997)
0.001077 0.001088 0.001119 0.001117 0.001122
1.25 (0.995) (0.995) (0.996) (0.997) (0.997)
*The values in the parenthesis are coverage probability.
Table 8: Average Length and Coverage Probability of Bayesian interval yt calculated for different values of parameters y0 and y± when execution time te = 20
1 2 3 4 5
Yi
0.001068 0.001079 0.001082 0.001085 0.001090
0.25 (0.994) (0.995) (0.995) (0.995) (0.995)
0.001068 0.001080 0.001084 0.001088 0.001092
0.50 (0.994) (0.995) (0.995) (0.995) (0.995)
0.001067 0.001081 0.001085 0.001089 0.001093
0.75 (0.995) (0.995) (0.995) (0.995) (0.996)
0.001066 0.001083 0.001085 0.001092 0.001094
1 (0.995) (0.995) (0.995) (0.995) (0.996)
0.001064 0.001086 0.001088 0.001093 0.001095
1.25 (0.995) (0.995) (0.995) (0.996) (0.996)
6. Conclusion
In this research paper, Bayesian interval has been proposed considering Poisson type Rayleigh class software reliability growth model as function with parameters total number of failures i.e. y0 and scale parameter y1. Bayesian analysis is carried out by considering non informative prior. The performance of two sided Bayesian interval is studied using Monte Carlo simulation technique. Average length and coverage probability of Bayesian interval is calculated for both the parameters y0 and y1 for different execution time te. From study it is concluded that proposed Bayesian interval has shorter average length for both parameters. Bayesian interval maintained coverage probability for both the parameters y0 and y1 for different execution time for different values of parameters. In future, confidence interval will be obtained for proposed model and will be compared with Bayesian interval on the basis of average length and coverage probability.
References:
[1] Chen, M. H., Ibrahim, J.G. and Kim, S. (2008). Properties and Implementation of Jeffrey's Prior in Binomial Regression Models. Journal of the American Statistical Association, 103:1959-1664.
[2] D'Cunha, J. G. and. Rao, K. A. (2016). Frequentist Comparison of the Bayesian Credible and Maximum Likelihood Confidence interval for the Median of the Lognormal Distribution for the Censored Data. International Journal of Scientific and Research Publications, 6:61-66.
[3] Dey, S. and Dey, T. (2012). Bayesian estimation and prediction intervals for Rayleigh distribution under a conjugate prior. Journal of Statistical Computation and Simulation, 82:16511660.
[4] Fang C.C. and Yeh C.W. (2016). Effective confidence interval estimation of fault detection process of software reliability growth model. International Journal of Systems science, 47: 2878-2892. http://dx.doi.org/10.1080/00207721.2015.1036474
[5] Gradshteyn, I. S and Ryzhik, I. M. Table of Integrals, Series, and Products, Alan Jeffrey (editor) 5 th Ed., Academic Press, New York, (1994).
[6] Jeffreys, H. Theory of Probability, Third Edition, Clarendon Press, Oxford(1961).
[7] Kale, B. K. A First Course on Parametric Inference (Second Edition) Narosa, (1999).
[8] Lee, T. Q. Fang, C. C. and Yeh C.W. (2013). Confidence Interval Estimation of Software reliability growth Models based on Obha's Inflection S shaped Model. Journal of Industrial and Intelligent Information, 1:196-200.
[9] Martz H.F. and Waller R.A. Bayesian Reliability Analysis, New York, Wiley (1982).
[10] Musa, J. D. and Okumoto, K. (1984), "A logarithmic Poisson execution time model for software reliability measurement", Proceedings of Seventh International conference on software engineering Orlando, pp 230-238.
[11] Ogura, T. and Yanagimoto, T. (2018). Improvement of Bayesian Credible interval for a small Binomial Proportion using Logit Transformation, Current Research in Biostatistics, 8.
[12] Rabie A. and Li J. (2019). E-Bayesian Estimation based on Burr-X Generalized Type II Hybrid censored data, Symmetry, 11: 626-640.
[13] Shrestha S.K. and Kumar V. (2014). Bayesian analysis of Extended Lomax distribution, International Journal of Mathematics Trends and Technology, 7: 33-41.
[14] Singh, R. Singh, P. and Kale, K. (2018). Bayes Estimators for the parameters of Binomial type Rayleigh class software reliability growth model using non- informative priors. Journal of Reliability and Statistical Studies, 11: 01-07.
[15] Sinha, S. K. Reliability and life testing, Wiley, New Delhi (1985).
[16] Song, K. Y. Chang I. H. and Pham H. (2017). A software Reliability Model with a Weibull Fault Detection rate Function Subject to Operating Environments. Journals of Applied Sciences, 7: 983-999.
[17] Wu, S.J. Chen, D.H. and Chen S.T. (2006). Bayesian inference for Rayleigh distribution under progressive censored Sample, Applied Stochastic models in business and industry, 22:269279.
[18] Xie, M. Hong G. Y. and Wohlin C. (1997). A practical method for the Estimation of Software Reliability Growth in the Early Stage of Testing, Proc. IEEE 7th International Symposium on software Reliability Engineering, Albuquerque, 116-123.
