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Estimation of Scale (Σ) and Shape (Θ) Parameters of Type I Generalized Half Logistic Distribution Using Median Ranks Method

Estimation of Scale (Σ) and Shape (Θ) Parameters of Type I Generalized Half Logistic Distribution Using Median Ranks Method

International Journal of and Systems ISSN 0973-2675 Volume 11, Number 1 (2016), pp. 9-18 © Research India Publications http://www.ripublication.com

Estimation of Scale (σ) and Shape (θ) parameters of Type I Generalized Half using Ranks Method

Ch. Rama Mohan1, O.V.Rajasekharam2 and G.V.S.R.Anjaneyulu3

1Lecturer cum , Community Medicine Department, A.S.R.Academy of Medical Sciences (ASRAM), Eluru, Andhra Pradesh-534004, India E-mail: [email protected] 2Research Scholar(BSRRFSMS), Department of Statistics, Acharya Nagarjuna University, Guntur, Andhra Pradesh, India. 3Professor & H.O.D, Department of Statistics, Acharya Nagarjuna University, Guntur, Andhra Pradesh, India. E-mail: [email protected]

Abstract

This paper considers an Type I Generalized Half Logistic Distribution. We discussed the scale (σ) and shape (θ) parameters estimation using the median ranks method (Benard‟s approximation). Rama Krishna(2008)1 studied the Type I Generalized Half Logistic Distribution scale (σ) and shape (θ) parameters estimation using the Least Square Method in two step estimation method. Also we computed Average Estimate (AE), (VAR), (STD), Square Error (MSE), Relative Absolute Bias (RAB), Relative Error (RE) for both the parameters under complete sample based on 10000 simulations to assess the performance of the .

Keywords: Type I Generalized Half Logistic Distribution, median ranks method(Benard‟s approximation), Least Square Method, Average Estimate (AE), Variance (VAR), Standard Deviation (STD), Mean Square Error (MSE), Relative Absolute Bias (RAB), Relative Error (RE).

Symbols: Estimated using Median Ranks Method-σ Estimated using Median Ranks Method-θ Estimated Scale parameter using Least Square Method- Estimated Shape parameter using Least Square Method-

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1. Introduction: In life testing and reliability studies, the of a phenomenon is generally categorizes as constant failure rate, increasing failure rate and decreasing failure rate. The last two categories are further called as monotone failure rates. A combination of monotone and constant failure rates over various segments of the of lifetime of the is also known as bath tub failure rate or non-monotone failure rate. Generally in biological and engineering sciences, we come across situations of non-monotone failure rates for handling the reliability analysis of the under consideration. In order to model such data, the generalization of of Stacy (1962)2, discrimination among some parametric models of Prentice (1975)3and analytical hazard representations for use in reliability, mortality and simulation studies of Gaver and Acar (1979)4 have been adopted. A reasonably comprehensive narration of the modles of the bath tub shaped failure rates was given in Rajarshi and Rajarshi (1988)5 but all these models often present some difficulties for statistical inferential procedures in the presence of . Balakrishnan (1985)6 considers half logistic probability models obtained as the distribution of the absolute value of standard logistic. Balakrishnan and Wong (1991)7 obtained approximate maximum likelihood estimate for location and scale parameters of half logistic distribution with type – II right sensoring. Madholkar et al (1995)8 presented an extension of the Weibull family that not only contains unimodel distribution with bath tub failure rates but also allows for a broader class of montone hazards rates and is computationally convenient for censored data. They named their extended version as “Exponentiated Weibull Family”. On similar lines Gupta and Kundu (2001b)9 proposed a new model called generalized . The half logistic distribution has not received much attention from researchers in terms of generalized. A generalized (type – II) version of logistic distribution was considered and some interesting properties of the distribution were derived by Balakrishnan and Hassain (2007)10. Rama Krishna (2008)1 studied the Type I Generalized Half Logistic Distribution scale (σ) and shape (θ) parameters estimation using the Least Square Method in two step estimation method. Torabi and Bagheri (2010)11 considers different parameter estimation methods in Extended Generalized Half Logistic Distribution for censored as well as complete sample. Kantam et al (2013)12 discussed the estimation and testing in type I Generalized Half Logistic Distribution. In the next section, we present the estimation of Scale (σ) and Shape (θ) parameters of type I Generalized Half Logistic Distribution using Median Ranks Method. Hence we called these estimators are Median Ranks Method estimators. In section-3, we discussed the estimation of Scale (σ) and Shape (θ) parameters of type I Generalized Half Logistic Distribution using Least Square Method proposed by Rama Krishna (2008)1. Hence we called these estimators are Least Square Method estimators. Section – 4 we made comparisons of Median Ranks Method estimators with Least Square Method estimators using Monte Carlo Simulation. In section – 5 presented observations and conclusions based on simulation results with numerical example. Half Logistic model obtained as the distribution of an absolute standard logistic variate is a probability model of recent origin (Balakrishnan, 1985)7. Its standard Estimation of Scale (σ) and Shape (θ) parameters of Type I Generalized 11 probability density function (pdf), cumulative distribution function (cdf) and hazard function (hdf) are given by 2e x f(x) , x 0 1.1 2 1 e x

x 1 e F(x) , x 0 1.2 x 1 e

1 h(x) , x 0 1.3 x 1 e The pdf, cdf and hazard function of the Type 1 Generalized Half Logistic Distribution are given by x x θ 1 2θe (1 e ) f(x) , x 0, θ 0 1.4 x θ 1 (1 e ) x θ 1 e F(x) , x 0, θ 0 1.5 x 1 e

x x θ 1 2θe (1 e ) h(x) , x 0, θ 0 1.6 x x θ x θ (1 e )[(1 e ) (1 e ) ] The pdf and cumulative distribution function of Type 1 Generalized Half Logistic Distribution with Scale parameter (σ) and shape parameter (θ) are given by x/σ x/σ θ 1 2θe (1 e ) f(x) , 0 x , σ 0, θ 0 1.7 x/σ θ 1 σ(1 e ) θ x/σ 1 e F(x) , 0 x , σ 0, θ 0 1.8 x/σ 1 e

2. Estimation of Scale(σ) and Shape(θ) parameters of Type 1 Generalized Half Logistic Distribution using Median Ranks Method. Let x1< x2< x3………..< xn. be an ordered sample of size „N‟ from Type I Generalized Half Logistic Distribution with the parameters Scale (σ) and Shape (θ). Then the cdf is given as in equcation (1.8) By neglecting the higher powers of e-x/σ can be written as

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x/σ x/σ 1 θe 1 θe 1 F(x) 2.1 x/σ 1 θe

x/σ 2 θ e 1 F(x) 2.2 x/σ 1 θe

1 F(x) θ e x/σ 2.3 1 F(x) 1 F(x) x σlogθ σlog 2.4 1 F(x) From the parameter estimation method (also known as ). Let us consider Aˆ 1 F(x) A σlogθ, θˆ exp , σˆ Bˆ and t log 2.5 1 F(x) σˆ Where F(X) is obtained by using the Median Ranks Method ( also called as Benard‟s Approximation) and is given by i 0.3 F(x) , i 1,2,3,...... n. 2.6 n 0.4 n n n n 2 t x ti xi ti i i Aˆ i 1 i 1 i 1 i 1 2.7 n n 2 n 2 ti ti i 1 i 1 n n n n t x i i ti xi Bˆ i 1 i 1 i 1 2.8 n n 2 2 n ti ti

i 1 i 1

Estimation of Scale (σ) and Shape (θ) parameters of Type I Generalized 13

3. Estimation of Scale (σ) and Shape (θ) parameters of Type I Generalized Half Logistic Distribution using Least Square Method. Let x1< x2< x3………..< xn. be an ordered sample of size „N‟ from Type I Generalized Half Logistic Distribution with the parameters Scale (σ) and Shape (θ) and from equations (2.1) and (2.2) can be written as 1 F(x) 1 3.1 1 F(x) x/σ θ e Taking Logerthams on bothsides, we have 1 F(x) x/σ 1 log log θ e 3.2 1 F(x)

1 F(x) x σ log θ σ log 3.3 1 F(x) From the least squares parameter estimation method (also known as regression analysis). Now let ~ ~ ~ A ~ 1 F(x) A σ log θ θ exp , σ~ B and t log 3.4 ~ 1 F(x) σ Where F(x) are obtained by the using the formula i F(x) , i 1,2,3,...... n. 3.5 n 1 n n n n 2 t x ti xi ti i i ~ A i 1 i 1 i 1 i 1 3.6 n n 2 n 2 ti ti i 1 i 1 n n n n t x i i ti xi ~ B i 1 i 1 i 1 3.7 n n 2 2 n ti ti

i 1 i 1

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4. Comparision of Median Ranks Method estimators with Least Square Method estimators. The Median Ranks Method estimates of Scale (σ) and Shape (θ) obtained from complete sample are compared with the Least Square Method estimates of Scale (σ) and Shape (θ) obtained from complete sample. Though the estimates from both the methods in closed form, it is very difficult to obtain the Average Estimate, Variance, Standard Deviation, Mean Square Error, Rrelative Absolute Bias and Relative Error of estimates mathematically, since the estimates are in non-liniar form. Hence we have resorted to Monte Carlo Simulation to compute the Average Estimate, Variance, Standard Deviation, Mean Square Error, Relative Absolute Bias and Relative Error of Median Ranks Method estimates as well as Least Square Method estimates, we simulate these values based on 10000 samples of size N = 10, 15,20, 25, 30, 35 generated from Generalized Half Logistic Distribution with scale parameter σ = 1 and shape parameter θ = 1. Here to asses the performance of Median Ranks Method estimates, we compare these with the Least Square Method estimates. Simulation study: In order to obtain the median ranks method estimators of Scale(σ) and Shape(θ) and study the properties of their estimates through the Average Estimate (AE), Variance (VAR), Standard Deviation (STD), Mean Square Error (MSE), Relative Absolute Bias (RAB) and Relative Error (RE) under In order to obtain the Median Ranks Method estimators of Scale (σ) and Shape (θ) and study the properties complete sample are given respectively by for given values of N, σ, θ. If ˆ , l 1,2 kl is Median Ranks Method estimate of l where l is a general , notation that can be replaced by σ and θ i.e 1 2 based on sapmle k,(k=1,2,3,……r) then The Average Estimate (AE), Variance (VAR), Standard Deviation (STD), Mean Square Error (MSE), Relative Absolute Bias (RAB) and Relative Error (RE) are given respectively by ^ 1 r ^ AverageEstimate( ) l kl r k 1 ^ 1 r 2 Variance( ) ( ˆ ˆ ) l kl kl r i 1 ^ 1 r 2 Standard Deviation( ) ( ˆ ˆ ) l kl kl r i 1 2 ^ 1 r ^ Mean Square Error l kl l r i 1 Estimation of Scale (σ) and Shape (θ) parameters of Type I Generalized 15

ˆ ^ 1 r kl l Relative Absolute Bias ( ) l r i 1 l 2 MSE( ˆ ) ^ 1 r l Relative Error l r i 1 l

5. Observations from Simulation results: 1. The Average Estimate (AE), Variance (VAR), Standard deviation ( STD), Mean Square Error (MSE), Relative Absolute Bias (RAB), Relative Error (RE) are independent of true values of the parameters of Scale (σ) and Shape (θ). 2. Average Estimate (AE) of Scale Parameter σ and Shape parameter θ by Median Ranks Method and Scale parameter and Shape parameter by Least Square Method were decreasing when sample size(N) is increasing. 3. Variance (VAR) of Scale parameter σ and Shape parameter θ by Median Ranks Method and Scale parameter and Shape parameter by Least Square Method were decreasing when sample size(N) is increasing. 4. Standard Deviation (STD) of Scale parameter σ and Shape parameter θ by Median Ranks Method and Scale parameter and Shape parameter by Least Square Method were decreasing when sample size(N) is increasing. 5. Mean Square Error (MSE) of Scale parameter σ and Shape parameter θ by Median Ranks Method and Scale parameter and Shape parameter by Least Square Method were decreasing when sample size(N) is increasing. 6. Relative Absolute Bias (RAB) of Scale parameter σ by Median Ranks Method and Scale parameter by Least Square Method were increasing when sample size (N) is increasing where as Relative absolute bias of Shape parameter θ by Median Ranks Method and Shape parameter by Least Square Method were decreasing when sample size (N) is increasing. 7. Relative Error (RE) of Scale parameter σ and Shape parameter θ by Median Ranks Method and Scale parameter and Shape parameter by Least Square Method were decreasing when sample size (N) is increasing. 8. The Average Estimate (AE) of scale parameter by Least Square Method is more nearer to the original value of scale parameter (σ) than to the Scale parameter σ by Median Ranks Method where as in shape parameter θ by Median Ranks Method is more nearer to the original value of Shape parameter (θ) than to the Shape parameter by Least Square Method. 9. Scale parameter σ by Median Ranks Method is having less variance (VAR) than to the scale parameter by Least Square Method where as in shape parameter by the Least Square Method is having less variance (VAR) than to the Shape Parameter θ by Median Ranks Method.

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10. Scale parameter σ by Median Ranks Method is having less Standard Deviation (STD) than to the scale parameter by Least Square Method where as in shape parameter by the Least Square Method is having less Standard Deviation (STD) than to the Scale parameter σ by Median Ranks Method. 11. when comparing to the Median Ranks Method estimators of Scale parameter σ and shape parameter θ, the Scale parameter and Shape parameter by Least Square Method are having less Mean Square Error. 12 when comparing to the Median Ranks Method estimators of Scale parameter σ and shape parameter θ, the Scale parameter and Shape parameter by Least Square Method are having less Relative Absolute Bias (RAB). 13. The Median Ranks Method estimates of Scale parameter σ and shape parameter θ are having less Relative Error (RE) than to the Scale parameter and Shape parameter by Least Square Method estimates.

Conclusion: The Average Estimate of Shape Parameter (θ) by Median Ranks Method are more accurate than to the Average Estimate of Shape Parameter ) by Least Square Method. If our interest is to find an efficient of for shape parameter (θ) Median Ranks Method be the best method.

Numerical example: We generated a random sample of size 10 with scale (σ) = 1 and shape parameter (θ)=3 by using the easy fit software 5.5 of GHLD distribution and the ordered sample is given by 0.27. 2.57, 2.68, 3.16, 3.42, 3.51, 3.86, 4.35, 4.58, 7.16 solving the equations (2.7), (2.8),(3.6) and (3.7) for this sample data and the resulting estimators of the Scale (σ) and Shape (θ) would be as follows: ˆ Scale parameter by Median Ranks Method = 1.6112 ˆ Shape parameter by median ranks method = 2.4390 ~ Scale parameter by Least Square Method = 1.7873 ~ Shape parameter by Least Square Method = 2.0386

Estimation of Scale (σ) and Shape (θ) parameters of Type I Generalized 17

Table – 1: The Average Estimate(AE), Variance(VAR), Standard deviation( STD), Mean Square Error(MSE), Relative Absolute Bias(RAB), Relative Error(RE) of Median ranks method estimators and least square method estimators of scale and shape parameters under complete sample of 10000 simulations. population parameters scale =1 and shape =1

Median ranks estimates Least square estimators size para AE VAR STD MSE RAB RE para AE VAR STD MSE RAB RE 10 σ 0.4282 0.0208 0.1438 0.3478 0.5721 0.6452 0.4753 0.0252 0.1582 0.3005 0.5256 0.6799 θ 1.0922 0.3161 0.5159 0.3238 0.2994 0.6452 0.9752 0.1687 0.3861 0.1688 0.2549 0.9734 15 σ 0.4263 0.0137 0.1165 0.3428 0.5737 0.6468 0.4631 0.0158 0.1253 0.304 0.5369 0.6743 θ 1.0246 0.1069 0.3201 0.1072 0.2192 0.6468 0.9439 0.0703 0.2604 0.0733 0.2013 0.9637 20 σ 0.4243 0.0102 0.1006 0.3416 0.5757 0.6468 0.4551 0.0115 0.1068 0.3084 0.5449 0.67 θ 1.0008 0.0679 0.2568 0.0678 0.1878 0.6468 0.9356 0.0481 0.2164 0.0522 0.1789 0.9614 25 σ 0.4218 0.008 0.0894 0.3424 0.5782 0.6458 0.4486 0.0089 0.0942 0.3129 0.5514 0.6661 θ 0.9903 0.0502 0.2223 0.0502 0.1683 0.6458 0.9341 0.0372 0.1916 0.0415 0.1636 0.9618 30 σ 0.4201 0.0068 0.082 0.343 0.5799 0.6451 0.4441 0.0074 0.0858 0.3164 0.5559 0.6633 θ 0.9846 0.0427 0.2047 0.043 0.1565 0.6451 0.9343 0.0326 0.1789 0.037 0.1545 0.9625 35 σ 0.4191 0.0057 0.0755 0.3431 0.5809 0.6448 0.4409 0.0062 0.0786 0.3189 0.5591 0.6613 θ 0.9762 0.034 0.183 0.0345 0.1445 0.6448 0.9311 0.0266 0.1619 0.0313 0.1449 0.9615 40 σ 0.4197 0.005 0.0707 0.3418 0.5803 0.6455 0.4396 0.0054 0.0734 0.3194 0.5604 0.6607 θ 0.9678 0.0277 0.1653 0.0288 0.1342 0.6455 0.9269 0.0221 0.1476 0.0274 0.1368 0.9598 45 σ 0.4177 0.0045 0.0668 0.3435 0.5823 0.6443 0.4363 0.0048 0.0691 0.3226 0.5637 0.6584 θ 0.9706 0.0257 0.1594 0.0265 0.1289 0.6443 0.9319 0.0208 0.1434 0.0254 0.1314 0.9626 50 σ 0.4172 0.004 0.0633 0.3436 0.5828 0.6441 0.4346 0.0043 0.0653 0.324 0.5654 0.6574 θ 0.9667 0.0229 0.1506 0.024 0.1233 0.6441 0.9306 0.0187 0.1362 0.0236 0.1271 0.9622

REFERENCES

[1]. Ramakrishna, V. 2008, “Generalizations to half logistic distribution and related inference”, PhD thesis, Acharya Nagarjuna University (AP), India. [2]. Stacy, E.W. 1962, “ A generalization of the gamma distribution”, Ann. Math. Statisti., Vol. 33, PP. 1187-1192. [3]. Prentice, R.L. 1975, “Discrimination among some parametric models”, Biometrika, Vol. 62, PP. 607-614. [4]. Gaver, D.P., and Acar, M. 1979, “Analytical hazard representations for use in reliability, mortality and simulation studies”, Commun. In statistics, part-B- Simulation and Computation, Vol. 8, 91-111. [5]. Rajarshi, S. and Rajarshi, M.B. 1988, “Bathtub distributions: A Review”, Commun. Statisti. – Theory & Methods, Vol. 17, PP. 2597-2621. [6]. Balakrishnan, N. 1985, “Order statistics from the half logistic distribution”, J. Statis. Comput. Simul. 20, 287-309. [7]. Balakrishnan, N. and Wong, K.M.T. 1991, “Approximate MLEs for the location and scale parameters of the half logistic distribution with type – II right censoring”, IEEE Trans. On Reliability, Vol. 40, 140-145.

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[8]. Mudholkar, G.S., Srivastava, D. and Freimer, M. 1995, “Exponentiated weibull family: A reanalysis of the bus-motor failure data”, Technometrics, Vol. 37, No. 4, PP. 436-445. [9]. Gupta, R.D. and Kundu, D. 2001b, “Generalized exponential distribution: Different methods of estimation”, Journal of Statistical Computation and Simulation, Vol. 69, PP. 315-338. [10]. Balakrishnan, N. and Hossain, A. 2007, “Inference for the type-II generalized logistic distribution under progressive type-II censoring”, Statistical Computation and Simulation, Vol. 77, PP. 1013-1031. [11]. Torabi, H. and Bagheri, F.L. 2010, “Estimation of parameters for an extended generalized half logistic distribution based on complete and censored data”, JIRSS, Vol. 9, No. 2, PP. 171-195. [12]. Kantam, R.R.L, Ramakrishna, V. and Ravikumar, M.S. 2013, “Estimation and testing in type 1 generalized half logistic distribution”, Modern applied statistical methods, Vol. 12, No. 1, PP. 198-206.

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