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Research article

On Weighted Lindley Distribution and its Applications to Model Lifetime Data

Rama Shanker1*, Kamlesh Kumar Shukla1, Hagos Fesshaye2

1Department of Statistics, Eritrea Institute of Technology, Asmara, Eritrea
2Department of Economics, College of Business and Economics, Halhale, Eritrea

*Corresponding author: Dr. Rama Shanker, Department of Statistics, Eritrea Institute of Technology, Asmara, Eritrea, Email: shankerrama2009@gmail.com

Submitted: 07-21-2016 Accepted: 08-18-2016 Published:  09-01-2016

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 {tab=Article}

Abstract

In the present paper, the expressions for coefficient of variation, skewness, kurtosis and index of dispersion of weighted Lindley distribution (WLD), of which Lindley distribution is a particular case, have been derived. The nature and behavior of coefficient of variation, skewness, kurtosis, and index of dispersion of WLD have also been discussed for varying values of its parameters. The stochastic ordering of the distribution has been studied. The applications and goodness of fit of the distribution have been discussed with several lifetime data sets and the fit has been compared with one parameter Lindley and exponential distributions.

Keywords: Weighted Lindley distribution; Coefficient of variation; Skewness; Kurtosis; Index of dispersion; Stochastic ordering;
Goodness of fit

Introduction

The probability density function (p.d.f.) of weighted Lindley distribution (WLD), introduced by Ghitany et al. [1] with parameters
α and ϴ , is given by

biostat frm 2.1

where

is the complete gamma function. The corresponding cumulative distribution function (c.d.f.) of WLD (1.1) is given by

biostat_frm_2.2

where

is the upper incomplete gamma function.

It can be easily shown that at α=1 WLD reduces to Lindley [2, 3] distribution having p.d.f.

biostat_frm_2.3

It can be easily verified that the p.d.f. (1.3) is a two – component mixture of exponential (ϴ) and gamma (2, ϴ)distributions. Sankaran [4] has obtained discrete Poisson-Lindley distribution by mixing Poisson-distribution with Lindley distribution and studied its properties, estimation of parameter and applications to model count data. Shanker et al. [5] have discussed comparative study of Lindley and exponential distributions  or modeling lifetime data sets from biomedical science and engineering. Further, p.d.f. (1.1) can also be expressed as a two-component mixture of gamma (α, ϴ) and gamma (α+1, ϴ) distributions. We have

biostat_frm_2.4
where

Ghitany et al. [1] have discussed the structural properties of WLD including nature of its p.d.f., hazard rate function, mean residual life function and applications for survival data using maximum likelihood estimation. It seems that some of its structural properties including central moments, coefficient of variation, skewness, kurtosis, index of dispersion, and stochastic ordering have not been studied. It also seems that not much works have been done on the applications of WLD for modeling lifetime data.

In the present paper, firstly central moments of WLD have been obtained and the expression for coefficient of variation, skewness, kurtosis, and index of dispersion has been given. The nature and behavior of coefficient of variation, skewness, kurtosis, and index of dispersion have been discussed using tables for varying values of its parameters. The stochastic ordering of the distribution has been explained. Finally, the applications and goodness of fit of the WLD have been discussed for several lifetime data using maximum likelihood estimate and the fit has been compared with one parameter Lindley and exponential distributions.

Moments and Associated Measures

Using mixture representation (1.4), the th moment about origin of WLD (1.1) can be obtained as

biostat_frm_2.5
The expressions for coefficient of variation (C.V.) coefficient of skewness , (√β1) coefficient of kurtosis (β2) , and index of dispersion (γ)of WLD are thus obtained as

biostat_frm_2.6

A table for studying the nature of coefficient of variation (C.V.), coefficient of skewness (√β1), coefficient of kurtosis (β2) and index of dispersion (γ) of WLD have been prepared for varying values of the parameters ϴ and α and presented in tables 1, 2, 3 and 4.

biostat table 2.1

For a given α value, the coefficient of variation decreases as θ increases, that is, as θ increases the C.V decreases. Similarly, for a given θ, the coefficient of variation decreases as α increases. 

biostat_table_2.2

Table 2. (√β1) of WLD for varying values of θ and α.

For a given α value, the coefficient of skewness increases as θ increases. On the other hand, for a given θ, the coefficient of skewness decreases as α increases.

biostat table 2.3

 For a given α, the coefficient of kurtosis decreases as θ increases. And, for a given θ the coefficient of kurtosis decreases as α increases.

biostat_table_2.4

For a given α the index of dispersion decreases as θ increases. Similarly, for a given θ the index of dispersion decreases as increases.

Stochastic Ordering

Stochastic ordering of positive continuous random variables is an important tool for judging their comparative behavior. A random variable X is said to be smaller than a random variable in the

biostat eqtn 2.1

 biostat thrm 2.1

Applications and Goodness of Fit

In this section, the applications and goodness of fit of the weighted Lindley distribution (WLD) has been discussed for several lifetime data and the fit is compared with Lindley and exponential distributions. In order to compare WLD, Lindley and exponential distributions,—2ln L , AIC (Akaike Information Criterion), K-S Statistics (Kolmogorov-Smirnov Statistics) and p-value for thirteen data sets data-sets have been computed and presented in table 5. The formulae for computing AIC and K-S Statistics are as follows:

biostat_frm_2.7

where k = the number of parameters is, n = the sample size and Fn(x) is the empirical distribution function. The best distribution corresponds to lower values of —2ln L, AIC, and K-S statistics and higher p- value.

The fitting of WLD, Lindley and exponential distributions are based on maximum likelihood estimates (MLE). The data sets 1 to 13 for which WLD, Lindley and exponential distributions have been fitted are given in the Appendix.

Concluding Remarks

In this paper firstly a general expression for r th moment about origin has been obtained and thus first four moments about origin and moments about mean have been given. The expressions for coefficient of variation, skewness, kurtosis and index of dispersion of weighted Lindley distribution, of which Lindley distribution is a particular case, have been obtained and their nature and behavior have been discussed by preparing a table for varying values of its parameters. The stochastic ordering of the distribution has been studied. The applications and goodness of fit of the distribution have been discussed with several lifetime data sets and the fit has been compared with one parameter Lindley and exponential distributions. The fitting of WLD shows that in almost all data sets, it gives better fit than one parameter Lindley and exponential distributions and hence it can be considered as an important distribution for modeling lifetime data over Lindley and exponential distributions.

biostat_table_2.5

biostat table 2.6

Table 5. MLE’s, -2ln L, AIC, AICC, BIC, K-S Statistics of the fitted distributions of data sets 1-13.

Acknowledgments

The authors thank the editor and the reviewer for their comments which led to improvement in the quality of the paper.

Appendix

Data Set 1. The data is given by Birnbaum and Saunders [7] on the fatigue life of 6061 – T6 aluminum coupons cut parallel to the direction of rolling and oscillated at 18 cycles per second. The data set consists of 101 observations with maximum stress per cycle 31,000 psi. The data ( x 10-3 ) are presented below (after subtracting 65). 

biostat data 2.1

Data Set 2. The data set is from Lawless [8]. The data given arose in tests on endurance of deep groove ball bearings. The data are the number of million revolutions before failure for each of the 23 ball bearings in the life tests and they are:

biostat_data_2.2

Data Set 3. The data is from Picciotto [9] and arose in test on the cycle at which the Yarn failed. The data are the number of cycles until failure of the yarn and they are: 

biostat_data_2.3

Data Set 4. This data represents the survival times (in days) of 72 guinea pigs infected with virulent tubercle bacilli, observed
and reported by Bjerkedal [10]. 

biostat_data_2.4

Data Set 5. This data is related with behavioral sciences, collected by Balakrishnan N et al [11] The scale “General Rating of
Affective Symptoms for Preschoolers (GRASP)” measures behavioral and emotional problems of children, which can be classified
with depressive condition or not according to this scale. A study conducted by the authors in a city located at the south part of Chile has allowed collecting real data corresponding to the scores of the GRASP scale of children with frequency in parenthesis, which are:

biostat_data_2.5

Data Set 6. The data set reported by Efron [12] represent the survival times of a group of patients suffering from Head and
Neck cancer disease and treated using radiotherapy (RT).

biostat_data_2.6

Data Set 7. This data set is given by Linhart and Zucchini [13] which represents the failure times of the air conditioning system
of an airplane.

biostat_data_2.7

Data Set 8. This data set used by Bhaumik et al. [14] is vinyl chloride data obtained from clean upgradient monitoring wells in
mg/l.

biostat_data_2.8

Data Set 9. This data set represents the waiting times (in minutes) before service of 100 Bank customers and examined and
analyzed by Ghitany et al [3] for fitting the Lindley [2] distribution.

biostat_data_2.9

Data Set 10. This data is for the times between successive failures of air conditioning equipment in a Boeing 720 airplane,
Proschan [15].

biostat_data_2.10

Data Set 11. This data set represents the lifetime’s data relating to relief times (in minutes) of 20 patients receiving an analgesic
and reported by Gross and Clark [16].

biostat_data_2.11

Data Set 12. This data set is the strength data of glass of the aircraft window reported by Fuller et al [17].

biostat_data_2.12

Data Set 13. The following data represent the tensile strength, measured in GPa, of 69 carbon fibers tested under tension at
gauge lengths of 20mm, Bader and Priest [18].

biostat_data_2.13

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2. Lindley DV. Fiducial distributions and Bayes’ Theorem. Journal of the Royal Statistical Society, Series B. 1958, 20(1): 102–107.

3. Ghitany ME, Atieh B, Nadarajah S. Lindley distribution and its Applications. Mathematics and Computers in Simulation. 2008, 78(4): 493–506.

4. Sankaran M. The discrete Poisson-Lindley distribution. Biometrics. 1970, 26(1), 145–149.

5. Shanker R, Hagos F, Sujatha S. On modeling of Lifetimes data using exponential and Lindley distributions. Biometrics &  Biostatistics International Journal. 2015, 2(5): 1-9.

6. Shaked M, Shanthikumar JG. Stochastic Orders and Their Applications. Academic Press, New York, 1994.

7. Birnbaum ZW, Saunders SC. Estimation for a family of life distributions with applications to fatigue. Journal of Applied Probability. 1969, 6(2), 328-347.

8.Lawless JF. Statistical models and methods for lifetime data. John Wiley & Sons. NewYork, 1982.

9. Picciotto R. Tensile fatigue characteristics of a sized polyester/ viscose yarn and their effect on weaving performance, Master thesis, North Carolina State, University of R a l e i g h , 1970.

10. Bjerkedal T. Acquisition of resistance in guinea pigs infected with different doses of virulent tubercle bacilli. Am J Hyg. 1960, 72: 130-148.

11. Balakrishnan N, Victor L, Antonio S. A mixture model based on Birnhaum-Saunders Distributions, A study conducted by Authors regarding the Scores of the GRASP (General Rating of Affective Symptoms for Preschoolers), in a city located at South Part of the Chile, 2010.

12. Efron B. Logistic regression, survival analysis and the Kaplan- Meier curve. Journal of the American Statistical Association. 1988, 83(402), 414-425.

13. Linhart H, Zucchini W. Model Selection, John Wiley, NewYork , 1986

14. Bhaumik DK, Kapur K, Gibbons RD. Testing Parameters of a Gamma Distribution for Small Samples. Technometrics. 2009, 51(3), 326-334.

15. Proschan F. Theoretical explanation of observed decreasing failure rate. Technometrics. 1963, 5(3), 375-383.

16. Gross AJ, Clark VA. Survival Distributions: Reliability Applications in the Biometrical Sciences, John Wiley, New York, 1975.

17. Fuller EJ, Frieman S, Quinn J, Quinn G, Carter W. Fracture mechanics approach to the design of glass aircraft windows: A case study. SPIE Proc. 1994, 2286: 419-430.

18. Bader MG, Priest AM. Statistical aspects of fiber and bundlestrength in hybrid composites, In; hayashi, T., Kawata, K. Umekawa, S. (Eds), Progress in Science in Engineering Composites, ICCM-IV, 1982. Tokyo, 1129 – 1136.

{/tabs}

Cite this article:  Rama Shanker. On Weighted Lindley Distribution and its Applications to Model Lifetime Data. J J Biostat. 2016, 1(1): 002

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