Submit manuscript...
eISSN: 2378-315X

Biometrics & Biostatistics International Journal

Research Article Volume 3 Issue 6

On modeling of lifetime data using akash, shanker, lindley and exponential distributions

Rama Shanker,1 Hagos Fesshaye2

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

Correspondence: Rama Shanker, Department of Statistics, Eritrea Institute of Technology, Asmara, Eritrea

Received: May 03, 2016 | Published: June 24, 2016

Citation: Shanker R, Fesshaye H. On modeling of lifetime data using akash, shanker, lindley and exponential distributions. Biom Biostat Int J. 2016;3(6):214-224. DOI: 10.15406/bbij.2016.03.00084

Download PDF

Abstract

The statistical analysis and modeling of lifetime data are crucial for statisticians and research workers in almost all applied sciences including engineering, biomedical science, insurance, and finance, amongst others. The two important and popular one parameter distributions for modeling lifetime data are exponential and Lindley distributions. Shanker et al.1 observed that there are many lifetime data where these distributions are not suitable from theoretical and applied point of view. Recently Shanker2,3 has introduced two one parameter Lifetime distribution namely “Akash distribution” and “Shanker distribution” for modeling lifetime data.

In the present paper the relationships and comparative studies of Akash, Shanker, Lindley and exponential distributions, their distributional properties and estimation of parameter have been discussed. The applications, goodness of fit and theoretical justifications of these distributions for modeling life time data through various examples from engineering, medical science and other fields have been discussed and explained.

Keywords: akash distribution, shanker distribution, lindley distribution, exponential distribution, statistical properties, estimation of parameter, goodness of fit

Introduction

In reliability analysis the time to the occurrence of event of interest is known as lifetime or survival time or failure time. The event may be failure of a piece of equipment, death of a person, development (or remission) of symptoms of disease, health code violation (or compliance). The modeling and statistical analysis of lifetime data are crucial for statisticians, research workers and policy makers in almost all applied sciences including engineering, medical science/biological science, insurance and finance, amongst others.

In statistics literature a number of lifetime distributions for modeling lifetime data-sets have been proposed. In this paper, the main objective is to have a critical and comparative study on one parameter lifetime distributions namely, Akash, Shanker, Lindley and exponential and their applications for modeling lifetime dats-sets from engineering, medical sciences, and other fields of knowledge.

Akash, shanker, lindley and exponential distributions

Akash distribution introduced by Shanker2 for modeling lifetime data from engineering and medical science is a two-component mixture of an exponential (θ)(θ) distribution and a gamma (3,θ)(3,θ) distribution with their mixing proportions θ2θ2+2θ2θ2+2  and 2θ2+22θ2+2 respectively. Shanker2 has discussed its various mathematical and statistical properties including its shape, moment generating function, moments, skewness, kurtosis, hazard rate function, mean residual life function, stochastic orderings, mean deviations, distribution of order statistics, Bonferroni and Lorenz curves, Renyi entropy measure, stress-strength reliability , amongst others. Shanker et al.3 has detailed study about modeling of various lifetime data from different fields using Akash, Lindley and exponential distributions and concluded that Akash distribution gives better fit in most of the lifetime data. Shanker5 has also obtained a Poisson mixture of Akash distribution named, “Poisson-Akash (PAD)” for modeling count data.

Shanker distribution introduced by Shanker2 for modeling lifetime data from engineering and medical science is a two- component mixture of an exponential (θ)(θ)  distribution and a gamma (2,θ)(2,θ) distribution with their mixing proportions θ2θ2+1θ2θ2+1  and 1θ2+11θ2+1 respectively. Shanker3 has discussed its various mathematical and statistical properties including its shape, moment generating function, moments, skewness, kurtosis, hazard rate function, mean residual life function, stochastic orderings, mean deviations, distribution of order statistics, Bonferroni and Lorenz curves, Renyi entropy measure, stress-strength reliability , amongst others. Shanker6 has also obtained a Poisson mixture of Shanker distribution named, “Poisson-Shanker (PSD)” for modeling count data.

Lindley7 distribution is a two-component mixture of an exponential (θ)(θ)  distribution and a gamma (2,θ)(2,θ) distribution with their mixing proportions θθ+1θθ+1  and 1θ+11θ+1 respectively. A detailed study about its various mathematical properties, estimation of parameter and application showing the superiority of Lindley distribution over exponential distribution for the waiting times before service of the bank customers has been done by Ghitany et al.8 A number of researchers have studied in detail the generalized, extended, mixtures and modified forms of Lindley distribution including Sankaran,9 Zakerzadeh & Dolat,10 Nadarajah et al.,11 Bakouch et al.,12 Shanker & Mishra,13,14 Shanker & Amanuel,15 Shanker et al.,16,17 Ghitany et al.,18 are some among others.

In statistical literature, exponential distribution was the first widely used lifetime model in areas ranging from studies on the lifetimes of manufactured to research involving survival or remission times in chronic diseases. The main reason for its wide usefulness and applicability as lifetime model is partly because of the availability of simple statistical methods for it and partly because it appeared to be suitable for representing the lifetimes of many things such as various types of manufactured items.

Let TT be a continuous random variable representing the lifetimes of individuals in some population The expressions for probability density function, f(t)f(t) , cumulative distribution function, F(t)F(t) ,  hazard rate function, h(t)h(t) , mean residual life function, m(t)m(t) , mean μ1 , variance μ2 , coefficient of variation (C.V.), coefficient of Skewness (β1) , coefficient of Kurtosis (β2) , and index of dispersion (γ)  of Akash and Shanker distributions introduced by Shanker2,3 are summarized in Table 1 and that of Lindley and exponential distributions are in Table 2.

A table of values for coefficient of variation (C.V.), coefficient of Skewness (β1) , coefficient of Kurtosis (β2) , and index of dispersion (γ)  for Akash , Shanker and Lindley distributions for varying values of their parameter are summarized in the Table 3.

The conditions under which Akash, Shanker and Lindley distributions are over-dispersed (μ<σ2) , equi-dispersed (μ=σ2)  , and under-dispersed (μ>σ2)  are summarized in Table 4.

The graphs of C.V, β1 , β2 and γ  of Akash, Shanker and Lindley distributions for varying values of the parameter θ  are shown in Figure 1.

Parameter estimation

Estimation of the parameter of akash distribution

Assuming (t1,t2,t3,...,tn)  be a random sample of size n  from Akash distribution, the maximum likelihood estimate (MLE) ˆθ  and the method moment estimate (MOME) ˜θ  of θ  is the solution of following cubic equation.

ˉtθ3θ2+2ˉtθ6=0 , where ˉt is the sample mean                                                       

Estimation of the parameter of shanker distribution                              

Let (t1,t2,t3,...,tn)  be a random sample of size n  from Shanker distribution. The maximum likelihood estimate (MLE) ˆθ  of θ  is the solution of the following non-linear equation.

2nθ(θ2+1)+ni=11θ+tinˉt=0                                         

The method of moment estimate (MOME) ˜θ  of θ  is the solution of the following cubic equation

ˉtθ3θ2+ˉtθ2=0 , where ˉt is the sample mean.                   

Estimation of the parameter of lindley distribution

Assuming  (t1,t2,....,tn)  be a random sample of size n  from Lindley distribution, the maximum likelihood estimate (MLE) ˆθ  and the method moment estimate (MOME) ˜θ  of θ  is given by

ˆθ=(ˉt1)+(ˉt1)2+8ˉt2ˉt;ˉt>0 , where ˉt is the sample mean.

Estimation of the parameter of exponential distribution

Assuming (t1,t2,....,tn)  be a random sample of size n from exponential distribution, the maximum likelihood estimate (MLE) ˆθ  and the method moment estimate (MOME) ˜θ  of θ  is is given by ˆθ=1ˉt , where ˉt is the sample mean.

Applications and goodness of fit

In this section the goodness of fit test of Akash, Shanker, Lindley and exponential distributions for following sixteen real lifetime data- sets using maximum likelihood estimate have been discussed.

In order to compare the goodness of fit of Akash, Shanker, Lindley and exponential distributions, 2lnL , AIC (Akaike Information Criterion), AICC (Akaike Information Criterion Corrected), BIC (Bayesian Information Criterion), K-S Statistics ( Kolmogorov-Smirnov Statistics)  for all sixteen real lifetime data- sets have been computed and presented in Table 5.  The formulae for computing AIC, AICC, BIC, and K-S Statistics are as follows:

AIC=2lnL+2k , AICC=AIC+2k(k+1)(nk1) , BIC=2lnL+klnn and D=Supx|Fn(x)F0(x)| , where k = the number of parameters, n = the sample size and Fn(x) is the empirical distribution function. The best distribution is the distribution which corresponds to lower values of 2lnL , AIC, AICC, BIC, and K-S statistics.

The best fitting has been shown by making -2ln L, AIC, AICC, BIC, and K-S Statistics in bold.

Conclusions

In this paper an attempt has been made to find the suitability of Akash, Shanker, Lindley and exponential distributions for modeling real lifetime data from engineering, medical science and other fields of knowledge. A table for values of the various characteristics of Akash, Shanker, and Lindley distributions has been presented for varying values of their parameter which reflects their nature and behavior. The conditions under which Akash, shanker, Lindley and exponential distributions are over-dispersed, equi-dispersed, and under-dispersed have been given. The goodness of fit test of Akash, Shanker, Lindley and exponential distributions for sixteen real lifetime data-sets have been presented using Kolmogorov-Smirnov test to test their suitability for modeling lifetime data.

Akash Distribution

Shanker Distribution

f(t)=θ3θ2+2(1+t2)eθt

f(t)=θ2θ2+1(θ+t)eθt

F(t)=1[1+θt(θt+2)θ2+2]eθt

F(t)=1[1+θtθ2+1]eθt

h(t)=θ3(1+t2)θt(θt+2)+(θ2+2)

h(t)=θ2(θ+t)(θ2+1)+θt

m(t)=θ2t2+4θt+(θ2+6)θ[θt(θt+2)+(θ2+2)]

m(t)=θ2+θt+2θ(θ2+θt+1)

μ1=θ2+6θ(θ2+2)

μ1=θ2+2θ(θ2+1)

μ2=θ4+16θ2+12θ2(θ2+2)2

μ2=θ4+4θ2+2θ2(θ2+1)2

C.V=σμ1=θ4+16θ2+12θ2+6

C.V=σμ1=θ4+4θ2+2θ2+2

β1=2(θ6+30θ4+36θ2+24)(θ4+16θ2+12)3/2

< β1=2(θ6+6θ4+6θ2+2)(θ4+4θ2+2)3/2

β2=3(3θ8+128θ6+408θ4+576θ2+240)(θ4+16θ2+12)2

β2=3(3θ8+24θ6+44θ4+32θ2+8)(θ4+4θ2+2)2

γ=σ2μ1=θ4+16θ2+12θ(θ2+2)(θ2+6)

γ=σ2μ1=θ4+4θ2+2θ(θ2+1)(θ2+2)

Table 1 Characteristics of Akash and Shanker Distributions

Lindley Distribution

Exponential Distribution

f(t)=θ2θ+1(1+t)eθt

f(t)=θeθt

F(t)=1θ+1+θtθ+1eθt

F(t)=1eθt

h(t)=θ2(1+t)θ+1+θt

h(t)=θ

m(t)=θ+2+θtθ(θ+1+θt)

m(t)=1θ

μ1=θ+2θ(θ+1)

μ1=1θ

μ2=θ2+4θ+2θ2(θ+1)2

μ2=1θ2

C.V=σμ1=θ2+4θ+2θ+2

C.V=σμ1=1

β1=2(θ3+6θ2+6θ+2)(θ2+4θ+2)3/2

β1=2

β2=3(3θ4+24θ3+44θ2+32θ+8)(θ2+4θ+2)2

β2=9

γ=σ2μ1=θ2+4θ+2θ(θ+1)(θ+2)

γ=σ2μ1=1θ

Table 2 Characteristics of Lindley and Exponential Distributions

Values of θ for Akash Distribution

0.01

0.05

0.1

0.3

0.5

1

1.5

2

μ1'

299.990

59.950

29.900

9.713

5.556

2.333

1.294

0.833

μ2

30001.000

1200.996

300.985

34.208

12.691

3.222

1.306

0.639

CV

0.577

0.578

0.580

0.602

0.641

0.769

0.883

0.959

β1

1.155

1.153

1.149

1.115

1.084

1.165

1.388

1.614

β2

5.000

4.997

4.987

4.897

4.785

4.834

5.473

6.391

γ

100.007

20.033

10.066

3.522

2.284

1.381

1.009

0.767

Values of θ  for Shanker Distribution

0.01

0.05

0.1

0.3

0.5

1

1.5

2

μ1'

199.990

39.950

19.901

6.391

3.600

1.500

0.872

0.600

μ2

20000.000

799.998

199.990

22.146

7.840

1.750

0.676

0.340

CV

0.707

0.708

0.711

0.736

0.778

0.882

0.943

0.972

β1

1.414

1.414

1.414

1.421

1.452

1.620

1.779

1.876

β2

6.000

6.000

6.000

6.020

6.121

6.796

7.593

8.159

γ

100.005

20.025

10.049

3.465

2.178

1.167

0.775

0.567

Values of θ  for Lindley Distribution

0.01

0.05

0.1

0.3

0.5

1

1.5

2

μ1'

199.010

39.048

19.091

5.897

3.333

1.500

0.933

0.667

μ2

19999.020

799.093

199.174

21.631

7.556

1.750

0.729

0.389

CV

0.711

0.724

0.739

0.789

0.825

0.882

0.915

0.935

β1

1.414

1.417

1.422

1.464

1.512

1.620

1.699

1.756

β2

6.000

6.007

6.025

6.162

6.343

6.796

7.173

7.469

γ

100.493

20.465

10.433

3.668

2.267

1.167

0.781

0.583

Table 3 Values of μ1' , μ2 , CV, β1 , β2 and γ of Akash, Shanker and Lindley distributions for varying values of the parameter θ

Distribution

Over-Dispersion
(μ<σ2)

Equi-Dispersion
(μ=σ2)

Under-Dispersion
(μ>σ2)

Akash

θ<1.515400063

θ=1.515400063

θ>1.515400063

Shanker

θ<1.171535555

θ=1.171535555

θ>1.171535555

Lindley

θ<1.170086487

θ=1.170086487

θ>1.170086487

Exponential

θ<1

θ=1

θ>1

Table 4 Over-dispersion, equi-dispersion and under-dispersion of Akash, Shanker , Lindley and exponential distributions for varying values of their parameter θ

Figure 1 Graphs of C.V, β1 , β2 and γ  of Akash, Shanker and Lindley distributions for varying values of the parameter θ .

Model

Parameter Estimate

-2ln L

AIC

AICC

BIC

K-S Statistic

Data 1

Akash

1.355445

163.73

165.73

165.79

169.93

0.355

Shanker

0.956264

162.28

164.28

164.34

166.42

0.346

Lindley

0.996116

162.56

164.56

164.62

166.70

0.371

Exponential

0.663647

177.66

179.66

179.73

181.80

0.402

Data 2

Akash

0.043876

950.97

952.97

953.01

955.58

0.184

Shanker

0.029252

980.97

982.97

983.01

985.57

0.238

Lindley

0.028859

983.11

985.11

985.15

987.71

0.242

Exponential

0.014635

1044.87

1046.87

1046.91

1049.48

0.357

Data 3

Akash

0.041510

227.06

229.06

229.25

230.20

0.107

Shanker

0.027675

231.06

233.06

233.25

234.19

0.145

Lindley

0.027321

231.47

233.47

233.66

234.61

0.149

Exponential

0.013845

242.87

244.87

245.06

246.01

0.263

Data 4

Akash

0.013514

1255.83

1257.83

1257.87

1260.43

0.110

Shanker

0.009009

1251.19

1253.34

1253.38

1255.60

0.097

Lindley

0.008970

1251.34

1253.34

1253.38

1255.95

0.098

Exponential

0.004505

1280.52

1282.52

1282.56

1285.12

0.190

Data 5

Akash

0.030045

794.70

796.70

796.76

798.98

0.184

Shanker

0.020031

788.57

790.57

790.63

792.28

0.133

Lindley

0.019841

789.04

791.04

791.10

793.32

0.134

Exponential

0.010018

806.88

808.88

808.94

811.16

0.198

Data 6

Akash

0.119610

981.28

983.28

983.31

986.18

0.393

Shanker

0.079746

1033.10

1035.10

1035.13

1037.99

0.442

Lindley

0.077247

1041.64

1043.64

1043.68

1046.54

0.448

Exponential

0.040060

1130.26

1132.26

1132.29

1135.16

0.525

Data 7

Akash

0.013263

803.96

805.96

806.02

810.01

0.298

Shanker

0.008843

764.62

766.62

766.69

768.06

0.246

Lindley

0.008804

763.75

765.75

765.82

767.81

0.245

Exponential

0.004421

744.87

746.87

746.94

748.93

0.166

Data 8

Akash

0.013423

609.93

611.93

612.02

613.71

0.280

Shanker

0.008949

579.51

581.51

581.60

583.29

0.220

Lindley

0.008910

579.16

581.16

581.26

582.95

0.219

Exponential

0.004475

564.02

566.02

566.11

567.80

0.145

Data 9

Akash

0.310500

887.89

889.89

889.92

892.74

0.198

Shanker

0.210732

847.37

849.37

849.40

852.22

0.132

Lindley

0.196045

839.06

841.06

841.09

843.91

0.116

Exponential

0.106773

828.68

830.68

830.72

833.54

0.077

Data 10

Akash

0.050293

354.88

356.88

357.02

358.28

0.421

Shanker

0.033569

325.74

327.74

327.88

329.14

0.351

Lindley

0.033021

323.27

325.27

325.42

326.67

0.345

Exponential

0.016779

305.26

307.26

307.40

308.66

0.213

Data 11

Akash

1.165719

115.15

117.15

117.28

118.68

0.156

Shanker

0.853374

112.91

114.91

115.03

116.44

0.131

Lindley

0.823821

112.61

114.61

114.73

116.13

0.133

Exponential

0.532081

110.91

112.91

113.03

114.43

0.089

Data 12

Akash

0.295277

641.93

643.93

643.95

646.51

0.100

Shanker

0.198317

635.26

637.26

637.30

639.86

0.042

Lindley

0.186571

638.07

640.07

640.12

642.68

0.058

Exponential

0.101245

658.04

660.04

660.08

662.65

0.163

Data 13

Akash

0.024734

194.30

196.30

196.61

197.01

0.456

Shanker

0.016492

181.58

183.58

183.89

184.29

0.388

Lindley

0.016360

181.34

183.34

183.65

184.05

0.386

Exponential

0.008246

173.94

175.94

176.25

176.65

0.277

Data 14

Akash

1.156923

59.52

61.52

61.74

62.51

0.320

Shanker

0.803867

59.78

61.78

61.22

62.77

0.325

Lindley

0.816118

60.50

62.50

62.72

63.49

0.341

Exponential

0.526316

65.67

67.67

67.90

68.67

0.389

Data 15

Akash

0.097062

240.68

242.68

242.82

244.11

0.266

Shanker

0.064712

252.35

254.35

254.49

255.78

0.326

Lindley

0.062988

253.99

255.99

256.13

257.42

0.333

Exponential

0.032455

274.53

276.53

276.67

277.96

0.426

Data 16

Akash

0.964726

224.28

226.28

226.34

228.51

0.348

Shanker

0.658029

233.01

235.01

235.06

237.24

0.355

Lindley

0.659000

238.38

240.38

240.44

242.61

0.390

Exponential

0.407941

261.74

263.74

263.80

265.97

0.434

Table 5 MLE’s, -2ln L, AIC, AICC, BIC, K-S Statistics of the fitted distributions of data-sets 1-16

0.55

0.93

1.25

1.36

1.49

1.52

1.58

1.61

1.64

1.68

1.73

1.81

2.00

0.74

1.04

1.27

1.39

1.49

1.53

1.59

1.61

1.66

1.68

1.76

1.82

2.01

0.77

1.11

1.28

1.42

1.50

1.54

1.60

1.62

1.66

1.69

1.76

1.84

2.24

0.81

1.13

1.29

1.48

1.50

1.55

1.61

1.62

1.66

1.70

1.77

1.84

0.84

1.24

1.30

1.48

1.51

1.55

1.61

1.63

1.67

1.70

1.78

1.89

Data Set 1 The data set represents the strength of 1.5cm glass fibers measured at the National Physical Laboratory, England. Unfortunately, the units of measurements are not given in the paper, and they are taken from Smith & Naylor19

5

25

31

32

34

35

38

39

39

40

42

43

43

43

44

44

47

48

48

49

49

49

51

54

55

55

55

56

56

56

58

59

59

59

59

59

63

63

64

64

65

65

65

66

66

66

66

66

67

67

67

68

69

69

69

69

71

71

72

73

73

73

74

74

76

76

77

77

77

77

77

77

79

79

80

81

83

83

84

86

86

87

90

91

92

92

92

92

93

94

97

98

98

99

101

103

105

109

136

147

Data Set 2 The data is given by Birnbaum & Saunders20 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 (x10-3 ) are presented below (after subtracting 65)

17.88

28.92

33.00

41.52

42.12

45.60

48.80

51.84

51.96

54.12

55.56

67.80

68.44

68.64

68.88

84.12

93.12

98.64

105.12

105.84

127.92

128.04

173.40

Data Set 3 The data set is from Lawless (1982, p-228). 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

86

146

251

653

98

249

400

292

131

169

175

176

76

264

15

364

195

262

88

264

157

220

42

321

180

198

38

20

61

121

282

224

149

180

325

250

196

90

229

166

38

337

65

151

341

40

40

135

597

246

211

180

93

315

353

571

124

279

81

186

497

182

423

185

229

400

338

290

398

71

246

185

188

568

55

55

61

244

20

284

393

396

203

829

239

236

286

194

277

143

198

264

105

203

124

137

135

350

193

188

Data Set 4 The data is from Picciotto 21 and arose in test on the cycle at which the Yarn failed. The data are the number of cycles until failure of the yarn

12

15

22

24

24

32

32

33

34

38

38

43

44

48

52

53

54

54

55

56

57

58

58

59

60

60

60

60

61

62

63

65

65

67

68

70

70

72

73

75

76

76

81

83

84

85

87

91

95

96

98

99

109

110

121

127

129

131

143

146

146

175

175

211

233

258

258

263

297

341

341

376

Data Set 5 This data represents the survival times (in days) of 72 guinna pigs infected with virulent tubercle bacilli, observed and reported by Bjerkedal22

19(16)

20(15)

21(14)

22(9)

23(12)

24(10)

25(6)

26(9)

27(8)

28(5)

29(6)

30(4)

31(3)

32(4)

33

34

35(4)

36(2)

37(2)

39

42

44

Data Set 6 This data is related with behavioral sciences, collected by Balakrishnan N et al.23 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

6.53

7

10.42

14.48

16.10

22.70

34

41.55

42

45.28

49.40

53.62

63

64

83

84

91

108

112

129

133

133

139

140

140

146

149

154

157

160

160

165

146

149

154

157

160

160

165

173

176

218

225

241

248

273

277

297

405

417

420

440

523

583

594

1101

1146

1417

Data Set 7 The data set reported by Efron24 represent the survival times of a group of patients suffering from Head and Neck cancer disease and treated using radiotherapy (RT)

12.20

23.56

23.7

25.9

31.98

37

41.35

47.38

55.46

58.36

63.47

68.46

78.3

74.5

81.43

84

92

94

110

112

119

127

130

133

140

146

155

159

173

179

194

195

209

249

281

319

339

432

469

519

633

725

817

1776

Data Set 8 The data set reported by Efron24 represent the survival times of a group of patients suffering from Head and Neck cancer disease and treated using a combination of radiotherapy and chemotherapy (RT+CT)

0.08

2.09

3.48

4.87

6.94

8.66

13.11

23.63

0.20

2.23

3.52

4.98

6.97

9.02

13.29

0.40

2.26

3.57

5.06

7.09

9.22

13.80

25.74

0.50

2.46

3.64

5.09

7.26

9.47

14.24

25.82

0.51

2.54

3.70

5.17

7.28

9.74

14.76

6.31

0.81

2.62

3.82

5.32

7.32

10.06

14.77

32.15

2.64

3.88

5.32

7.39

10.34

14.83

34.26

0.90

2.69

4.18

5.34

7.59

10.66

15.96

36.66

1.05

2.69

4.23

5.41

7.62

10.75

16.62

43.01

1.19

2.75

4.26

5.41

7.63

17.12

46.12

1.26

2.83

4.33

5.49

7.66

11.25

17.14

79.05

1.35

2.87

5.62

7.87

11.64

17.36

1.40

3.02

4.34

5.71

7.93

11.79

18.10

1.46

4.40

5.85

8.26

11.98

19.13

1.76

3.25

4.50

6.25

8.37

12.02

2.02

3.31

4.51

6.54

8.53

12.03

20.28

2.02

3.36

6.76

12.07

21.73

2.07

3.36

6.93

8.65

12.63

22.69

Data set 9 This data set represents remission times (in months) of a random sample of 128 bladder cancer patients reported in Lee & Wang25

23

261

87

7

120

14

62

47

225

71

246

21

42

20

5

12

120

11

3

14

71

11

14

11

16

90

1

16

52

95

Data Set 10 This data set is given by Linhart & Zucchini,26 which represents the failure times of the air conditioning system of an airplane

5.1

1.2

1.3

0.6

0.5

2.4

0.5

1.1

8

0.8

0.4

0.6

0.9

0.4

2

0.5

5.3

3.2

2.7

2.9

2.5

2.3

1

0.2

0.1

0.1

1.8

0.9

2

4

6.8

1.2

0.4

0.2

Data Set 11 This data set used by Bhaumik et al.,27 is vinyl chloride data obtained from clean up gradient monitoring wells in mg/l

0.8

0.8

1.3

1.5

1.8

1.9

1.9

2.1

2.6

2.7

2.9

3.1

3.2

3.3

3.5

3.6

4.0

4.1

4.2

4.2

4.3

4.3

4.4

4.4

4.6

4.7

4.7

4.8

4.9

4.9

5.0

5.3

5.5

5.7

5.7

6.1

6.2

6.2

6.2

6.3

6.7

6.9

7.1

7.1

7.1

7.1

7.4

7.6

7.7

8.0

8.2

8.6

8.6

8.6

8.8

8.8

8.9

8.9

9.5

9.6

9.7

9.8

10.7

10.9

11.0

11.0

11.1

11.2

11.2

11.5

11.9

12.4

12.5

12.9

13.0

13.1

13.3

13.6

13.7

13.9

14.1

15.4

15.4

17.3

17.3

18.1

18.2

18.4

18.9

19.0

19.9

20.6

21.3

21.4

21.9

23.0

27.0

31.6

33.1

38.5

Data set 12 This data set represents the waiting times (in minutes) before service of 100 Bank customers and examined and analyzed by Ghitany et al.8 for fitting the Lindley7 distribution.

74

57

48

29

502

12

70

21

29

386

59

27

153

26

326

Data Set 13 This data is for the times between successive failures of air conditioning equipment in a Boeing 720 airplane, Proschan28

1.1

1.4

1.3

1.7

1.9

1.8

1.6

2.2

1.7

2.7

4.1

1.8

1.5

1.2

1.4

3

1.7

2.3

1.6

2

Data Set 14 This data set represents the lifetime’s data relating to relief times (in minutes) of 20 patients receiving an analgesic and reported by Gross & Clark29

18.83

20.8

21.657

23.03

23.23

24.05

24.321

25.5

25.52

25.8

26.69

26.77

26.78

27.05

27.67

29.9

31.11

33.2

33.73

33.76

33.89

34.76

35.75

35.91

36.98

37.08

37.09

39.58

44.045

45.29

45.381

Data Set 15 This data set is the strength data of glass of the aircraft window reported by Fuller et al.30

1.312

1.314

1.479

1.552

1.700

1.803

1.861

1.865

1.944

1.958

1.966

1.997

2.006

2.021

2.027

2.055

2.063

2.098

2.140

2.179

2.224

2.240

2.253

2.270

2.272

2.274

2.301

2.301

2.359

2.382

2.382

2.426

2.434

2.435

2.478

2.490

2.511

2.514

2.535

2.554

2.566

2.570

2.586

2.629

2.633

2.642

2.648

2.684

2.697

2.726

2.770

2.773

2.800

2.809

2.818

2.821

2.848

2.880

2.954

3.012

3.067

3.084

3.090

3.096

3.128

3.233

3.433

3.585

3.858

Data Set 16 The following data represent the tensile strength, measured in GPa, of 69 carbon fibers tested under tension at gauge lengths of 20mm Bader & Priest.31,32

Acknowledgments

None.

Conflicts of interest

Author declares that there are no conflicts of interest.

References

  1. Shanker R, Hagos F, Sujatha S. On modeling of Lifetimes data using exponential and Lindley distributions. Biometrics & Biostatistics International Journal. 2015a ;2(5):1‒9.
  2. Shanker R. Akash distribution and Its Applications. International Journal of Probability and Statistics. 2015a;4(3):65‒75.
  3. Shanker R. Shanker distribution and Its Applications. International Journal of Statistics and Applications. 2015b;5(6):338‒348.
  4. Shanker R, Hagos F, Sujatha S. On modeling of Lifetimes data using one parameter Akash, Lindley and exponential distributions. Biometrics & Biostatistics International Journal. 2015b;3(2):1‒10.
  5. Shanker R. The discrete Poisson-Akash distribution. Communicated, 2016a.
  6. Shanker R. The discrete Poisson-Shanker distribution. Communicated, 2016b.
  7. Lindley DV. Fiducial distributions and Bayes’ Theorem. Journal of the Royal Statistical Society. 1958;20(1):102‒107.
  8. Ghitany ME, Atieh B, Nadarajah S. Lindley distribution and its Applications. Mathematics Computing and Simulation. 2008;78:493‒506.
  9. Sankaran M. The discrete Poisson-Lindley distribution. Biometrics. 1970;26(1):145‒149.
  10. Zakerzadeh H, Dolati A. Generalized Lindley distribution. Journal of Mathematical extension. 2009;3(2):13‒25.
  11. Nadarajah S, Bakouch HS, Tahmasbi R. A generalized Lindley distribution. The Indian Journal of Statistics. 2011;73(2):331‒359.
  12. Bakouch SH, Al-Zahrani BM, Al-Shomrani AA, et al. An extended Lindley distribution. Journal of Korean Statistical Society. 2012;41(1):75‒85.
  13. Shanker R, Mishra A. A quasi Lindley distribution. African journal of Mathematics and Computer Science Research. 2013a ;(4):64‒71.
  14. Shanker R, Mishra A. A two-parameter Lindley distribution. Statistics in transition new series. 2013b;14(1):45‒56.
  15. Shanker R, Amanuel AG. A new quasi Lindley distribution. International Journal of Statistics and systems. 2013;9(1):87‒94.
  16. Shanker R, Sharma S, Shanker R. A two-parameter Lindley distribution for modeling waiting and survival times data. Applied Mathematics. 2013;4:363‒368.
  17. Shanker R, Hagos F, Sharma S. On Two Parameter Lindley distribution and Its Applications to model lifetime data. Biometrics & Biostatistics International Journal. 2015c;3(1):1‒8.
  18. Ghitany M, Al-Mutairi D, Balakrishnan N, et al. Power Lindley distribution and associated inference. Computational Statistics and Data Analysis. 2013;64:20‒33.
  19. Smith RL, Naylor JC. A comparison of Maximum likelihood and Bayesian estimators for the three parameter Weibull distribution. Applied Statistics. 1987;36(3):358‒369.
  20. Birnbaum ZW, Saunders SC. Estimation for a family of life distributions with applications to fatigue. Journal of Applied Probability. 1969;6(2):328‒347.
  21. Picciotto R. Tensile fatigue characteristics of a sized polyester/viscose yarn and their effect on weaving performance, Master thesis, North Carolina State, University of Raleigh, USA, 1970.
  22. Bjerkedal T. Acquisition of resistance in guinea pigs infected with different doses of virulent tubercle bacilli. Am J Hyg. 1960;72(1):130‒148.
  23. 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.
  24. Efron B. Logistic regression, survival analysis and the Kaplan-Meier curve. Journal of the American Statistical Association. 1988;83(402):414‒425.
  25. Lee ET, Wang JW. Statistical methods for survival data analysis, 3rd edition, John Wiley and Sons, New York, USA, 2003.
  26. Linhart H, Zucchini W. Model Selection. John Wiley, USA: New York; 1986.
  27. Bhaumik DK, Kapur K, Gibbons RD. Testing Parameters of a Gamma Distribution for Small Samples. Technometrics. 2009;51(3): 326‒334.
  28. Proschan F. Theoretical explanation of observed decreasing failure rate. Technometrics. 1963;5(3):375‒383.
  29. Gross AJ, Clark VA. Survival Distributions: Reliability Applications in the Biometrical Sciences, John Wiley, USA: New York; 1975.
  30. Fuller EJ, Frieman S, Quinn J, Quinn G, Carter W (1994) Fracture mechanics approach to the design of glass aircraft windows: A case study. SPIE Proc 2286, 419‒430.
  31. Lawless JF. Statistical models and methods for lifetime data, John Wiley and Sons, New York, USA.
  32. Bader MG, Priest AM (1982) Statistical aspects of fiber and bundle strength in hybrid composites. In: Hayashi T, editor, Progress in Science in Engineering Composites. ICCM-IV, Tokyo, 1982. p. 1129‒1136.
Creative Commons Attribution License

©2016 Shanker, et al. This is an open access article distributed under the terms of the, which permits unrestricted use, distribution, and build upon your work non-commercially.