Journal of Probability and Statistics
Volume 2019 (2019), Article ID 8024769, 13 pages
https://doi.org/10.1155/2019/8024769

Research Article

On the Alpha Power Transformed Power Lindley Distribution

Amal S. Hassan,¹ M. Elgarhy,² Rokaya E. Mohamd,¹ and Sharifah Alrajhi³

¹Institute of Statistical Studies and Research, Cairo University, Egypt
²Vice Presidency for Graduate Studies and Scientific Research, University of Jeddah, Jeddah, Saudi Arabia
³Statistics Department, Faculty of Science, King AbdulAziz University, Jeddah, Saudi Arabia

Correspondence should be addressed to M. Elgarhy; m_elgarhy85@yahoo.com

Received 26 September 2018; Accepted 13 November 2018; Published 1 January 2019

Academic Editor: Ramón M. Rodríguez-Dagnino

Copyright © 2019 Amal S. Hassan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

In this paper, we introduce a new generalization of the power Lindley distribution referred to as the alpha power transformed power Lindley (APTPL). The APTPL model provides a better fit than the power Lindley distribution. It includes the alpha power transformed Lindley, power Lindley, Lindley, and gamma as special submodels. Various properties of the APTPL distribution including moments, incomplete moments, quantiles, entropy, and stochastic ordering are obtained. Maximum likelihood, maximum products of spacings, and ordinary and weighted least squares methods of estimation are utilized to obtain the estimators of the population parameters. Extensive numerical simulation is performed to examine and compare the performance of different estimates. Two important data sets are employed to show how the proposed model works in practice.

1. Introduction

The one-parameter Lindley distribution was primarily proposed by Lindley [1] as an alternative model for data with a nonmonotone hazard rate shape. Ghitany et al. [2] provided the properties of the Lindley distribution under a careful mathematical treatment. They also displayed in a numerical application that the Lindley distribution provides a better model than the exponential distribution. A random variable Y is said to follow the Lindley distribution with scale parameter θ > 0, if its probability density function (pdf) is given by

which is two-component mixture of exponential () and gamma (). The Lindley distribution has only increasing failure rate which has been identified as a major difficulty in lifetime analysis. To overcome this situation, many generalizations of the Lindley distribution have been introduced in literature. Nadarajah et al. [3] proposed a generalized Lindley distribution. Bakouch et al. [4] introduced the extended Lindley (EL) distribution. Shanker et al. [5] introduced the two-parameter Lindley distribution. Ghitany et al. [6] proposed the power Lindley (PL) as an extension of Lindley distribution. Based on pdf (1) and by using the transformation X = , the pdf of the PL distribution is given by

The cumulative distribution function (cdf) corresponding to (2) is given by

Many other generalizations of Lindley distribution have been proposed by several authors; see for example, a new generalized Lindley distribution [7], transmuted quasi Lindley distribution [8], exponentiated power Lindley [9], transmuted generalized Lindley distribution [10], transmuted generalized quasi Lindley distribution [11], and transmuted Kumaraswamy quasi Lindley distribution [12].

The alpha power transformation (APT) is one of the procedures that make the distributions richer and flexible to model the real life data. The APT has been proposed by Mahdavi and Kundu [13] with the parameter to incorporate skewness to the base distribution. The cdf of APT is specified by

Mahdavi and Kundu [13] applied the APT method to the exponential distribution and they studied various properties of the alpha power exponential distribution. Based on APT method, some new distributions are considered like APT Weibull [14], APT generalized exponential [15], APT Lindley [16], APT extended exponential [17], and APT inverse Lindley [18].

Following the similar idea of Mahdavi and Kundu [13], we introduce a new three-parameter distribution, the so-called APTPL distribution. The main purpose of the new model is that the additional parameter can give several desirable properties and more flexibility in the form of the hazard and density functions. The APTPL distribution contains a number of known lifetime submodels such as the APT Lindley (APTL) distribution, PL distribution, Lindley distribution, and gamma distribution. Properties, estimation of the parameters and applications with real data, are considered.

This paper is constructed as follows. We introduce and study the properties of APTPL in Sections 2 and 3, respectively. In Section 4, we obtain the maximum likelihood (ML) estimator, least square (LS) estimator, weighted least squares (WLS) estimator, and maximum product of spacing (MPS) estimator of APTPL distribution. In the same section, a simulation study is executed to investigate the effectiveness of the estimates. The analyses of two real data sets are employed in Section 5. The paper ends with concluding remarks.

2. The APTPL Model

Here, we now introduce the notion of APTPL distribution.

Definition 1. A random variable X is said to have a three-parameter APTPL distribution with the scale parameter θ>0 and shape parameters α, β >0, if its cdf is of the form

where is a set of parameters. The pdf of the APTPL distribution is given by

Also, the reliability function, say and hazard rate function (hrf), say of X are given, respectively, as follows:

and

A random variable X with distribution (6) is denoted by X ~ APTPL ( Submodels of the APTPL distribution are as follows:

(i)For β =1, the pdf (6) reduces to APTL [16].(ii)For =1, the pdf (6) reduces to PL [6].(iii)For β =1 and =1, the pdf (6) reduces to Lindley [1].(iv)For β =1 and =1, the pdf (6) reduces to gamma (2, θ).

Some descriptive pdf and hrf plots of X are illustrated Figures 1 and 2 for specific parameter choices of (see Figures 1 and 2).

Figure 1: The density plots of APTPL distribution.

Figure 2: The hrf plots of APTPL distribution.

We conclude from Figure 1 that the pdf of the APTPL distribution can be reversed J-shaped, uni-model, and right and left skewed for some values of . Also, we conclude that, for some values of , the hrf of APTPL distribution can be increasing, decreasing, and reversed J-shaped.

3. Statistical Properties

Here, some statistical properties of the APTPL distribution including quantile, moments, moment generating function, incomplete moments, Bonferroni and Lorenz curves, probability weighed moments, Rényi entropy, and stochastic ordering are derived.

3.1. Quantile Function

The quantile function, say Q(u) =F⁻¹(u), u∈ (0, 1), is obtained by inverting (5) as follows:

which yields

Multiply both sides by , then we have the Lambert equation

Hence, we have the negative Lambert W function of the real argument

i.e.,

where u∈ (0, 1) and W₋₁(.) is the negative Lambert W function. The first quantile (25%), median (50%), and third quantile (75%) for some choices values of the parameters are computed numerically (see Table 1).

Table 1: First quantile, median, and third quantile for some values of , , and .

25%
50%
75%
25%
50%
75%

1
0.5
0.045
0.735
6.662
0.394
0.912
1.766

2
0.279
3.175
19.709
0.682
1.414
2.446

5
0.852
6.908
33.388
0.953
1.786
2.865

7
1.225
8.780
39.174
1.063
1.919
3.005

10
1.743
11.040
45.625
1.181
2.055
3.146

1.5
0.5
0.007
0.134
1.339
0.228
0.547
1.091

2
0.049
0.617
4.151
0.403
0.865
1.533

5
0.156
1.390
7.183
0.572
1.104
2.330

7
0.228
1.786
8.481
0.641
1.190
1.899

10
0.329
2.268
9.935
0.717
1.279
1.991

2
0.5
0.002
0.040
0.428
0.157
0.381
0.775

2
0.014
0.193
1.366
0.279
0.610
1.098

5
0.047
0.444
2.398
0.399
0.784
1.2999

7
0.069
0.575
2.843
0.449
0.847
1.368

10
0.101
0.735
3.343
0.503
0.912
1.436

We detect form Table 1 that as the value of increases, for fixed values of and , the value of percentage points increases. As the value of increases, for fixed values of and , the percentage value points decrease. Also, as the value of increases, for fixed values of and , the percentage value points decrease.

3.2. Moments and Related Measures

The kth moment of a random variable X having the APTPL distribution is obtained as follows:

Since the power series can be written as

hence (14) can be expressed as follows:

Using the binomial expansion, then

Hence, the th moment of the APTPL distribution takes the form

where

Particularly, the mean is as follows:

Similarly, the central moment of a given random variable can be defined as

The coefficient of skewness (CS) and coefficient of kurtosis (CK) are defined by

Thus, numerical values of the , variance (), coefficient of variation (CV), CS, and CK of the APTPL distribution for some certain values of parameters are obtained and recorded in Table 2.

Table 2: Summary statistics of moments of the APTPL distribution.

1.344
0.713
0.366
0.391
0.371

22.379
1.994
0.369
0.393
0.342

874.97
7.675
0.418
0.427
0.333

6085
37.375
0.520
0.493
0.340

20.574
1.486
0.234
0.240
0.204

CV
3.376
1.709
1.322
1.252
1.219

CS
8.461
2.284
0.980
0.726
0.596

CK
133.188
9.423
2.728
2.011
1.704

Furthermore, the moment generating function of the APTPL distribution is given by

3.3. Incomplete Moments

The sth lower incomplete moment, say , of the APTPL distribution is given by

Using the power expansion (15) and binomial expansion, then (24) will be

So, the sth lower incomplete moment of the APTPL distribution is

where , is the lower incomplete moments. The first incomplete moment of the APTPL distribution can be obtained by setting s =1 in (26). The first incomplete moment is related to the Bonferroni and Lorenz curves, the mean residual, and mean waiting times. The Bonferroni and Lorenz curves are important in economics, reliability, demography, insurance, and medicine. The Lorenz curves, say , and Bonferroni curve, say , are defined by

For the proposed model, they can be determined from (27) as

and,

3.4. The Probability-Weighted Moments

The class of probability-weighted moments (PWMs), denoted by , for a random variable X, is defined as follows:

Substituting (5) and (6) in (30), then the PWM of the APTPL distribution is

Using the binomial expansion, then

Using (15) and binomial expansions, then (32) will be

Hence, the PWM of the APTPL distribution is given by

where

3.5. Rényi Entropy

The Rényi entropy of a random variable represents a measure of variation of the uncertainty. The Rényi entropy is defined by

Substituting (6) in (36), then we have

Using (15), then is converted to

Using binomial expansions, then we have

The above integral is complicated to obtain, so it will be solved numerically. Some of the numerical values of Rényi entropy for some certain values of parameters are given in Table 3.

Table 3: Rényi entropy for some values of , and .

1
0.5
1.116
0.8085

2
1.359
0.8855

5
1.479
0.8996

7
1.512
0.8989

10
1.5407
0.8990

1.5
0.5
0.832
0.623

2
1.090
0.8282

5
1.221
0.9179

7
1.258
0.9387

10
1.2901
0.9547

2
0.5
0.634
0.4637

2
0.899
0.6773

5
1.037
0.7738

7
1.076
0.7967

10
1.1104
0.8146

3.6. Stochastic Ordering

Here, we compare the APTPL₁ and the APTPL₂ ( with respect to stochastic ordering information. Let and be two random variables with cdfs, reliability functions and pdfs and and ; and and , respectively, where and . A random variable is said to be smaller than in the following ordering (see [19]), if the following holds:

(i)Stochastic order if for all .(ii)Likelihood ratio order if is decreasing in .(iii)Hazard rate order if is decreasing in .(iv)Mean residual life order if .

We have the following chain of implications among the various partial orderings discussed above:

Theorem 2. Let ~ APTPL₁ ( and X₂ ~ APTPL₂. If , , and , then , , , and .

Proof. It is sufficient to show is a decreasing function of ; the likelihood ratio is

therefore,

Hence, if , , and , then , which implies that is stochastically greater than with respect to likelihood ratio order. Similarly, we can conclude , , and .

4. Estimation

The population parameters of the APTPL distribution can be estimated using ML, MPS, LS, and WLS methods of estimation.

4.1. Maximum Likelihood Estimators

To obtain the ML estimators of the APTPL distribution with set of parameters , let X₁,…, Xn be observed values from this distribution. Hence, the log-likelihood function for the vector of parameters, say , can be written as

The ML equations of the APTPL distribution are obtained as follows:

and

Equating and with zeros and solving simultaneously, we obtain the ML estimators of α, , and θ.

4.2. Maximum Product of Spacings Estimators

The MPS method is a powerful alternative to ML method for estimating the population parameters of continuous distributions (see [20]). Let

be the uniform spacings of a random sample from the APTPL distribution, where

The MPS estimator is obtained by maximizing the geometric mean (GM) of the spacings

with respect to , , and . Equivalently, the MPS estimator of , , and can be obtained by maximizing the logarithm of the GM of sample spacings (48). There is no closed solution, so the numerical technique is applied to find the required estimates.

4.3. Least Squares and Weighted Least Squares Estimators

Suppose is a random sample of size n from APTPL distribution and suppose is the corresponding ordered sample. Then, the LS estimators can be obtained by minimizing the sum of squares errors,

with respect to the unknown parameters. So, the LS estimators of the population parameters of APTPL distribution are obtained by minimizing the following quantity

with respect to , and Also, the WLS estimators of APTPL distribution can be obtained by minimizing the following with respect to the population parameters , and

where

4.4. Numerical Illustration

As seen from the three previous subsections, that the ML, MPS, LS, and WLS estimators of the APTPL are very hard to obtain. Therefore, a simulation study is implemented to study the behavior of estimates, based on certain measures, which are mean square errors (MSEs) and absolute biases (ABs). 1000 random sample X₁,…Xn of sizes n = 10, 20, 30, and 100 are generated from the APTPL distribution. Exact values of parameters are considered as

and . The ML MPS, LS, and WLS estimates of and their measures are calculated. Numerical outcomes are listed in Table 4 and we conclude the following.

(i)As the sample sizes increase, the MSEs and ABs of the ML, MPS, LS, and WLS estimates of parameters decrease.(ii)Based on ML and MPS methods, for fixed values of , and as the value of increases, the MSEs and ABs increase for and estimates and decrease for estimates.(iii)Based on the ML method, as the values of and increase while the value of decreases, the MSEs and ABs for and estimates increase, whereas the values of increase for approximately sample size and different sets.(iv)The MSEs and ABs of the MPS estimates are smaller than the corresponding for the other estimators.(v)The MSEs and ABs of the WLS estimates for and estimates are smaller than the corresponding for LS estimates in set 1 at most sample sizes.(vi)Based on the LS method, as the values of and increase, while the value of decreases, the MSEs for and estimates decrease for all sample sizes and different set of parameters.(vii)Generally, the MPS estimates of all parameters have the smallest MSEs followed, respectively, by ML, LS, and WLS estimates. Also, the MSEs of LS estimates are better than the MSEs of WLS estimates for most sample sizes.

Table 4: MSEs and ABs of the APTPL distribution.

n
Methods
Properties
set1≡(α=1.5, θ=0.5, β=0.5)
set2≡(α=2, θ=0.5, β=0.8)
set3≡(α=0.5, θ=1.5, β=1.25)
set4≡(α=0.5, θ=1.5, β=1.75)

α
θ
β
α
θ
Β
α
θ
β
α
θ
β

10
ML
MSE
0.463
0.250
0.503
0.370
0.508
0.656
0.236
2.036
1.563
0.510
2.250
0.941

AB
0.547
0.500
0.704
0.580
0.707
0.800
0.486
1.427
1.250
0.645
1.500
0.948

MPS
MSE
0.230
0.160
0.250
0.202
0.247
0.640
0.118
0.573
1.546
0.250
0.845
0.940

AB
0.466
0.387
0.500
0.375
0.476
0.797
0.318
0.725
1.241
0.500
0.917
0.924

LS
MSE
2.250
1.398
0.546
0.402
0.868
0.659
1.195
2.480
2.548
2.095
2.681
1.391

AB
1.000
2.304
1.444
0.308
1.790
0.999
1.695
0.702
0.661
2.115
0.434
0.260

WLS
MSE
2.249
1.510
0.521
0.412
0.915
0.660
1.295
2.394
2.282
2.406
2.610
1.666

AB
1.000
2.425
1.413
0.312
1.855
1.000
1.806
0.757
0.673
2.315
0.534
0.254

20
ML
MSE
0.326
0.250
0.480
0.361
0.514
0.647
0.131
2.034
1.543
0.455
2.250
0.933

AB
0.510
0.500
0.690
0.591
0.712
0.797
0.362
1.426
1.248
0.629
1.500
0.932

MPS
MSE
0.213
0.147
0.250
0.163
0.239
0.637
0.114
0.556
1.562
0.250
0.710
0.740

AB
0.455
0.375
0.500
0.355
0.473
0.473
0.694
1.497
1.240
0.500
0.840
0.828

LS
MSE
2.249
1.345
0.523
0.384
0.887
0.658
0.971
2.205
1.013
1.515
2.532
1.235

AB
1.000
2.240
1.414
0.310
1.884
1.000
1.689
0.850
0.533
2.067
0.497
0.275

WLS
MSE
2.247
1.400
0.483
0.411
0.901
0.659
1.227
2.186
1.015
2.036
2.495
1.122

AB
1.000
2.336
1.358
0.314
1.722
1.000
1.937
0.928
0.586
2.416
0.754
0.105

30
ML
MSE
0.298
0.245
0.250
0.360
0.506
0.640
0.121
2.014
1.562
0.451
2.249
0.920

AB
0.512
0.499
0.693
0.589
0.707
0.798
0.348
1.419
0.284
0.632
1.500
0.949

MPS
MSE
0.212
0.128
0.244
0.157
0.235
0.636
0.079
0.345
1.542
0.250
0.696
0.721

AB
0.450
0.348
0.497
0.345
0.473
0.800
0.149
0.461
1.243
0.500
0.831
0.824

LS
MSE
2.249
1.258
0.490
0.370
0.789
0.649
0.899
2.133
0.990
1.225
2.444
1.148

AB
1.000
2.138
1.365
0.302
1.777
0.988
1.672
0.848
0.523
1.989
0.489
0.298

WLS
MSE
2.249
1.248
0.450
0.410
0.821
0.650
1.195
2.184
0.805
1.857
2.187
1.120

AB
1.000
2.219
1.332
0.312
1.712
0.997
2.011
0.968
0.586
0.020
0.886
2.459

100
ML
MSE
0.494
0.249
0.372
0.318
0.339
0.639
0.101
1.971
1.561
0.368
2.248
0.661

AB
0.449
0.499
0.601
0.482
0.567
0.799
0.318
1.404
1.241
0.597
1.499
0.811

MPS
MSE
0.214
0.125
0.243
0.156
0.233
0.634
0.028
0.036
1.541
0.245
0.589
0.591

AB
0.453
0.342
0.211
0.344
0.471
0.629
0.119
0.048
1.213
0.486
0.765
0.757

LS
MSE
2.248
1.147
1.147
0.368
0.769
0.640
0.774
2.116
0.892
1.130
2.067
0.664

AB
1.000
2.000
1.256
0.308
1.769
0.977
1.663
0.742
0.519
2.058
0.763
0.197

WLS
MSE
2.249
1.142
0.430
0.391
0.713
0.642
1.128
2.117
0.643
1.894
2.180
0.747

AB
1.000
2.381
1.351
0.307
1.231
0.642
2.002
0.953
0.518
2.665
0.977
0.734

5. Real Data Analysis

In this section, we analyze two real data sets to demonstrate the performance of the APTPL distribution in practice. One of the data sets is taken from Linhart and Zucchini [21, page 69]. The other data are taken from Aarset [22] representing the failure times of 50 devices. For the two data sets, we compare the results of the fits with the APTL, PL, EL, Lindley (L), and exponential (E) distributions. The pdfs of APTL and EL are given by

and

To test the superiority of the APTPL distribution, measures of goodness-of-fit can be applied in comparison to some other models. Mainly we use minus log-likelihood (-log L) Kolmogorov-Smirnov (KS) test statistic, Akaike information criterion (AIC), corrected Akaike information criterion (CAIC), Bayesian information criterion (BIC), and Hannan-Quinn information criterion (HQIC).

5.1. Linhart and Zucchini Data

The first data consists of a sample of 30 failure times of air-conditioned system of an airplane. The data are 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. The ML estimates, SEs of the model parameters, and analytical measures are provided in Table 5.

Table 5: Analytical results of the APTPL model and other competing models for the first data.

Model
ML Estimates (SE)
-Log L
AIC
BIC
CAIC
HQIC
KS

APTPL

176.725
359.45
357.881
360.373
360.795
0.13989

EL

217.635
441.27
439.701
442.193
442.615
0.86825

PL

195.999
395.999
394.953
396.443
396.895
0.4927

APTL

183.415
370.83
369.784
371.274
371.727
0.2803

L

276.715
555.429
554.906
555.572
555.877
0.9113

E

233.056
468.111
467.589
468.254
468.56
0.8929

According to the criteria mentioned in Table 5, we found that the APTPL model is the best fitted model compared to the other competitive models. The histogram of data and the estimated pdfs and cdfs for the fitted models are displayed in Figure 3. It is clear from Figure 3 that the APTPL model provides a better fit to this data.

Figure 3: Estimated pdf, cdf, and reliability function of APTPL model and other competing models for first data.

5.2. Aarset Data

The second data consists of 50 failure times of devices. The data are 0.1, 0.2, 1, 1, 1, 1, 1, 2, 3, 6, 7, 11, 12, 18, 18, 18, 18, 18, 21, 32, 36, 40, 45, 46, 47, 50, 55, 60, 63, 63, 67, 67, 67, 67, 72, 75, 79, 82, 82, 83, 84, 84, 84, 85, 85, 85, 85,85, 86, 86. The ML estimates, SEs of the model parameters, and analytical measures are provided in Table 6.

Table 6: Analytical results of the APTPL model and other competing models for the second data.

Model
ML Estimates (SE)
-LogL
AIC
BIC
CAIC
HQIC
KS

APTPL

282.257
570.515
569.612
571.037
572.699
0.17452

EL

348.441
702.883
701.98
703.405
705.067
0.86086

PL

305.67
615.34
614.738
615.595
616.796
0.3249

APTL

296.747
597.495
596.892
597.939
598.951
0.2186

L

440.025
882.05
881.749
882.133
882.778
0.9789

E

389.68
781.36
781.059
781.503
782.088
0.9645

As we see from Table 6 that the APTPL distribution gives better fit than the other models; therefore, it could be more adequate model for explaining the used data set. More information can be provided in Figure 4.

Figure 4: Estimated pdf, cdf, and survival function of the APTPL and other competing distributions for second data.

Also, from Figure 4, we conclude that the APTPL distribution provides better fits then we expect that the proposed model may be an interesting alternative model for a wider range of statistical research.

6. Concluding Remarks

In this paper a new three-parameter power Lindley distribution, called the APTPL distribution, based on alpha power transformation is proposed. The main objective behind this generalization is to give more flexibility for more data that can be analyzed using the proposed distribution. Various statistical properties are discussed such as moments, moment generating function, incomplete moments, and quantile function. The model parameters are estimated via the maximum likelihood, maximum product of spacing, and ordinary and weighted least squares methods. Simulation study is employed to examine the behavior of different estimates. The usefulness of the proposed model is illustrated motivated by two real data.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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