Several researchers are interested in establishing new generalized classes of univariate continuous distributions by modifying a baseline distribution with one or more parameters to generate new distributions. These extended distributions provide greater flexibility in certain applications to real data. Various methods of generating new statistical distributions were presented in the literature. [For more details see Marshall and Olkin (1997), Eugene et al. (2002), Cordeiro and Castro (2011), Alzaatreh et al. (2013), Lee et al. (2013), and Jones (2015)].
Sangsanit and Bodhisuwan (2016) introduced the ToppLeone family of distributions. The cumulative distribution function (cdf) and probability density function (pdf) of the ToppLeone family are given, respectively, by
(1)
and
(2)
Several ToppLeone families of distributions can be generated by specifying the cdf, . Many authors studied the ToppLeone family, for example the ToppLeone BurrXII distribution by Reyad and Othman (2017) and ToppLeone exponentiated power Lindley by Aryuyuen (2018). Also, Reyad et al. (2019) presented the ToppLeone generalized inverted Kumaraswamy distribution.
Mahdavi and Kundu (2016) proposed a method for introducing new lifetime distributions named as alpha power transformation (APT). If and are the cdf and pdf of a random variable X, then the cdf of APT is given by
(3)
and the corresponding pdf as
(4)
where α is the shape parameter.
The rest of the paper is organized as follows: In Section 2, ToppLeone Alpha power (TLAP) family is presented. Useful expansions of TLAP family are derived in Section 3. In Section 4, general expressions for some statistical properties of the general family are obtained. The ToppLeone alpha power Weibull (TLAW) distribution is introduced in Section 5. In Section 6, a simulation study is conducted to evaluate the performance of the estimators. Also, a real data set is analyzed in Section 7. Finally, the conclusion is given in Section 8.
2. The ToppLeone Alpha Power Family
Substituting the cdf of the APT family defined in (3) as the baseline cdf in (1). Then, the cdf of the TLAP family is
(5)
where
The pdf can be written as
(6)
The survival function (sf), , hazard rate function (hrf), , and reversed hazard rate function (rhrf), are, respectively, given by
(7)
(8)
and
(9)
3. Expansions of the ToppLeone Alpha Power Family
A useful linear representation for the pdf and cdf of the TLAP family are obtained using the binomial expansion and power series representation which are given below
(10)
Thus, the cdf in (5) can be written as
(11)
where
and
Hence, the pdf of the TLAP family can be written as follows:
(12)
where
and
4. Some Statistical Properties
In this section, some statistical properties of the TLAP family including quantile function, moments and moment generating function, mean residual life, and order statistics are presented.
4.1 The quantile function
The quantile function of TLAP random variable X is given by
(13)
4.2 Moments and moment generating function
Let , then using (12), the r^{th} moment of X is obtained as follows:
(14)
where
The moment generating function is
where
Then,
(15)
4.3 Residual life and reversed residual life
The residual lifetime function of TLAP random variable X is given by
(16)
The reversed residual lifetime function of TLAP random variable X is
(17)
4.4 Order statistics
Considering that is a random sample of size n from TLAP family and is the corresponding order statistics. Then, the pdf of the r^{th} order statistic is given by
(18)
where denotes the beta function.
The pdf of the smallest order statistics and the largest order statistics are given below
(19)
and
(20)
5. The ToppLeone Alpha Power Weibull Distribution
In this section, the TLAP family is specialized to the Weibull distribution. The cdf and pdf of the Weibull distribution are given respectively by
(21)
and
(22)
By substituting the cdf and pdf of the Weibull distribution, respectively, in (5) and (6). Hence, one can obtain the cdf and the pdf of the TLAW distribution as follows:
(23)
and
(24)
where
The sf and hrf are, respectively, given by
(25)
and
(26)
Some submodels of TLAW distribution (24) are given in Table 1 and the plots for the pdf and hrf of the TLAW distribution are displayed in Figure 1.
Table 1: Submodels of TLAW distribution
θ

α

λ

β

cdf

Submodel

ــــ

ــــ

ــــ

1


ToppLeone alpha power one parameter Weibull distribution

ــــ

ــــ

1

ــــ


ToppLeone alpha power exponential distribution
[See Rady et al. (2021)]

ــــ

1

ــــ

ــــ


ToppLeone Weibull distribution
[See Tuoyo et al. (2021)]

ــــ

ــــ

2

ــــ


ToppLeone alpha power Rayleigh distribution

ــــ

1

2

ــــ


ToppLeone Rayleigh distribution
[See Olayode (2019)]

ــــ

1

1

ــــ


ToppLeone exponential distribution
[See AlShomrani et al. (2016)]

Figure 1: Different plots of pdf and hrf of the TLAW distribution
The plots of Figure 1 indicate that the TLAW density can be decreasing, unimodal, right skewed and left skewed. Also, the plots of the hrf can be decreasing, upsidedown bathtub, increasing and J shaped. Therefore, the TLAW distribution is flexible for analyzing lifetime data.
5.1 Some statistical properties for ToppLeone alpha power Weibull distribution
In this subsection, some basic properties of the TLAW distribution are derived.
5.1.1 The quantile function
The quantile function of TLAW random variable X is given by using (13) or (23), then
(27)
Then, a random sample from TLAW distribution can be generated using (27) where is random sample from uniform.
5.1.2 Moments and moment generating function
If , then using (15) the r^{th} moment of is
(28)
The moment generating function is obtained by using (15) as follows:
(29)
5.1.3 Residual life and reversed residual life
The residual lifetime function of TLAW random variable X is given by using (16)
(30)
The reversed residual lifetime function of TLAW random variable X is given by using (17)
(31)
5.1.4 Order statistics
From (18), the pdf of the r^{th} order statistic of the TLAW distribution is given by
(32)
where denotes the beta function.
5.2 Maximum likelihood estimation
Let be a simple random sampling of size n from the TLAW distribution, then, from the pdf in (24), the likelihood function is
(33)
where .
The log likelihood function is
(34)
The maximum likelihood (ML) estimators of can be obtained by differentiating (34) with respect to then, equating the results to zeros as follows:
(35)
(36)
(37)
and
(38)
It is to be noted that the likelihood equations cannot be solved explicitly, so, the ML estimates of can be obtained numerically.
6. Simulation Study
In this section, a simulation study is carried out using Mathematica 9 to evaluate the performance of the ML estimates based on generated data from the TLAW distribution. Evaluating the performance of the estimates is considered through the relative absolute bias (RAB) and estimated risk (ER), where
(39)
and
(40)
The ML averages of the estimates, RAB, and ER for the TLAW distribution are displayed in Tables 2 and 3. The population parameter values used in this simulation study are (λ=4, β=1.5, α=2.5, θ=1.5) and (λ=3, β=2, α=3.5, θ=3), where the number of replications is N=1000 and the samples of size, n=30, 50, 100, and 150.
Table 2: ML averages, relative absolute biases, estimated risks and 95% confidence intervals of the parameters for TLAW distribution
(N=1000, λ=4, β=1.5, α=2.5, θ=1.5)
n

Parameter

Average

RAB

ER

LI

UI

Length

30

λ

2.0769

0.4808

3.6987

2.0385

2.1153

0.0768

β

1.5846

0.0563

0.0076

1.5427

1.6265

0.0839

α

2.7937

0.1175

0.0878

2.7152

2.8721

0.1568

θ

0.7532

0.4978

0.5577

0.7360

0.7705

0.0345









50

λ

2.0786

0.4804

3.6920

2.0512

2.1060

0.0548

β

1.5840

0.0560

0.0073

1.5515

1.6164

0.0649

α

2.7886

0.1154

0.0843

2.7265

2.8508

0.1243

θ

0.7535

0.4976

0.5571

0.7407

0.7665

0.0258









100

λ

2.0788

0.4802

3.6907

2.0614

2.0964

0.0350

β

1.5830

0.0553

0.0070

1.5595

1.6064

0.0469

α

2.7850

0.1140

0.0817

2.7421

2.8280

0.0859

θ

0.7539

0.4973

0.5566

0.7451

0.7628

0.0177









150

λ

2.0798

0.4800

3.6871

2.0648

2.0948

0.0300

β

1.5833

0.0555

0.0070

1.5640

1.6027

0.0386

α

2.7850

0.1140

0.0815

2.7502

2.8198

0.0696

θ

0.7539

0.4973

0.5566

0.7462

0.7616

0.0154

Table 3: ML averages, relative absolute biases, estimated risks, and 95% confidence intervals of the parameters for TLAW distribution
(N=1000, λ=3, β=2, α=3.5, θ=3)
n

Parameter

Average

RAB

ER

LI

UI

Length

30

λ

1.2054

0.5981

3.2204

1.1910

1.2198

0.0288

β

1.8318

0.0840

0.0284

1.8048

1.8588

0.0540

α

3.7224

0.0635

0.0494

3.7153

3.7295

0.0142

θ

1.5922

0.4692

1.9829

1.5220

1.6625

0.1405









50

λ

1.2063

0.5978

3.2170

1.1955

1.2173

0.0218

β

1.8311

0.0844

0.0286

1.8093

1.8529

0.0436

α

3.7218

0.0633

0.0492

3.7169

3.7269

0.0100

θ

1.5896

0.4701

1.9898

1.5335

1.6457

0.1122









100

λ

1.2064

0.5978

3.2168

1.1986

1.2142

0.0156

β

1.8316

0.0841

0.0284

1.8165

1.8468

0.0303

α

3.7211

0.0631

0.0489

3.7177

3.7246

0.0069

θ

1.5909

0.4696

1.9858

1.5514

1.6304

0.0789









150

λ

1.2064

0.5978

3.2167

1.2000

1.2129

0.0129

β

1.8317

0.0841

0.0283

1.8191

1.8442

0.0251

α

3.7209

0.0631

0.0488

3.7182

3.7237

0.0055

θ

1.5912

0.4695

1.9847

1.5588

1.6237

0.0649

From Tables 2 and 3, one can observe that the ERs of the ML averages of the estimates for the parameters 𝛼, 𝛽, λ, and θ decrease when the sample size n increases. Also, the lengths of the confidence interval become narrower as the sample size increases.
7. Application
In this section, a real data set of COVID19 is used to illustrate the applicability of the TLAW distribution and compared with the other lifetime models including ToppLeone exponential (TLEx), alpha power Weibull (APW), and Weibull distributions. The distribution functions of the competitive models are as follows:
 The TLEx distribution is introduced by AlShomrani et al. (2016)
 The APW distribution is proposed by Nassar et al. (2017)
 The cdf of the Weibull distribution is given in (21).
To verify which distribution fits better to the real data set, some means for model selection are obtained such as the Akaike information criterion (AIC), consistent Akaike information criterion (CAIC), Bayesian information criterion (BIC), and KolmogorovSmirnov (KS) statistic to compare the proposed model and other competitive models. The best model corresponds to the lowest values of AIC, CAIC, BIC and KS statistic, which indicates better fit to the data. Figures 24 display the PPplot, QQplot, and fitted pdf of the TLAW distribution respectively. Further, Table 4 presents the estimates of the parameters and standard errors (SE). The values for AIC, CAIC, BIC, and KS statistic are given in Table 5.
The data given by Mubarak and Almetwally (2021) represents drought mortality rate of COVID19 data belongs to the United Kingdom of 76 days, from 15 April to 30 June 2020. The data are given as follows:




0.1303 0.1652 0.2079


0.1303 0.1652 0.2079

0.1303 0.1652 0.2079

















































































Figure 2: PP plots of the TLAW distribution for the data set
Figure 3: QQ plots of the TLAW distribution for the data set
Figure 4: Fitted pdf of the TLAW distribution for the data set
Table 4: ML estimates of the parameters and
SE of TLAW distribution for the data set
Parameters

ML estimate

SE

λ

1.2383

0.0208

β

2.9587

0.0462

α

0.7889

0.0204

θ

0.3244

0.0051

Table 5: Criteria for comparison of the fitted models for the data set
Model

KS

pvalue

AIC

BIC

CAIC

TLAW

0.171

0.217

292.9

302.2

293.4

TLEx

0.184

0.152

304.2

308.8

304.3

APW

0.211

0.069

306.4

313.4

306.7

Weibull

0.197

0.103

305.2

309.8

305.3

It can be seen from Tables 4 and 5 that the TLAW distribution provides a better fit than the other distributions regarding the values of AIC, CAIC, BIC, and KS statistic.
8. Conclusion
In this paper, a general class of distributions called TLAP family is presented. Some statistical properties of the general family are studied. A special submodel is considered namely, TLAW distribution. Estimation of the parameters for TLAW distribution using the ML method is discussed, and a simulation study is carried out. A real data set is applied, and some certain accuracy measures are evaluated. These measures ensure that the TLAW distribution provides a better fit to the real data set than the TLEx, APW, and Weibull distributions.