J. Nig. Soc. Phys. Sci. 4 (2022) 848

Journal of the
Nigerian Society

of Physical
Sciences

The Type II Topp-Leone-G Power Series Distribution with
Applications on Bladder Cancer

Boikanyo Makubate∗, Broderick O. Oluyede, Marang Pearl Matsuokwane, Lesego Gabaitiri, Simbarashe Chamunorwa

Botswana International University of Science and Technology, P. Bag 16, Palapye, Botswana

Abstract

Statistical distributions are important in modeling the real life of an item and therefore proper distributions that provide useful information for
sound conclusions and decisions are needed. For that reason, the demand for developing new generalized distributions have become appropriate
for data that have both monotonic and non-monotonic hazard rate functions. In this paper, we develop a new distribution called the Type II
Topp-Leone-G Power Series (TIITLGPS) distribution by compounding the Type II Topp-Leone-G (TIITLG) distribution with the power series
distribution. Statistical properties of the TIITLGPS distribution are obtained. A variety of shapes for the densities and hazard rate are presented
of the considered special case. A simulation study to examine the efficiency of the maximum likelihood estimates is also conducted. Finally, the
bladder cancer data example is analyzed for illustrative purposes, it is displayed that the introduced distribution provides better fit when compared
to other non-nested distributions considered in this work.

DOI:10.46481/jnsps.2022.848

Keywords: Type II Topp-Leone-G Distribution, Power Series Distribution, Maximum Likelihood Estimation.

Article History :
Received: 12 May 2022
Received in revised form: 16 July 2022
Accepted for publication: 20 July 2022
Published: 15 August 2022

c© 2022 The Author(s). Published by the Nigerian Society of Physical Sciences under the terms of the Creative Commons Attribution 4.0 International license

(https://creativecommons.org/licenses/by/4.0). Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Communicated by: B. J. Falaye

1. Introduction

Statistical distributions are of tremendous importance in
real lifetime analysis in various fields such as clinical studies,
medical studies, biological studies and environmental studies,
just to mention few. Several classical distributions have been
derived over past years for modeling data. However, data aris-
ing from public health and other areas may not fit the classical
distributions. Therefore, modifications of the well-established
distributions are highly recommended to obtain more flexible
new families of distributions. These generalized distributions

∗Corresponding author tel. no:
Email address: makubateb@biust.ac.bw. (Boikanyo Makubate )

have proved to be crucial in capturing heavily tailed, and where
the hazard rate function is non-monotonic (uni-modal, bathtub,
upside bathtub or upside bathtub followed by bathtub) and im-
proving the goodness-of-fit in empirical distribution. Thus, in-
creased demand in generating new families of distributions that
provide flexibility in lifetime phenomenon data modeling.

The Topp-Leone (TL) distribution, proposed by Topp and
Leone [1] as a lifetime model, is one of the distributions used
within the theory and practice of statistics. It has proven to be a
useful lifetime model through its applicability in different fields
such as medical and actuarial sciences. Ghitany et al. [2] stud-
ied the TL distribution by providing some reliability measures
while Vicaria et al. [3] introduced a two-sided generalized ver-
sion of the TL distribution. Other various families of extended

1



Makubate et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 848 2

distributions have been introduced. For instance, the Topp-
Leone-Gompertz-G family by Oluyede et al. [4], the exponenti-
ated Odd Weibull-Topp-Leone-G family by Chamunorwa et al.
[5] and the type II Generalized Topp-Leone family by Hassan
et al. [6]. For any baseline distribution, G(x; ψ), the cdf of the
Type II Topp-Leone-G (TIITLG)[7, 8] family of distributions is
given by

FT I IT LG (x; b,ψ) = 1 − [1 − G
2(x; ψ)]b, (1)

where b > 0 and ψ is a parameter vector. The survival function
associated with 1 is given by

S T I IT LG (x; b,ψ) = [1 − G
2(x; ψ)]b. (2)

Suppose that a series system has N components at a given time
where N is a discrete random variable with a power series distri-
bution (truncated at zero) and probability mass function (pmf)

P(N = n) =
anθn

C(θ)
, n = 1, 2, ...,

where an ≥ 0 depends only on n, C(θ) =
∑
∞
n=1 anθ

n and
θ ∈ (0, s) (s can be ∞) is chosen such that C(θ) is finite
and its three derivatives with respect to θ are defined and
given by C′(.), C′′(.) and C′′′(.), respectively. The power se-
ries family of distributions [9, 10, 11] includes Binomial, Pois-
son, Geometric and Logarithmic distributions. Given N, sup-
pose (X1, X2, ...XN ) ∼ T I IT LG are the independent and iden-
tically distributed failure times of the N components. Let
X=min(X1, ..., XN ), then the conditional cdf of X|N = n is given
by

FX|N=n(x) = 1 − [S T I I−T L−G (x; ψ)]
n, x > 0. (3)

The Type II Topp-Leone-G Power Series (TIITLGPS) class of
distributions is defined by the marginal cdf of X. The cdf of
X ∼ T I IT LGPS is given by

FT I IT LGPS (x; θ, b,ψ) = 1 −
C(θ(1 − G2(x; ψ))b)

C(θ)
. (4)

Table 1 shows some sub-classes of the TIITLGPS distribution.
The density function associated with equation (4) is given

by

fT I IT LGPS (x; θ, b,ψ) = 2bθg(x; ψ)G(x; ψ)

×[1 − G2(x; ψ)]b−1
C′

(
θ[1 − G2(x; ψ)]b

)
C(θ)

, (5)

where C′(θ) =
∑
∞
n=1 nanθ

n−1. The hazard function (hrf) is given
by

hT I IT LGPS (x; θ, b,ψ)

=
2bθg(x; ψ)G(x; ψ)[1 − G2(x; ψ)]b−1

C(θ(1 − G2(x; ψ))b)

×C′
(
θ[1 − G2(x; ψ)]b

)
.

The quantile function of the TIITLGPS distribution is given by

Q(x) = G−1
(1 −

[
C−1(C(θ)(1 − u))

θ

]1/b )1/2 , (6)

where G−1 is the inverse of the baseline distribution.
Note that, the TIITLG family of distribution defined in equation
(1) is a limiting special case of the TIITLGPS class of distribu-
tions when θ → 0+.

Using the binomial expansion, we express the pdf in 5 as

fT I IT LGPS (x; θ, b,ψ) =
∞∑

m=0

Vmgm(x; ψ), (7)

where

gm(x; ψ) = (2m + 2)g(x; ψ)G
2m+1(x; ψ) (8)

is the Exp-G distribution with power parameter 2m + 2, and

Vm =
∞∑

n=0

(−1)m2b(n + 1)a(n+1)θ(n+1)

C(θ)(2m + 2)

(
b(n + 1) − 1

m

)
. (9)

Thus, the statistical properties of the TIITLGPS class of dis-
tributions including rth moment, incomplete moment, the mo-
ment generating function, the rth moment of residual life, the rth

moment of reversed residual life, mean deviations, Bonferroni
curves and Lorenz curves can be obtained directly from those
of the Exp-G family of the distributions. For, more informa-
tion regarding the properties of Exp-G distributions the reader
is refereed to [12-21].
The primary motivation for developing the Type II Topp-Leone-
G Power Series (TIITLGPS) family of distributions is the ver-
satility and flexibility derived from compounding continuous
distributions in public health field. The proposed distribution
exhibit both monotonic, non-monotic hazard rate functions and
heavy tailed data sets, which is a common phenomenon with
medical data especially cancer data. Figure 1b shows that the
special case TIITLWP distribution can capture upside-down
bathtub followed by bathtub and uni-modal shapes hazard rate
function(s). Furthermore, from Bladder Cancer data modeling
example presented, the introduced distribution captures well
heavy tailed data and has better fit compared to some of the
selected generalizations in this work.
This paper is organized as follows. In Section 2 some mathe-
matical properties of the new model are presented. Parameter
estimation via the method of the maximum likelihood is given
in Section 3. Some special models of TIITLGPS class of dis-
tributions are presented in Section 4. In Section 5, a simulation
study is done. Section 6 contain applications to bladder cancer
data set. Concluding remarks are given in Section 7.

2. Some mathematical properties of the TIITLGPS

2.1. Moments and moment generating function

This section discusses the rth moment, mth incomplete mo-
ment, moment generating function and residual and reversed
residual life of the TII-TL-GPS. The rth moment of the TII-TL-
GPS can be represented as

µr = E(X
r ) =

∞∑
m=0

Vm

∫
∞

0
xr gm(x; ψ)d x (10)

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Makubate et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 848 3

Table 1. Sub-Classes of the TIITLGPS Distribution
Distribution an C(θ) cdf

Type II Topp-Leone G Poisson (n!)−1 eθ − 1 1 - e
θ

(
[1−G2 (x;ψ)]b

)
−1

eθ−1

Type II Topp-Leone G Geometric 1 θ(1 − θ)−1 1 -
(
[1−G2 (x;ψ)]b

)
(1−θ)

1−θ
(
1−G2 (x;ψ)

)b
Type II Topp-Leone G Logarithmic n−1 − log(1 − θ) 1 -

log
(
1−θ[1−G2 (x;ψ)]b

)
log(1−θ)

Type II Topp-Leone G Binomial
(

m
n

)
(1 + θ)m − 1 1 -

(
1+θ[1−G2 (x;ψ)]b

)m
−1

(1+θ)m−1

where gm(x; ψ) and Vm are defined in (8) and (9) respec-
tively and 2p+2 is the power parameter.
The mth incomplete moment can be obtained in the following
way

µm(y) =
∞∑

p=0

Vp

∫ y
0

xr gp(x; ψ)d x. (11)

The moment generating function of the TII-TL-GPS is rep-
resented as:

Mx(t) = E(e
tx) =

∞∑
p=0

Vp

∫
∞

0
etxgp(x; ψ)d x. (12)

We state the following two theorems and provide the proofs
at https://drive.google.com/file/d/1wMwiSK60cj0unyRqA–
pwFKBbWGaP9ta/view?usp=sharing.

Theorem 2.1. Let X1, X2, .., Xm be a random sample of size
m from the TIITLGPS class of distributions and X1:m < X2:m <
... < Xm:m denote the corresponding ordered random sample of
size m. The pdf of the ith order statistic, Xi:m is given by

fi:m(x) =
∞∑

s=0

ηsgs(x; ψ),

where gs(x; ψ) = (2s + 2)g(x; ψ)G2s+1(x; ψ) is the Exp-G distri-
bution with power parameter 2s + 2 and

ηs =

∞∑
j,n,k,z=0

m!
(i − 1)!(m − i)!

×
2b(n + 1)a(n+1)θz+n+1(−1) j+k+sdz,k

Ck+1(θ)

×

(
m − i

j

)(
j + i − 1

k

)(
b(z + n + 1) − 1

s

)
(2s + 2).

Theorem 2.2. The Rényi entropy for TIITLGPS class of dis-
tributions is given by

IR(ν) =
1

1 −ν
log

 ∞∑
s=0

Wse
(1−ν)IREG

 ,

where

IREG =
1

1 −υ
log

( ∫ ∞
0

[
(
2s
ν

+ 2)g(x; ψ)G
2s
ν

+1(x; ψ)
]ν

d x
)

is the Rényi entropy of Exp-G distribution with power parame-
ter ( 2s

ν
+ 2) and

Ws =
∞∑

z=0

(2b)ν dz,νθ
z+ν(−1)s

(
b(ν + z) −ν

s

)
1

( 2s
ν

+ 2)ν
.

3. Maximum Likelihood estimation

This section uses the method of maximum likelihood to es-
timate unknown parameters of the TIITLGPS class of distribu-
tions. Let x1, x2, ..., xn be a random sample from a TIITLGPS
distribution given in equation (5), then the log-likelihood of the
parameter vector φ = (b,θ,ψ)T is given by:

`(φ) = nlog2 + nlogb + nlogθ +
n∑

i=1

logG(xi; ψ)

+(b − 1)
n∑

i=1

log[1 − G2(xi; ψ)]

−nlogC(θ) +
n∑

i=1

logC′
(
θ[1 − G2(xi; ψ)]

b
)
.

The components of the score function are obtained by finding
the partial derivatives (with respect to the parameters θ, b and
ψ, respectively). They are given by:

Uθ =
n
θ
−

nC′(θ)
C(θ)

+

n∑
i=1

(
C′′

(
θ[1 − G2(xi; ψ)]b

))
[1 − G2(xi; ψ)]

C′
(
θ[1 − G2(xi; ψ)]b

) ,
Ub =

n
b

+

n∑
i=1

[1 − G2(xi; ψ)]

+

n∑
i=1

(
C′′

(
θ[1 − G2(xi; ψ)]b

))
C′

(
θ[1 − G2(xi; ψ)]b

) [1 − G2(xi; ψ)]b
×log[1 − G2(xi; ψ)]

3



Makubate et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 848 4

and

Uψ = (b − 1)
n∑

i=1

1
[1 − G2(xi; ψ)]

∂[1 − G2(x; ψ)]
∂ψ

+

n∑
i=1

∂G(xi ; psi)
∂ψ

G(xi; ψ)
+

n∑
i=1

∂g(xi ; psi)
∂ψ

g(xi; ψ)

+

n∑
i=1

(
C′′

(
θ[1 − G2(xi; ψ)]b

))
C′

(
θ[1 − G2(xi; ψ)]b

)
×2bθ[1 − G2(xi,ψ)]

b−1G(xi; ψ)
∂G(xi; ψ)

∂ψ
.

Note that these equations are non-linear and can not be solved
analytically, but can be solved numerically using software like
R language.

4. Some Special Models

This section considers and presents some special cases of
the TIITLGPS class of distributions when the baseline distribu-
tions are Weibull and Burr XII distributions.

4.1. Type-II-Topp-Leone-Weibull-Poisson (TIITLWP) Distribu-
tion

The cdf and pdf of the TIITLWP distribution are given by

FT I IT LW P(x; θ, b,α,λ) = 1 −
e

(
θ
[
1−

(
1−e−λx

α )2]b)
−1

eθ − 1

and

fT I IT LW P(x; θ, b,α,λ) = 2αbθλx
α−1e−λx

α
(
1 − e−λx

α
)

[
1 −

(
1 − e−λx

α
)2]b−1

×
e

(
θ
[
1−

(
1−e−λx

α )2]b)
eθ − 1

,

respectively, for θ, b,α,λ and x > 0.

Figures 1(a) and 1(b) show the plots of the pdfs and hrfs, respec-
tively, for the TIITLWP distribution for selected parameter val-
ues. Plots of the TIITLWP pdf exhibit different shapes includ-
ing almost symmetric, left-skewed, right-skewed, and reverse-
J shapes. Plots of the hrf of the TIITLWP distribution show
different shapes including increasing, decreasing, upside-down
bathtub and uni-modal shapes.
The cdf, pdf and plots (pdf and hrfs plots) of Type-
II-Topp-Leone-Weibull-Geometric (TIITLWG), Type-II-Topp-
Leone-Burr XII-Poisson (TIITLBXIIP), Type-II-Topp-Leone-
Burr XII-Geometric (TIITLBXIIG) distributions can be
seen at https://drive.google.com/file/d/1wMwiSK60cj0unyRqA–
pwFKBbWGaP9ta/view?usp=sharing.

(a)

(b)

Figure 1. Pdfs and hrfs plots for the TIITLWP distribution

5. Simulation Study

In this section, we conduct Monte Carlo simulation study
to assess the performance of maximum likelihood estimates of
the TIITLWP distribution. We generated N=1000 samples of
size n= 100, 200, 400 and 800 with the help of the R package.
We consider simulations for the following sets of initial
parameters values (I: α = 1.0, b = 0.5,λ = 1.5,θ = 1.0),
(II: α = 0.5, b = 0.5,λ = 0.5,θ = 1.0), (III:
α = 1.5, b = 0.5,λ = 0.5,θ = 1.0), and (IV: α = 0.5, b =

4



Makubate et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 848 5

Table 2. Parameter estimates and goodness-of-fit statistics for various models fitted for cancer patients data set
Estimates Statistics

Model α b λ θ −2 log L AIC AICC BIC W∗ A∗ K-S p-value
TIITLWP 0.7361 3.0338 0.0749 3.6985 819.4 827.4 827.8 838.9 0.0226 0.1495 0.0371 0.9945

(0.1340) (8.2303) (0.1021) (2.4628)
α λ γ θ

OWTLLLoGL 1.0089 0.4757 9.7751 1.2905×10−9 826.4 834.4 834.7 845.8 0.0925 0.6058 0.0574 0.7930
(0.3164) (0.1060) (3.1140) (0.0033)

p α β λ
EGEL 1.0729×10−8 0.8922 1.1774 1.3292 826.2 834.2 834.6 845.6 0.1135 0.6819 0.0707 0.5442

(0.0095) (0.6760) (0.1426) (10.0710)
a b α β

EPLP 5.8448×10−8 0.5647 0.82441 2.7914 820.9 828.9 829.2 840.3 0.0391 0.2563 0.0428 0.9733
(0.0196) (0.1021) (0.3150) (1.3114)

c s α λ
EBXIIP 1.0223 1.5133 1.9338 7.8251×10−9 871.7 879.7 880.0 891.1 0.2452 1.600 0.2481 2.872×10−7

(0.0902) (0.2426) (0.2105) (0.0085)
a b λ θ

BOL-U 1.1799 2.8446 3.2800 ×105 7.6115 ×106 826.4 834.4 834.7 845.8 0.1186 0.7117 0.0734 0.4955
(0.1329) (0.3599) (3.1630 ×10−6) (1.3631 ×10−7)

β λ θ γ
ELOLLoGW 6.1173×10−5 0.0378 2.9053 1.0478 828.2 836.2 836.5 847.6 0.1314 0.7864 0.0700 0.5573

(0.4015) (0.0061) (8.4383 ×10−5) (0.0676)

(a)

(b)

Figure 2. Fitted densities and probability plots for cancer patients data

1.0,λ = 1.5,θ = 0.5). Simulation results are shown
at https://drive.google.com/file/d/1wMwiSK60cj0unyRqA–
pwFKBbWGaP9ta/view?usp=sharing.
The results show that as the sample size increases, the estimates
approach the true parameter values, since root mean square
errors (RMSE) and average bias decays toward zero for all the
parameters.

6. Applications and Discussion

In this section, we demonstrate the applicability of the
TIITLWP distribution to bladder cancer data set. We compared
the TIITLWP distribution to six non-nested models. We use
the nlm function in R software to estimate model parameters.
We present the model parameters estimates (standard errors
in parenthesis) and the goodness-of-fit-statistics. We assessed
model performance using -2loglikelihood (-2 log L), Akaike
Information Criterion (AIC), Consistent Akaike Information
Criterion (AICC), Bayesian Information Criterion (BIC),
Cramer-von-Mises (W∗), and Andersen-Darling (A∗) (see
Chen and Balakrishnan [23] for details). These statistics are
used to verify which model fits the given data well. The
smaller the values of these statistics, the better the model.
Kolmogorov-Smirnov (K-S) statistic (and its p-value) and
sum of squares (SS) from probability plots were also used to
assess the fit of the model. The smaller the KS value and the
higher the p-value for the K-S statistic the better the model.
Tables 2 shows model parameters estimates (standard errors
in parentheses) and several goodness-of-fit statistics. Figure
2 shows fitted densities and probability plots (as described by
Chambers et al. [22]) to demonstrate how well our model fits
the selected data sets.

The non-nested models considered in this paper are the
exponentiated generalized logarithmic (EGEL) distribution by

5



Makubate et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 848 6

Oluyede et al. [24], odd Weibull-Topp-Leone-log logistic log-
arithmic (OWTLLLoGL) distribution by Oluyede et al. [25],
exponentiated power Lindley Poisson (EPLP) distribution by
Pararai et al. [21], exponentiated Burr XII Poisson (EBXIIP)
distribution by da Silva et al. [12], beta odd Lindley-Uniform
(BOL-U) distribution by Chipepa et al. [26] and exponential
Lindley odd log-logistic Weibull (ELOLLoGW) distribution
by Korkmaz[15]. The pdfs of the non-nested models can be
seen at https://drive.google.com/file/d/1wMwiSK60cj0unyRqA–
pwFKBbWGaP9ta/view?usp=sharing. For the EBXIIP and
ELOLLW distributions we consider k = 1 and α = 1,
respectively.

6.1. Bladder Cancer Patients Data

This data set represents remission times of a random sample
of 128 bladder cancer patients. For more information regard-
ing the bladder cancer data set, Lee and Wang [27] is recom-
mended.
From the values of the goodness-of-fit statistics A∗, W∗, K-S
and the p-value of the K-S statistic as shown in Table 2, we
conclude that the TIITLWP model performs better than the non-
nested models considered in this paper. Figures 2(a) and 2(b)
show the fitted densities and probability plots for the TIITLWP
model. We observe that the TIITLWP model has better fit to ex-
treme tailed data compared to the selected competitive models.

7. Conclusions

The demand of developing new families of distributions
from classical ones has been of interest among authors in the
past years. A new distribution called the TIITLGPS distribu-
tion is developed which combines the TIITL-G with the power
series to provide compound TIITL-GPS family of distributions
with better performance compared to selected models. We
study some of mathematical properties of the new family of dis-
tributions, such as ordinary moment, quantile functions, order
statistics and Rényi entropy. We discuss maximum likelihood
estimates of the model parameters for bladder cancer data. Ad-
ditionally, the performance of the MLEs of the two selected
members is assessed via Monte Carlo simulation studies based
on two criteria; bias and RMSE. The study exhibits a good per-
formance when estimating the parameters of the proposed fam-
ily using the maximum likelihood method. Finally, the special
case of the new distribution applied to the bladder cancer data
set and a simulation study demonstrate its usefulness and poten-
tiality to analysis of lifetime data. We hope that the proposed
distribution will attract wider application in various fields such
as health, survival and lifetime data, finance, insurance, among
others.

Future Work

• Take into consideration bivariate extensions of proposed
model via copulas like the Farlie-Gumbel-Margestern
Copulas, Ali-Mikhail-Haq copula, Clayton copula and
Rényi entropy copula.

• Apply different parameter estimation techniques such as
Bayesian technique, weighted least squares (WLS), max-
imum product of spacings (MPS) on the proposed mod-
els.

Limitations

• We derive mathematical and statistical properties from
the exponentiated G distribution cause its a bit compli-
cated to derive them straight from the TIITLGPS pdf.

• The model can not perform well on some data sets.

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