Penalized Likelihood Parameter Estimation in the Quasi Lindley and
Nadarajah-Haghighi Distributions
MARWA MOHAMED HAMADA1,*, MAHMOUD RAID MAHMOUD 2,
RASHA MOHAMED MANDOUH3
1Department of Mathematical Statistics,
Cairo University, Faculty of Graduate Studies for Statistical Research,
Cairo,
EGYPT
2Department of Mathematical Statistics,
Cairo University, Faculty of Graduate Studies for Statistical Research,
Cairo,
EGYPT
3 Department of Mathematical Statistics,
Cairo University, Faculty of Graduate Studies for Statistical Research,
Cairo,
EGYPT
*Corresponding Author
Abstract: - The issues of performing inference on the parameters of quasi-Lindley (QL) distribution and
Nadarajah-Haghighi exponential distribution (N-H) is addressed. It is shown graphically that the likelihood
function of the quasi-Lindley distribution and Nadarajah-Haghighi exponential distribution is configured with a
flat monotone shape. Which makes it very difficult to pick the values of the parameters that maximize the
likelihood function. A penalization scheme is used to improve maximum likelihood point estimation. A
penalization scheme based on the Jeffreys prior.
Key-Words: - Jeffreys Prior, Maximum Likelihood Estimation, Monotone Likelihood, Penalized Likelihood
Estimation, Quasi Lindely Distribution, Nadarajah-Haghighi Exponential Distribution.
Received: February 2, 2024. Revised: February 16, 2024. Accepted: March 2, 2024. Published: March 8, 2024.
1 Introduction
In statistical inference, the maximum likelihood
method is used to estimate the parameters of a
probability distribution. In most cases, the
likelihood function is peaked and this searches for
the peak reasonably simple. A problem occurs when
the likelihood function is configured with a flat
monotone shape, causing difficulties in the search
for the point of maximization. A way around this
problem suggested in the literature (known as Firth
correction) is an adaptation of a method created to
reduce the bias of maximum likelihood estimates.
The method leads to finite estimates using
maximization of the penalized likelihood
procedure. In this case, the penalty might be
interpreted as a Jeffreys type prior widely used in
the Bayesian context, [1]. This problem appeared in
some distributions as shown by [2]. They presented
the maximization of penalized likelihood estimation
MPLEs in the modified extended Weibull
distribution.
In this article, we study quasi-Lindley
distribution QL(α, θ) and discuss the properties of
this distribution in section 2. We present the
maximization of penalized likelihood estimation for
quasi-Lindley distribution in section 2.1. In section
2.2, real data is introduced to compare the maximum
likelihood estimation with the maximization of
penalized likelihood estimation. In section 2.3, a
simulation study is presented for quasi-Lindley
distribution QL (α,θ) by the maximum likelihood
estimation and maximization of penalized likelihood
estimation. In section 3, we introduce the
Nadarajah-Haghighi exponential distribution N-H
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2024.23.16
Marwa Mohamed Hamada,
Mahmoud Raid Mahmoud, Rasha Mohamed Mandouh
E-ISSN: 2224-2880
132
Volume 23, 2024
(α, λ) and the properties of this distribution. We
present the maximization of penalized likelihood
estimation for the Nadarajah-Haghighi exponential
distribution in section 3.1. In section 3.2, a real data
set is used to compare between the maximum
likelihood estimation and maximization of penalized
likelihood estimation. In section 3.3, a simulation
study is presented for Nadarajah-Haghighi
exponential distribution N-H (α, λ) by the
maximum likelihood estimation and maximization
of penalized likelihood estimation
2 Quasi Lindley Distribution
The quasi-Lindley distribution QL (α, θ) was
introduced by [3]. The probability density function
(pdf) of the quasi-Lindley distribution is defined by
)
( , ) 0,
(
,00 ,,
1
x
f x x
xe
(1)
Its cumulative distribution function (cdf) is
obtained as:
1
1 , 0, 0, 1
1
x
x
F x e x
(2)
They obtained the rth moment about the origin of
quasi-Lindley distribution as:
'11
; 1, 2
1
rr
rr
r
(3)
Let x1, x2, …, xn be a random sample of size n from
the quasi Lindley distribution. The log-likelihood
function for the vector of parameters α, θ is:
11
, log log 1 log
nn
ii
ii
l l n n x X
(4)
The maximum likelihood estimator ( and
)
of α and θ can be obtained by maximizing the log-
likelihood function given in equation (4) concerning
α and θ. This can be done by setting the score
function equal to zero and solving the resulting
system of equations. The components of the score
function are given
11
nn
ii
ii
i
x
ln X
x
(5)
1
1
1
n
ii
ln
x
(6)
Figure 1 indicates plots of the density function of
quasi-Lindley distribution for some parameter
values . Figure 2 shows plots of the hazard
function of quasi-Lindley distribution for some
parameter Values α,θ
Fig. 1: Plots of the density function for some
parameter values α and θ.
Fig. 2: Plots of the hazard function for some
parameter values α and θ.
2.1 Maximization of Penalized Likelihood
Estimation
In [4], the author proposed to modify the score
function to reduce the bias of the MLE. His
proposal is an alternative to the usual approach of
subtracting from the MLE its estimated bias. The
underlying idea is that since the parameter estimate
may not exist, it is safer to transform the estimating
equations to correct for bias prior toestimation. For
the canonical parameter of the exponential family
model, the rth component of the modified score
equation is given by:
*
r r r
U U A
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2024.23.16
Marwa Mohamed Hamada,
Mahmoud Raid Mahmoud, Rasha Mohamed Mandouh
E-ISSN: 2224-2880
133
Volume 23, 2024
Where is the rth component of
. Here, is the
first-order term in the bias expansion of the MLE:
2
11
/ / ...B B n B n
For an exponential family in canonical form,
1log
2
r
r
AI
Here, the correction amounts to finding the mode
of the posterior distribution obtained by using the
Jeffreys invariant prior, see [2]. i.e., it amounts to
finding the mode of ,where
is the likelihood function. Equivalently,
estimation can be carried out by maximizing
Notice that the penalization term is
the Jeffreys invariant prior.
[2], introduced maximization of penalized
likelihood estimation in the modified extended
Weibull distribution and they used Jeffrey prior to
find maximization of penalized likelihood
estimation MPLEs. In particular, we consider the
following family of penalized log likelihoods:
Numerical maximization of the quasi-Lindley
log-likelihood function can be problematic. A
penalization scheme is used to improve maximum
likelihood point estimation. A penalization scheme
based on the Jeffreys invariant prior.
To estimate the parameters of quasi-Lindley
distribution by maximization of penalized likelihood
estimation, we use equation (7) and let x1, x2,…, xn
be a random sample of size n from the quasi Lindley
distribution. The elements of Fisher's (expected)
information matrix are obtained as follows
11 11
I E J
22 22
I E J
21 12 12
I I E J
Using equations (5) and (6), we get
2
11 22
2
1
1
1
n
ii
ln
Jx
(8)
2
2
22 2
22
1
ni
ii
x
ln
Jx
(9)
2
12 2
1
ni
ii
x
l
Jx
(10)
The information matrix is given by:
22
2
22
2
ll
IE
ll
(11)
The penalized likelihood function is:
*1log
2
l l I
where
11 12
21 22
II
III
Notice that the penalization term is the
Jeffreys prior.
To use the Jeffreys prior for penalizing the log-
likelihood function, we need to compute the
expected values of the quantities given in the
equations from (8) to (10).
To illustrate the use of the proposed modified
penalized likelihood function, we return to the four
samples that gave rise to the log-likelihood
functions presented in Figure 3. This figure
contains plots of the log-likelihood function as a
function of α for four different samples of size 50
obtained from a quasi Lindley distribution with α =
0.15, θ = 0.35. Notice that the likelihood function is
configured with a flat monotone shape. Table 1
contains the quasi-Lindley maximum likelihood
parameter estimates and the corresponding the log-
likelihood values. One can notice that from Table 1,
the large values of obtained using samples 3 and
4 are approximately 6882.21 and 49271.9,
respectively. Table 2 contains the quasi-Lindley
maximum penalized likelihood parameter estimates
and one can notice that the values of obtained
using samples 3 and 4 are approximately 0.9207 and
0.8715, respectively. The parameter estimates
obtained using the log-likelihood function are
presented in Table 1. These results are to be
compared to those in Table 2. Notice that the log-
likelihood functions for samples 3 and 4 contain
large estimates of α.
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2024.23.16
Marwa Mohamed Hamada,
Mahmoud Raid Mahmoud, Rasha Mohamed Mandouh
E-ISSN: 2224-2880
134
Volume 23, 2024
We note that there is an improvement in the
results when using maximization of penalized
likelihood estimation (Table 1 and Table 2).
Notice that the values of samples 3 and 4 are
reduced in Table 2 when using maximization of
penalized likelihood estimation.
It is possible to compute a confidence interval
for parameters of distribution in the case of the
penalized likelihood estimation. That can be
computed by using the bootstrap method.
The following steps were followed to obtain the
confidence interval:
1. Data: x1,…, x n are drawn from a distribution F(θ)
with unknown parameter θ.
2. Compute
that estimates θ.
3. Our bootstrap samples are drawn from F(
)
4. For each bootstrap sample
x1*, ..., xn*
We compute
*
and the bootstrap difference
**
.
5. Use the bootstrap differences to make a bootstrap
confidence interval for θ, [5].
Now, we introduce real data for quasi-Lindley
distribution in the next section.
Table1. MLEs of α and θ for 4 Samples of size n =
50; from quasi Lindley (0.15,0.35) distribution.
Maximum Likelihood Estimation MLEs
Sample
log-
likelihood
1
2
3
4
0.02524
0.30688
6882.21
4 49271.9
0.38891
0.41098
0.18832
0.21510
-127.58500
-119.95500
-133.48700
-126.8330
Table 2. MPLEs of α and θ for 4 Samples of size n =
50; from quasi Lindley (0.15,0.35) distribution.
Maximum Penalized Likelihood Estimation MPLEs
Sample
log-
likelihood
1
2
3
4
0.00625
0.25714
0.92077
0.87156
0.38856
0.41201
0.28230
0.32239
-122.46100
-115.20600
-130.40300
-123.15000
2.2 Application for Real Data
The following data from [6], is used to compare
parameter estimation in quasi-Lindley distribution
between maximum likelihood estimation and
penalized likelihood approach.
Data Set:
Complete data: All 50 items are put into use at t = 0
and failure times are in weeks
0.013, 0.065, 0.111, 0.111, 0.163, 0.309, 0.426,
0.535, 0.684, 0.747, 0.997, 1.284,1.304,1.647,1.829,
2.336, 2.838, 3.269, 3.977, 3.981, 4.520, 4.789,
4.849, 5.202, 5.291, 5.349, 5.911, 6.018, 6.427,
6.456, 6.572, 7.023, 7.087, 7.291, 7.787, 8.596,
9.388, 10.261, 10.713, 11.658, 13.006, 13.388,
13.842, 17.152, 17.283, 19.418, 23.471, 24.777,
32.795, 48.105
Parameters of quasi Lindley distribution are
estimated by maximum likelihood estimation
follows
= 74897.9,
= 0.127862,
The estimate
of
seems unreasonably large.
Figure 4 indicates plots of empirical and fitted
cumulative distribution function (cdf) for quasi-
Lindley distribution QL(α, θ) at two values of
parameters F(x,74897.9,0.127862) and
F(x,2.16711,0.162343), it appears that the fitted
distributions both from maximum likelihood and
penalized likelihood estimation fit the data
reasonably well, but the extreme value of the
maximum likelihood estimate seems strange and the
penalized likelihood would be preferable.
Now, the Confidence interval (CI) for alpha and
theta parameters of the quasi-Lindley distribution is
given by:
CI for α = [59074, 149791]
CI for θ = [0.0707601, 0.155148]
Using the same real data set for the same
distribution, the penalized likelihood estimation of
the parameter is:
= 2.16711;
= 0.162343
The confidence intervals for parameters of the quasi
Lindley distribution in the case of penalized
likelihood estimation are given by:
CI for α = [1.56153, 3.12993]
CI for θ = [0.105837, 0.197349]
2.3 Simulation Study
We used a simulation study to assess the
performance of the penalized likelihood estimation
of the point estimate, of which estimates two
parameters of QL(α, θ) form=1000, the sample size
n is 50,75,100, and different parameter values. The
following steps were followed to obtain the results:
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2024.23.16
Marwa Mohamed Hamada,
Mahmoud Raid Mahmoud, Rasha Mohamed Mandouh
E-ISSN: 2224-2880
135
Volume 23, 2024
1. Select initial values for parameters α, and θ to be
used for generating data.
2. Specify the sample size n to be used for the study.
3. Generate pseudo-random samples with size n
from QL(α, θ) for selected value of α and θ
4. Obtain the maximum likelihood estimates and
maximization of penalized likelihood estimates for
α and θ for different sample sizes.
5. For each sample size, obtain the mean, bias,
relative bias, variance and mean squared error
(MSE) for each estimator of quasi Lindley (α, θ) for
different values of
by using maximization of
penalized likelihood, see equation (7).
6. Choose the value of
with the smallest mean
square error and obtain the mean, bias and relative
bias, variance and mean squared error for each
estimator for different sample sizes.
Table 3 is shown the mean, bias, relative bias, and
MSE for α and θ; by sample size from quasi Lindley
(0.15,0.35) distribution using the maximum
likelihood esttimation.
Data generation was carried out using the
computational software Mathematica [10].
Simulation results are shown in Table 4, Table 5,
Table 6 and Table 7 in Appendix.
Fig. 3: Log-likelihood function for α. For each
sample n = 50 from quasi Lindley distribution
(0.15; 0.35)
Fig. 4: Plots of empirical and fitted cumulative
distribution function (cdf) for quasi-Lindley
distribution QL(α, θ)
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2024.23.16
Marwa Mohamed Hamada,
Mahmoud Raid Mahmoud, Rasha Mohamed Mandouh
E-ISSN: 2224-2880
136
Volume 23, 2024
Table 3. Mean, bias, relative bias, and MSE of
MLEs of α and θ; by sample size from quasi Lindley
(0.15,0.35) distribution
For each estimator considered, the following
quantities: the mean, bias, relative bias, variance,
and Mean square error were computed.
Table 5 (Appendix) shows the Mean, bias,
relative bias, and MSEs of α and θ using the Jeffreys
prior penalization with η = 0.1,..., 0.9, 2; samples of
size n = 100 drawn from quasi Lindley(0.15,0.35)
distribution.
Table 8 and Table 9 in Appendix show that the
smallest MSE of ˆα and ˆθ correspond to η = 2.4.
Figure 5 indicates MSE of ˆα and ˆθ and the selected
value η for QL distribution.
From Table 10 (Appendix), We found the mean,
bias, relative bias, and variance for ˆα and ˆθ using
maximization of penalized likelihood estimation
with η = 2.4 and samples of size n = 50, 75, 100
from quasi Lindley distribution for different values
of parameters α, θ.
The results of the simulation study show that the
bias for any estimator decreases when the sample
size increases. Also, the relative bias decreases
when the sample size increases. The mean square
error (MSE) decreases when the sample size
increases.
In the next section, we introduce the
maximization of penalized likelihood estimation in
another distribution called the Nadarajah-Haghighi
exponential distribution.
Fig. 5: MSE of ˆα and ˆθ and selected values η for
QL distribution
3 Nadarajah-Haghighi Exponential
Distribution
Nadarajah-Haghighi exponential distribution N-H
(α, λ) was introduced in [7]. The cumulative
distribution function (cdf), probability density
function (pdf), and the quantile function are follows:
1 exp 1 1 ,F x x
(12)
1
1 exp 1 1f x x x
(13)
and
1/
11 log 1 1 ,0 1Q p p p
(14)
Respectively.
Fig. 6: Plots of the density function for some
parameter values α and λ
aximum Likelihood Estimation MLEs
n
para
meter
Mean
Bias
Relative
bias
Variance
MSE
25
α
911.274
-911.124
-607416
7.9515*
E07
8.034*
E07
θ
0.37190
-0.02195
-6.2738
0.0032
0.003
50
α
252.221
-252.071
-168047
2.12176
E07
2.121E07
θ
0.36579
-0.01579
-4.51319
0.0016
0.001
100
α
17.8852
0
-17.7352
-11823.5
31361
313929
θ
0.36069
-0.01069
- 3.05490
0.0006
0.00071
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2024.23.16
Marwa Mohamed Hamada,
Mahmoud Raid Mahmoud, Rasha Mohamed Mandouh
E-ISSN: 2224-2880
137
Volume 23, 2024
Fig. 7: Plots of the hazard function for some
parameter values α and λ
Now consider estimation by the method of
maximum likelihood. The log-likelihood function
(LL) of the two parameters is as follows:
, log , log 1l L n n
11
log 1 1
nn
ii
ii
xx
(15)
It follows that the maximum-likelihood
estimators (MLEs) are the simultaneous solutions of
the equations:
11
log 1 1 log 1 0
nn
i i i
ii
nx x x
(16)
and
11
11
1 1 1 0
nn
i i i i
ii
nx x x x
(17)
For interval estimation of (α, λ), one requires
the Fisher information matrix. The elements of this
matrix in equation (15) can be worked out as
follows:
22
22
log 1 log 1
Ln
E nE X X
(18)
22
2
22
log 11
Ln
E n E X X
2
2
11n E X X
(19)
and
2log L
E
11
1 1 log 1nE X X n E X X X
1
1nE X X
(20)
Where
2
1 log 1 0, ,2 ,E X X eJ
2
21 2, 2,1 ,E X X eI
2
21 2, 2,1 ,E X X eI
1
1 1, 1,1 ,E X X eI
1
1 log 1 1, 1,1 ,E X X X eJ
and
1
1 1, 1,1 ,E X X eI
Since
/1
1
0
1
, , 1 1,
nai bi
i
abi
I a b c c c
i
1
11
0/1
1
, , 1 ,1
c
aa
c a c
ibi
a
J a b c i
the information matrix is given by:
22
2
22
2
log log
log log
LL
IE LL
(21)
Figure 6 indicates plots of the density function
of the Nadarajah-Haghighi exponential distribution
for some parameter values α and λ. Figure 7 shows
plots of the hazard function of the Nadarajah-
Haghighi exponential distribution for some
parameter values α and λ.
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2024.23.16
Marwa Mohamed Hamada,
Mahmoud Raid Mahmoud, Rasha Mohamed Mandouh
E-ISSN: 2224-2880
138
Volume 23, 2024
3.1 Maximization of Penalized Likelihood
Estimation for N-H distribution
To calculate the penalized likelihood function, we
need to calculate the Jeffreys prior, see equation (7).
To use the Jeffreys prior for penalizing the log-
likelihood function, we need to compute the
expected values of the quantities given in equation
(18) to equation (20).
To illustrate the use of the proposed modified
penalized likelihood function, we shall return to the
two samples that gave rise to the log-likelihood
functions presented in Figure 8. This figure contains
plots of log-likelihood function versus λ for two
different samples of size 50 obtained from
Nadarajah-Haghighi exponential distribution with α
= 0.5, λ = 1 and α = 1, λ = 1. Notice that the
likelihood function is configured with a flat
monotone shape. Table 11 contains the Nadarajah-
Haghighi exponential maximum likelihood
parameter estimates and the corresponding log-
likelihood values. One can notice that from Table 11
, the large values of ˆλ obtained using samples 1 and
2 are approximately 9.54432 × 109 and 1.01713 ×
109, respectively. Table 12 contains the
maximization of penalized likelihood estimation for
Nadarajah-Haghighi exponential distribution and
one can notice that the values of ˆλ obtained using
samples 1 and 2 are approximately 3.67773 and
0.98556, respectively. The parameter estimates
obtained using the log-likelihood function are
presented in Table 11. These results are to be
compared to those in Table 12. Notice that the log-
likelihood functions for samples 1 and 2 contain
large estimates of λ.
Table 11. MLEs of α and λ of size n = 50; the
samples were drawn from the Nadarajah-Haghighi
exponential distribution.
Maximum Likelihood Estimation MLEs
Sample
1) α = 0.5,λ = 1
0.01768
9.54432 × 10^9
2)α = 1,λ = 1
0.01992
1.01713× 10^9
Table 12. MPLEs of α and λ of size n = 50; the
samples were drawn from the Nadarajah-Haghighi
exponential distribution
Maximum Penalized Likelihood Estimation MPLEs
Sample
1) α = 0.5,λ = 1
0.34103
3.67773
2)α = 1,λ = 1
0.71565
0.98556
We note that there is an improvement in the
results when using maximization of penalized
likelihood estimation, see Table 11 and Table 12.
Notice that the values of ˆλ in samples 1 and 2 are
reduced in Table 12 when using maximization of
penalized likelihood estimation.
We introduce real data for the Nadarajah-
Haghighi exponential distribution in the next
section.
Fig. 8: Log-likelihood function for α. For each
sample n = 50 observations were obtained from
Nadarajah-Haghighi exponential distribution
sample1 (α = 0.5, λ = 1) and sample2 (α = 1, λ = 1)
3.2 Real Data for Nadarajah-Haghighi
Exponential Distribution
The following data from [6], was used to compare
parameter estimation in the Nadarajah-Haghighi
exponential distribution between maximum
likelihood estimation and penalized likelihood
approach.
Dataset:
0.03, 0.12, 0.22, 0.35, 0.73, 0.79, 1.25, 1.41, 1.52,
1.79, 1.80, 1.94, 2.38, 2.40, 2.87, 2.99, 3.14, 3.17,
4.72, 5.09
Parameters of Nadarajah-Haghighi exponential
distribution are estimated by maximum likelihood
estimation as follows:
10
0.0160311, 4.96749 10
The estimate of ˆλ of λ seems unreasonably large.
(b) Sample 2
(a) Sample 1
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2024.23.16
Marwa Mohamed Hamada,
Mahmoud Raid Mahmoud, Rasha Mohamed Mandouh
E-ISSN: 2224-2880
139
Volume 23, 2024
Now, the confidence interval (CI) for α and λ
parameters of the Nadarajah-Haghighi exponential
distribution in the case of maximum likelihood
estimation is given by
CI for
-0.0527535, 0.435248
CI for
10 10
4.88508 10 , 4.96749 10
Using the same real data, see [4], parameters of the
Nadarajah-Haghighi exponential distribution are
estimated by penalized likelihood estimation as
follows
0.439155, 1.08643
And confidence interval for parameters of the
Nadarajah-Haghighi exponential distribution in the
case of penalized likelihood estimation is given by
CI for
0.397233, 0.682527
CI for
-7.24384, 1.20497
3.3 Simulation Study for Nadarajah-
Haghighi Exponential Distribution
We used a simulation study to assess the
performance of the penalized likelihood estimation
of the point estimate for several cases, of which
estimates two parameters of N-H (α, λ) for m=2000,
the sample size n are 25, 50, and different parameter
values. The following steps were followed to obtain
the results:
1. Select initial values for parameters α, λ to
be used for generating data.
2. Specify the sample size n to be used for
generating data.
3. Generate pseudo-random samples with size
n from N-H (α, λ) for selected values of α
and λ.
4. Obtain the maximum likelihood estimates
and the maximization of penalized
likelihood estimates for α, and λ for
different sample sizes.
5. For each sample size, obtain the mean, bias,
relative bias, variance, and mean squared
error (MSE) for each estimator for different
values of (η) by using maximization of
penalized likelihood, see equation (7).
6. Choose the value of (η) with the smallest
mean square error and obtain the mean,
bias, relative bias, variance, and mean
squared error for each estimator for
different sample sizes.
Data generation was carried out using the
computational software Mathematica[10]. The
simulation result is shown in Table 13 (Appendix).
For each estimator considered, we computed the
following quantities: mean, bias, relative bias,
variance, and mean square error. Table 13
(Appendix) shows that the smallest MSE of α and λ
correspond to η = 1.2. From Table 14 (Appendix),
we found mean, bias, relative bias, and variance for
the estimation of α and λ using maximization of
penalized likelihood estimation with η = 1.2;
samples of size n = 25; 50 from Nadarajah-Haghighi
exponential distribution for different values of
parameters α, λ. In Table 15 (Appendix), we found
the confidence interval for the estimator by the
bootstrap method. Table 16 and Table 17 in
Appendix, the results of the simulation study
showing that the bias for any estimator decreases
when the sample size increases. Also, the relative
bias decreases when the sample size increases.
Mean square error (MSE) decreases when the
sample size increases.
4 Summary and Conclusion
In this article, we have shown that the maximum
likelihood estimation of the parameters that index
the Quasi-Lindley distribution can be problematic.
The MLE estimates of parameters for the Quasi-
Lindley distribution are large, so a penalized
likelihood function is proposed. The penalization
term is a modified version of the Jeffreys invariant
prior. We used a simulation study to assess the
performance of the penalized likelihood estimation
of the point estimate for several cases, of which
estimates two parameters of Quasi Lindley
distribution QL(α, θ) for m=1000, the sample size n
are 50,75,100 and different parameter values by
using penalized likelihood estimation, see equation
(7). We choose the value of η with the smallest
mean square error. Simulation results on point
estimation are found in Table 4, Table 5, Table 6
and Table 7, Table 8, Table 9 and Table 10 in
Appendix. We used a simulation study to estimate
parameters for several cases, of which estimates two
parameters of N-H (α, λ) for m=2000, the sample
size n are 25, 50, and different parameter values and
different the value of η, see equation (7). We choose
the value of η with the smallest mean square error
(Table 13, Appemdix). A real data set from [6] was
used to compare parameter estimation in the Quasi-
Lindley distribution and N-H distribution between
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2024.23.16
Marwa Mohamed Hamada,
Mahmoud Raid Mahmoud, Rasha Mohamed Mandouh
E-ISSN: 2224-2880
140
Volume 23, 2024
maximum likelihood estimation and maximization
of the penalized likelihood approach. The results
indicate that both the fitted distributions from
maximum likelihood and penalized likelihood
estimation reasonably well but the extreme value of
the maximum likelihood estimate seems strange and
the penalized likelihood would be preferable, see
Figure 5. The results of the simulation study show
that the bias for any estimator decreases when the
sample size increases. Also, the relative bias
decreases when the sample size increases. The mean
square error (MSE) decreases when the sample size
increases.
References:
[1] F.M. Almeida , E.A. Colosimo, V.D.
Mayrink, Prior specifications to handle the
monotone likelihood problem in the Cox
regression model, Statistics and Its Interface,
Vol.11, pp.687-698, 2018.
[2] V.M.C. Lima, F. Cribari-Neto, Penalized
Maximum Likelihood Estimation in the
Modified Extended Weibull Distribution,
Communications in Statistics - Simulation and
Computation, Vol.48, pp.334-349, 2017.
DOI: 10.1080/03610918.2017.1381735.
[3] R. Shanker, A. Mishra, A quasi Lindley
distribution, African Journal of Mathematics
and Computer Science Research, Vol.6, No.4,
pp.64-71,2013. DOI:
10.5897/AJMCSR12.067.
[4] D. Firth, Bias reduction of maximum
likelihood estimates, Biometrika, Vol.80,
No.1, pp.2738, 1993.
[5] J. Orloff, J. Bloom, Bootstrap confidence
intervals, Retrieved from MIT Open Course
Ware, 2014, [Online].
https://ocw.oouagoiwoye.edu.ng/courses/math
ematics/18-05-introduction-to-probability-
and-statistics-spring-
2014/readings/MIT18_05S14_Reading24.pdf
(Accessed Date: March 2, 2024).
[6] D.N. Murthy, Prabhakar, Min Xie, Renyan
Jiang, Weibull models, John Wiley & Sons,
2004.
[7] S. Nadarajah, F. Haghighi, An extension of
the exponential distribution, Statistics, Vol.45,
No.6, pp.543-558, 2011. DOI:
10.1080/02331881003678678.
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The authors equally contributed to the present
research, at all stages from the formulation of the
problem to the final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflicts of Interest
The authors have no conflicts of interest to declare.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
_US
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2024.23.16
Marwa Mohamed Hamada,
Mahmoud Raid Mahmoud, Rasha Mohamed Mandouh
E-ISSN: 2224-2880
141
Volume 23, 2024
APPENDIX
Table 4. Mean, bias, relative bias, and MSEs of α and θ using the Jeffreys prior penalization with η =0.1, ...,
0.9, 2.1; samples of size n = 50 drawn from quasi Lindley(0.15,0.35) distribution
Maximization of penalized likelihood estimation MPLEs
η
parameter
Mean
Bias
Relative bias(%)
Variance
MSE
0.1
α
0.18483
-0.03482
-23.21680
0.06308
0.06429
θ
0.36631
-0.01630
-9.95005
0.00134
0.00161
0.3
α
0.17809
-0.02809
-18.73160
0.04392
0.04471
θ
0.36648
-0.01648
-4.71001 8
0.00131
0.00158
0.5
α
0.16947
-0.01948
-12.98500
0.03483
0.03521
θ
0.36734
-0.01734
-4.95467
0.00138
0.00168
0.7
α
0.16152 -
-0.01152
-7.67770
0.02776
0.02789
θ
0.36706
-0.01706
-4.87368
0.00127
0.00155
0.9
α
0.15504
-0.00504
-3.36441
0.01924
0.01927
θ
0.36685
-0.01685 -
-4.81674
0.00115
0.00143
1
α
0.15144
-0.00144
-0.96075
0.01732
0.01732
θ
0.36733
-0.01733
-4.95077
0.00130
0.00160
1.1
α
0.14793
0.00207
1.38346
0.01409
0.01410
θ
0.36607
-0.01606
-4.59021
0.00098
0.00124
1.3
α
0.13945
0.01055
7.03518
0.00983
0.00995
θ
0.36611
-0.01611
-4.60323
0.00095
0.0012
1.5
α
0.13501
0.01498
9.98851
0.00654
0.00677
θ
0.36478
-0.01478
-4.22347
0.00088
0.00110
1.7
α
0.13563
0.01437
9.58050
0.00358
0.00379
θ
0.36209
-0.01209
-3.45639
0.00073
0.00088
1.9
α
0.13912
0.01088
7.25475
0.00191
0.00203
θ
0.35856
-0.00856
-2.44487
0.00052
0.00059
2
α
0.14250
0.00749
4.99671
0.00157
0.00163
θ
0.35630
-0.00631
-1.80224
0.00058
0.00062
2.1
α
0.14980
0.00019
0.12878
0.00001
0.00001
θ
0.35118
-0.00118
-0.33779
0.00001
0.00001
Table 5. Mean, bias, relative bias, and MSEs of α and θ using the Jeffreys prior penalization with η = 0.1,...,
0.9, 2; samples of size n = 100 drawn from quasi Lindley(0.15,0.35) distribution
Maximization of penalized likelihood estimation MPLEs
η
parameter
Mean
Bias
Relative bias(%)
Variance
MSE
0.1
α
0.16601
-0.01601
-10.67440
0.01355
0.013810
θ
0.36077
-0.01077
-4.57475
0.00055
0.00067
0.3
α
0.16480
-0.01480
-9.86835
0.01290
0.01312
θ
0.36079
-0.01079
-3.08368
0.00055
0.00067
0.5
α
0.16277
-0.01278
-8.51867
0.01219
0.01236
θ
0.36085
-0.01085
-3.09970
-0.00054
0.00066
0.7
α
0.15992
-0.00992
-6.61390
0.01136
0.01146
θ
0.36101
-0.01101
-3.14560
0.00055
0.00067
0.9
α
0.15627
-0.00627
-4.18472
0.00993
0.00997
θ
0.36113
-0.01112
-3.17966
0.00053
0.00065
1
α
0.15397
-0.00397
-2.64632
0.00926
0.00928
θ
0.36120
-0.01120
-3.20120
0.00926
0.00065
1.1
α
0.15180
-0.00180
-1.20134
0.00820
0.00820
θ
0.36110
-0.01110
-3.17157
0.00051
0.00063
1.3
α
0.14724
0.00276
1.84130
0.00587
0.00588
θ
0.36042
-0.01041
-2.97690
0.00043
0.00054
1.5
α
0.14086
0.00914
6.09684
0.00445
0.00454
θ
0.36010
-0.01010
-2.88679
0.00042
0.00052
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2024.23.16
Marwa Mohamed Hamada,
Mahmoud Raid Mahmoud, Rasha Mohamed Mandouh
E-ISSN: 2224-2880
142
Volume 23, 2024
Maximization of penalized likelihood estimation MPLEs
1.7
α
0.13962
0.01038
6.91927
0.00243
0.00253
θ
0.35867
-0.00867
-2.47602
0.00033
0.00041
1.9
α
0.14055
0.00945
6.29844
0.00124
0.00134
θ
0.35659
-0.00659
-1.88260
0.00024
0.00028
2
α
0.14533
0.00467
3.11477
0.00050
0.00053
θ
0.35356
-0.00356
-1.01647
0.00012
0.00013
Table 6. Mean, bias, relative bias, and MSEs of α and θ using the Jeffreys prior penalization with η = 1, ..., 2.4;
samples of size n = 50 drawn from quasi Lindley(0.15,0.35) distribution
Maximization of penalized likelihood estimation MPLEs
η
parameter
Mean
Bias
Relative bias(%)
Variance
MSE
1
α
0.15950
0.00950
-6.31090
0.02020
0.02030
θ
0.36500
0.01500
-4.29160
0.00110
0.00130
1.5
α
0.14090
0.00900
6.02230
0.00680
0.00680
θ
0.36270
0.01270
2.58090
0.00070
0.00090
2
α
0.14490
0.00500
3.35890
0.00120
0.00120
θ
0.35490
0.00490
-1.38710
0.00030
0.00040
2.1
α
0.14760
0.00234
1.56130
0.00059
0.00060
θ
0.35270
0.00270
-0.77530
0.00016
0.00017
2.3
α
0.14940
0.00060
0.38790
0.00023
0.00023
θ
0.35130
0.00130
-0.37980
0.00006
0.00006
2.4
α
0.14920
0.00084
0.56130
0.00014
0.00014
θ
0.35160
0.00160
-0.45850
0.00005
0.00006
Table7. Mean, bias, relative bias, and MSEs of α and θ using the Jeffreys prior penalization with η = 1, ..., 2.4;
samples of size n = 100 drawn from quasi Lindley(0.15,0.35) distribution
Maximization of penalized likelihood estimation MPLEs
η
parameter
Mean
Bias
Relative bias(%)
Variance
MSE
1
α
0.154600
-0.004560
-3.045800
0.011270
0.011290
θ
0.360900
-0.010870
-3.106900
0.000520
0.000640
1.5
α
0.140300
0.009700
6.464300
0.005210
0.005310
θ
0.360100
-0.010100
2.770400
0.000420
0.000520
2
α
0.143700
0.006300
4.198600
0.001090
0.001130
θ
0.354600
-0.004600
-1.302100
0.000210
0.000230
2.1
α
0.145800
0.004130
2.755360
0.000700
0.000720
θ
0.353200
-0.003250
-0.928070
0.000120
0.000130
2.3
α
0.149700
0.000320
0.211830
0.000020
0.000020
θ
0.351200
-0.001190
-0.341040
0.000009
0.000011
2.4
α
0.149900
0.000099
0.066653
0.000009
0.000010
θ
0.351100
-0.001049
-0.299887
0.000002
0.000003
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2024.23.16
Marwa Mohamed Hamada,
Mahmoud Raid Mahmoud, Rasha Mohamed Mandouh
E-ISSN: 2224-2880
143
Volume 23, 2024
Table 8. Mean, bias, relative bias, and MSEs of α and θ using the Jeffreys prior penalization with η = 0.5, ..., 3;
samples of size n = 50 drawn from quasi Lindley (0.2,1) distribution
Maximization of penalized likelihood estimation MPLEs
η
parame
ter
Mean
Bias
Relative
bias(%)
Varian
ce
MSE
0.5
α
0.450304
-0.150304
-50.10140
0.108400
0.130991
θ
2.009500
-0.009497
-7.51521
0.055801
0.055892
1
α
0.38009
-0.08009
-26.69830
0.041559
0.047975
θ
2.02771
-0.02771
-1.73201
0.049741
0.050509
1.5
α
0.326186
-0.026185
-8.72861
0.012007
0.0126931
θ
2.03464
-0.034640
-1.73201
0.037606
0.0388056
1.8
α
0.311974
-0.011970
-3.99145
0.003673
0.0038159
θ
2.025710
-0.025706
-1.28530
0.022137
0.0227987
1.9
α
0.308770
-0.008778
-2.92613
0.003085
0.0031621
θ
2.020240
-0.020241
-1.01203
0.018151
0.0185607
2
α
0.309824
-0.009824
-3.27473
0.001081
0.0011772
θ
2.013650
-0.013651
-0.682575
0.009564
0.0097504
2.2
α
0.19608
0.00392
1.96120
0.003428
0.003444
θ
0.98696
0.01303
1.30359
0.004404
0.004574
2.3
α
0.19926
0.00074
0.37008
0.002220
0.002221
θ
0.98919
0.01091
1.08139
0.003566
0.003683
2.4
α
0.20058
-0.00058
-0.29079
0.002316
0.002317
θ
0.99251
0.00749
0.74922
0.002258
0.002314
2.5
α
0.19975
0.00024
0.12271
0.002909
0.002909
θ
0.99229
0.00771
0.77109
0.002054
0.002114
2.6
α
0.19845
0.001546
0.77301
0.003736
0.003738
θ
0.99162
0.008382
0.83829
0.002035
0.002106
2.7
α
0.19587
0.004133
2.06671
0.011602
0.011619
θ
0.99081
0.009188
0.91879
0.003187
0.003272
3
α
0.20026
-0.000268
-0.13400
0.003575
0.003575
θ
0.99119
-0.008812
0.88120
0.002959
0.003036
Table 9. Mean, bias, relative bias, and MSEs of α and θ using the Jeffreys prior penalization with η = 0.5, ..., 3;
samples of size n = 50 drawn from quasi Lindley (0.8,1.5) distribution
Maximization of penalized likelihood estimation MPLEs
η
parame
ter
Mean
Bias
Relative
bias(%)
Varia
nce
MSE
0.5
α
0.90498
-0.104976
-13.1220
0.36014
0.37116
θ
1.51909
-0.019092
-6.9984
0.04841
0.04877
1
α
0.70489
0.095103
11.8879
0.11717
0.12622
θ
1.54101
-0.041015
-2.7343
0.04637
0.04805
1.5
α
0.54894
0.251058
31.3823
0.04286
0.10589
θ
1.54450
-0.044498
-2.9665
0.05463
0.05661
1.8
α
0.44183
0.358173
44.77170
0.03299
0.16128
θ
1.53350
-0.033496
-2.23309
0.06908
0.07019
1.9
α
0.40854
0.391457
48.9321
0.03138
0.18462
θ
1.51763
-0.017631
-1.17541
0.07183
0.07214
2
α
0.38089
0.419102
52.38770
0.03326
0.18462
θ
1.50055
-0.000550
-0.03667
0.07109
0.07109
2.2
α
0.38767
0.412329
51.54110
0.03783
0.20784
θ
1.42797
0.072031
4.80204
0.05028
0.05547
2.3
α
0.41584
0.384157
48.0196
0.03749
0.18507
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2024.23.16
Marwa Mohamed Hamada,
Mahmoud Raid Mahmoud, Rasha Mohamed Mandouh
E-ISSN: 2224-2880
144
Volume 23, 2024
Maximization of penalized likelihood estimation MPLEs
θ
1.38042
0.119577
7.97179
0.03891
0.05321
2.4
α
0.42318
0.376817
47.1021
0.04113
0.18312
θ
1.33976
0.160236
10.6824
0.02986
0.05554
2.5
α
0.35983
0.44017
55.0215
0.04649
0.24025
θ
1.29949
0.20051
13.3675
0.02914
0.06935
2.6
α
0.27353
0.52647
65.8086
0.03977
0.31694
θ
1.23240
0.26759
17.8398
0.03891
0.11052
Table 10. Mean, bias, relative bias, Variance, and MSEs of α and θ using the Jeffreys prior penalization with η
= 2.4; by sample size n drawn from QL(α, θ)
Maximization of penalized likelihood estimation MPLEs
n
parameter
Mean
Bias
Relative bias(%)
Variance
MSE
50
α =2
1.17230
0.82770
41.38610
0.40310
1.08830
θ =2.5
2.32470
0.17530
7.01080
0.39680
0.42750
75
α =2
1.18497
0.81503
40.75150
0.360749
1.02502
θ =2.5
2.35645
0.14355
5.74201
0.279884
0.30049
100
α =2
1.20750
0.79260
39.62770
0.23990
0.86810
θ =2.5
2.38680
0.11320
4.52980
0.22120
0.23400
50
α=3
2.88895
0.11105
0.16535
3.70154
0.17768
θ=0.5
0.50559
-0.00559
0.00179
-1.11774
0.00183
75
α=3
2.90021
0.00979
0.15757
3.32639
0.16753
θ=0.5
0.50697
-0.00697
0.00128
-1.39373
0.00133
100
α=3
2.91664
0.08335
0.11667
2.77850
0.12362
θ=0.5
0.50763
-0.00763
0.00092
-1.52563
0.00098
50
α=0.3
0.30982
-0.00982
-3.27270
0.00003
0.0001300
θ=2
2.00085
-0.00085
-0.04256
0.00065
0.0006500
75
α=0.3
0.30996
-0.00996
-3.32054
0.000015
0.0001144
θ=2
1.99976
0.00023
0.01193
0.00039
0.0003900
100
α=0.3
0.31002
-0.01001
-3.33827
0.000017
0.0001140
θ=2
1.99975
0.00025
0.01272
0.00017
0.0001700
Table 13. Mean, bias, relative bias, and MSEs of α and λ using the Jeffreys prior penalization with η = 0.1, ...,
2.5; samples of size n = 50 drawn from N-H (1,1) distribution
Maximization of penalized likelihood estimation MPLEs
η
parameter
Mean
Bias
Relative bias(%)
Variance
MSE
0.1
α
1.3735
-0.3735
-37.3513
0.1098
0.2479
λ
1.4420
-0.4420
-44.2004
0.0479
0.2433
0.3
α
1.3303
-0.3303
-33.0334
0.11707
0.2262
λ
1.4420
-0.4473
-44.7334
0.04875
0.2489
0.5
α
1.2508
-0.2508
-25.0841
0.1118
0.1747
λ
1.4590
-0.4590
-45.9007
0.0503
0.2609
0.7
α
1.1644
-0.1644
-16.4447
0.0910
0.1181
λ
1.4690
-0.4690
-46.9044
0.0501
0.2701
0.9
α
1.0803
-0.0802
-08.0255
0.0728
0.0792
λ
1.4760
-0.4760
-47.6049
0.0501
0.2767
1
α
1.0364
-0.0364
-3.6436
0.0646
0.0659
λ
1.4797
-0.4797
-47.9681
0.0503
0.2804
1.1
α
0.9923
0.0077
0.7735
0.0584
0.0584
λ
1.4832
-0.4832
-48.3228
0.0518
0.2853
1.2
α
0.9494
0.0506
05.0582
0.0519
0.05448
λ
1.4858
-0.4858
-48.5765
0.0545
0.2876
1.3
α
0.9109
0.0891
8.9068
0.0471
0.0551
λ
1.4869
-0.4869
-48.6989
0.0541
0.2912
1.5
α
0.8009
0.1990
19.9022
0.0458
0.0854
λ
1.4964
-0.4964
-49.6403
0.0556
0.3020
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2024.23.16
Marwa Mohamed Hamada,
Mahmoud Raid Mahmoud, Rasha Mohamed Mandouh
E-ISSN: 2224-2880
145
Volume 23, 2024
Maximization of penalized likelihood estimation MPLEs
2.1
α
0.5595
0.4405
44.0494
0.0182
0.2123
λ
1.4502
-0.4502
-45.0206
0.0489
0.2515
2.3
α
0.5291
0.4709
47.0890
0.0219
0.2453
λ
1.3348
-0.3347
-33.4756
0.0453
0.1574
2.5
α
0.4737
0.5263
52.6331
0.0365
0.3135
λ
1.1460
-0.1460
-14.6047
0.0361
0.0575
Table 14. Mean, bias, relative bias, and MSEs of α and λ using the Jeffreys prior penalization with η = 1.2;
samples of size n = 25, 50 drawn from N-H (0.5,1) distribution
Maximization of penalized likelihood estimation MPLEs
n
parameter
Mean
Bias
Relative bias(%)
Variance
MSE
25
α =0.5
0.514848
-0.014848
-2.969670
0.001464
0.001684
λ =1
1.121760
-0.121758
-12.175800
1.645470
1.660300
50
α =0.5
0.519715
-0.019716
-3.943090
0.000097
0.000486
λ =1
1.012780
-0.012777
-1.277740
0.320775
0.320939
Table 15. Mean, Confidence Interval CI for estimation of α and λ using the Jeffreys prior penalization with η =
1.2; by sample size drawn from N-H(0.5,1) distribution
n
parameter
Mean
Bias
25
α =0.5
0.514848
[ 0.513834,0.515385]
λ =1
1.12176
[1.10871,1.14927]
50
α =0.5
0.519715
[0.509496,0.520078]
λ =1
1.01278
[1.00012,1.02556]
Table 16. Mean, bias, relative bias, and MSEs of α and λ using the Jeffreys prior penalization with η = 1.2; by
sample size drawn from N-H (0.5,0.5) distribution
Maximization of penalized likelihood estimation MPLEs
n
parameter
Mean
Bias
Relative bias(%)
Variance
MSE
25
α =0.5
0.477388
0.022612
4.522420
0.008701
0.009213
λ =0.5
0.863929
-0.363929
-72.785800
2.813260
2.945700
50
α =0.5
0.490334
0.009666
1.933210
0.005776
0.005869
λ =0.5
0.744447
-0.244447
-48.889300
2.620230
2.679980
100
α =0.5
0.507113
-0.007113
-1.422650
0.002242
0.002293
λ =0.5
0.552156
-0.052156
-10.431100
0.442650
0.445370
150
α =0.5
0.513666
-0.013666
-2.733170
0.001005
0.001192
λ =0.5
0.514280
-0.014280
-2.856060
0.070553
0.070757
Table 17. Mean, bias, relative bias, and MSEs of α and λ using the Jeffreys prior penalization with η = 1.2; by
sample size drawn from N-H (1,1) distribution
Maximization of penalized likelihood estimation MPLEs
n
parameter
Mean
Bias
Relative bias(%)
Variance
MSE
25
α =1
0.591546
0.408454
40.845400
0.0185191
0.185354
λ =1
2.235650
-1.235650
-123.565000
19.977700
21.504600
50
α =1
0.625168
0.374832
37.483200
0.009941
0.150440
λ =1
1.671450
-0.671453
-67.145300
15.553100
16.003900
100
α =1
0.646386
0.353614
35.361400
0.005107
0.130150
λ =1
1.444810
-0.444815
-44.481500
11.893600
12.091500
150
α =1
0.658156
0.341844
34.184400
0.003197
0.120054
λ =1
1.201460
-0.201456
-20.145600
9.629080
9.669660
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2024.23.16
Marwa Mohamed Hamada,
Mahmoud Raid Mahmoud, Rasha Mohamed Mandouh
E-ISSN: 2224-2880
146
Volume 23, 2024
