![](data:image/png;base64,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)
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
And confidence interval for parameters of the
Nadarajah-Haghighi exponential distribution in the
case of penalized likelihood estimation is given by
CI for
CI for
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