A Discrete Odd Lindley Half-Logistic Distribution with Applications

In the reality, the most of lifetime data is discrete in nature, despite it is known that this data is continuous. As a consequence, the tendency to convert the continuous lifetime distributions to its discrete counterpart has raised. The nature of the most lifetime data in real life is discrete, thus, many of researchers interested to convert. This article aims to introduce a new one-parameter discrete distribution, namely Discrete Odd Lindley Half-Logistic (DOLiHL) distribution. The DOLiHL distribution is derived by discretizing the analogue continuous distribution, using a survival discretization technique. An intensive study of this distribution is provided, including characteristic functions and statistical properties. Moreover, the unknown parameter is estimated using different methods of estimation namely, the maximum likelihood method, method of moments, least squares method and Cramer-von Mises minimum distance estimation method. A simulation study is conducted to examine the distribution’s behavior and compare between the estimation methods at different sample sizes. To evaluate the efficiency of this distribution against other competitive distributions, an implementation on real data in different fields is performed. In conclude, the DOLiHL distribution has assured its efficiency of fitting in lifetime and count data from various fields in real life.


Introduction
In the real-life, the data sets are usually discrete so it is more appropriate to deal with discrete distribution.In the recent decade, driving a discrete distribution from its analogue continuous distribution has great attention in the statistical literature, according to its significant role in analyzing lifetime, reliability and counts data sets.As the demand of discretization is increasing, new methods of discretization are arising by researchers.Accordingly, there are many methods of discretization developed in the literature.Furthermore, [1] studied a new method called the Minimum Optimal Description Length (MODL) based on Bayesian approached which has an important role in data mining , machine learning, and analyzing data.[2] presented different methods of generating discrete probability distributions as analogues of continuous probability distributions with their applications in construction of new discrete distributions.As an extension [3] introduced three new two-stage composite discretization methods to meet the demand of fitting lifetime distributions.
Several researches studied the generated approach of discertization of the continuous distributions to utilise as lifetime distributions to meet the need of lifetime models.[4] converted the continuous Burr and Pareto distributions to their discrete analogues.They presented their properties and the estimation of their unknown parameters.Moreover, applications in series variables are grouped into the unit intervals then the DRV of Y is equal to the largest integer less than or equal Y .The generated random variable is a DRV and denoted by X, its PMF could be constructed by using SF technique.Then we will have the probability mass function PMF as the following: P (X = x) = P (x ≤ X < x + 1) = F (x + 1) − F (x) = S(x) − S(x + 1); x = 0, 1, 2, . . .
Consider the CRV Y follows an OLiHL distribution with parameter λ, then its PDF, and CDF are given as follows respectively: Also, the corresponding SF of the continuous OLiHL distribution is given by The analogue DOLiHL distribution is derived here using the SF discertization technique.By using (1) the PMF of the DOLiHL distribution is The corresponding CDF is given by The SF of x follows DOLiHL distribution is given by The HRF is given by The alternative hazard function (AHRF) is given by using the alternative definition giving by [14].
The reversed hazard rate function (RHRF) is defined as For more details see [15] and [16].
The possible shapes of the PMF, HRF, AHRF and RHRF are presented in Figure 1, Figure 2, Figure 3 and Figure 4 which show that the DOLiHL distribution could be a proper distribution for right-skewed, negative-skewed and unimodal.
Figure 1 shows the behavior of the PMF of the DOLiHL distribution at different values of λ.The curve of the distribution affected clearly by changing the value of the shape parameter λ.The distribution is positively skewed as λ is increased.In particular, at λ = 0.65 the distribution tend to be almost symmetric.
Figure 2 and Figure 3 show the behavior of the HRF and AHRF of the DOLiHL distribution at different values of λ.The distribution is negatively skewed as λ is decreased.
Figure 4 shows the behavior of the RHRF of the DOLiHL distribution at different values of λ.The distribution is positively skewed as λ is increased, whereas, the distribution is negatively skewed as λ is decreased.

Moments and related measures
The non-central moments of the DOLiHL distribution is given by x r i (λe Thus, the mean of the DOLiHL distribution is obtained when r = 1 in equation (11) as the following, Figure 1: The PMF of DOLiHL distribution using different values of λ.
Furthermore, the next three non-central moments are x 2 i (λe As a result, the variance for the DOLiHL distribution is giving by calculating the following  formula, x 2 i (λe IC-MSQUARE-2023 Journal of Physics: Conference Series 2701 (2024) 012034 The central moments of the DOLiHL distribution reached using the giving formula Several remarkable relationships between tow types of moments are written below:

Skewness and Kurtosis
The skewness and kurtosis of the DOLiHL can be found by using the moment precept formula and the relationships between them as the following From Table 1 and Figure 5, it is clear that when our shape parameter λ increased to reaches unity, the mean and variance for the DOLiHL distribution decreased.On the other hand, the skewness is increased along with the increasing of λ which indicates that our model is more appropriate for negative skewed data.Furthermore, kurtosis is usually less than three which means Platykurtic or (short-tailed) distributions appropriate for our proposed model.

Index of Dispersion and Coefficient of Variation Measures
The index of dispersion (IOD) is to evaluated the dataset either under or over dispersed.If the IOD is grater than one, it indicates the over dispersion.In contrast, if the IOD is less than one it indicates under dispersion.The IOD for the DOLiHL distribution is defined as below, Another measurement to evaluate the variability in data is the coefficient of variation (COV) by comparing the independent samples variability, the higher COV the higher variability.The COV for the DOLiHL distribution is giving by, Table 2 shows, that the IOD and COV increased with the increasing in our parameter.However, the increasing in COV means that we have higher variability.In addition, the IOD is less than one which indicates that the DOLiHL model is more appropriate for under dispersion dataset.

The Moment Generating Function
The moment generating function of the DOLiHL is given by e tx i (λe

Order Statistics
Let x 1 , x 2 , . . ., x n are DRV follow the DOLiHL distribution, then the corresponding order statistics are x 1:n , x 2:n , . . ., x n:n .Then, the CDF Order statistic of the DOLiHL distribution is given by, The corresponding PMF The rth moments of (25)

Quantile Function
The quantile function QF of the DOLiHL distribution is given by solving the equation F (x) = u for x, then the QF is the inverse of the CDF x = F −1 (u) which is given by where u ∈ (0, 1) and W (.) is the negative branch for Lambert-W function.
Moreover, the first quartile is Q 1 (0.25), the second quartile is Q 2 (0.50) which is also called the median, and the third quartile is Q 3 (0.75) can be obtained by substituting u = 0.25 , u = 0.5 and u = 0.75 in equation ( 23) respectively as follow,

The Parameter Estimation
In this section, we propose the estimation of DOLiHL model parameter by the methods of moment (MOM), maximum likelihood estimation method (MLE), least squares estimation method (LSE) and finally Cramer-von Mises minimum distance method (CvME).

Method of Moments (MOM)
The basic concept for the moments method is equating the population moments to the corresponding sample moments and subsequently solving the equations.As a result of having only one parameter we equating the first population mean with the sample moment respectively as following: Then by equating equations ( 27) and (28) we get, The estimator of λ in (29) can be found by numerical methods as nleqslv function of R program.

Maximum Likelihood Estimation(MLE)
Let X 1 , X 2 , . . ., X n are random variables follow the DOLiHL distribution with unknown parameter λ.The maximum likelihood estimation (MLE) of λ is λ is given by maximizing the log-likelihood function The log likelihood function can be obtain as Then the first partial derivatives of the log likelihood function in Equation (31) with respect to λ and equating the result to zero as follow: The estimator of λ in (32) can be solved numerically such as Newton-Raphson, Nelder-Mead, or by R with optim function, maxLik function and nlm function.

Confidence Interval
The approximate confidence interval for the parameter λ is where Z α 2 is the 100(α/2) th percentile of a standard normal distribution, σ x the standard deviation and the upper limit is U L = λ + Z α 2 σ x on the other hand, the lower limit is

Least Squares Estimation Method (LSE)
Let x (1) , x (2) , . . ., x (n) be the ordered statistics of a DRV from the DOLiHL distribution.Let F (X (j) ) represents the jth order statistics from standard uniform distribution U (0, 1), the jth order statistics of U (0, 1) is distributed as Beta(j, n − j + 1) which is one of the features of beta distribution see [17] and [18].Then we have for Beta(j, n − j + 1) The least square estimation (LSE) of the parameter is given by minimizing the following, The least square estimation (LSE) is given by solving the equation The estimator of λ in (36) can be found by numerical methods as lsmeans package and nlm function in R.

Cramer-von Mises Minimum Distance Method (CvME)
The basic concept of Cramer-von Mises(CvME) is to minimize the difference between the estimate of the cumulative distribution function and the empirical distribution function for more details see [19].Then,the Cramer-von Mises estimator is obtained by minimizing the following function The estimator of λ in (37) can be solved numerically by software's packages and functions in R such as lsmeans, optimise and nlm.

Simulation Study
In this section, a Monto Carol simulation study is carried out, to estimate and examine the unknown distribution parameter numerically.For this purpose, four estimation methods are used which are MOM, MLE, LSE and CvME.To compare and evaluate the efficiency of the proposed estimator of the mentioned methods.Absolute Biases (ABs) and mean square errors (MSEs) are used.Random samples of different sizes (n=20,50,100,200,500,600) are generated from DOLiHL distribution at different values of parameter (λ= 0.5,1.5),with number of iteration (m=1000) using R Program.Moreover, the approximate confidence intervals with confidence level 95% are calculated.The algorithm of this simulation is presented as follows: Step 1: Generating m = 1000 samples of size n = 20, 100, 200, 500, 600 from the discrete DOLiHL distribution with true parameter value λ = 0.5, 1.5.
Step 2: The parameter is estimated by using the MOM, MLE, LSE and CvME for each samples respectively.
Step 3: For each simulation, the average of estimates (AEs), the average biases (ABs), and the mean square errors (MSEs) are computed by: The results are organized in Table 3 and Table 4.
From Table 3 and Table 4 show that the bias and MSE of the parameter approximate to zero as the sample size increased.Although that all estimation methods obtain a good estimation results, the CvME method achives good results in the case of small and large sample sizes.

Applications
In this Section, the importance of the DOLiHL distribution is illustrated by using two count datasets.Dataset I represent the number of deaths from COVID-19 in Saudi Arabia, while dataset II regards hydrology data of the Wheaton River in Canada.
Furthermore, the comparison of goodness of fit of the proposed model has been conducted using some criteria, namely, the log-likleihood value ( l), Akaike information criterion (AIC),Bayesian information criterion (BIC) and Kolmogorov-Smirnov (KS).The efficiency of this model against other competitive's distributions, such as discrete Burr-Hatke (DBH), discrete Rayleigh (DR), discrete inverse Rayleigh (DIR) and discrete Pareto (DPr) is proven.
From Table 5, it is obvious that the DOLiHL distribution has the best results among the other tested distributions, since it has minimum values of AIC and BIC criteria besides, maximum l Furthermore, Table 6 presents the estimator values of λ, KS and P.value for the DOLiHL for Dataset I using different estimation methods.The results indicate that the MOM, LSE and CvME methods are good methods for Dataset I.
The results for the competitive distributions taken from [21].According to Table 7, it is clear that the DOLiHL distribution has better results among the other competitive models, because it has minimum values of AIC and BIC criteria and maximum l.Furthermore, Table 8 present the estimation methods of the DOLiHL distribution which indicate that the LSE and CvME methods behave good fit methods for Dataset II .

Conclusion
This paper introduced a new one parameter distribution called the DOLiHL is driven from its counterpart continuous distribution.Furthermore, the characteristic functions and properties for this distribution are studied.Four estimation methods are used to estimate the DOLiHL parameter.Moreover, a simulation study is performed to illustrate the behavior of the proposed model, it is found that all four methods (MOM, MLE, LSE and CvME) are achieved well as the sample size is increased, however, the CvME method is performed better with the small sample size than other methods.Finally, the efficiency of the DOLiHL distribution against other competitive distributions is proven, using two real data sets from different fields.The proposed distribution consistently better fits than other competing distributions.It is concluded that the DOLiHL distribution is a good and competitive model for modeling real-life data in different fields.

Suggestions for Future Work
For further researches we suggest several ideas which are • Study different discretization methods.
• Estimate DOLiHL distribution's parameter by other estimation methods.
• Create a regression model for DOLiHL distribution with applications.

Figure 3 :
Figure 3: The AHRF of DOLiHL distribution using different values of λ.

Figure 4 :
Figure 4: The RHRF of DOLiHL distribution using different values of λ.

Table 1 :
4. Some descriptive measures for DOLiHL distribution at different values of λ and when n = 100.

Table 2 :
The IOD and COV for DOLiHL distribution at different values of λ and when n = 100.

Table 3 :
Simulation results of DOLiHL distribution using MOM, MLE, LSE and CvME methods at λ = 0.5

Table 4 :
Simulation results of DOLiHL distribution using MOM, MLE, LSE and CvME methods at λ = 1.5

Table 5 :
The goodness of fit for DOLiHL distribution and the other competitive discrete models for Dataset I.

Table 6 :
The estimator values of λ, KS and P.value for the DOLiHL distribution for Dataset I using different estimation methods.

Table 7 :
The goodness of fit for DOLiHL distribution and the other competitive discrete models for Dataset II.

Table 8 :
The estimator values of λ, KS and P.value for the DOLiHL distribution for Dataset II using different estimation methods.