Abstract
The type II half logistic length-biased exponential distribution, which extends the length-biased exponential model, is introduced and examined. This model's statistical features, such as moments, conditional moments, mean order statistics, and entropy, are derived. To estimate distribution parameters, statistical inference employs three estimation methods: maximum likelihood, Cramer–von Mises, and Anderson–Darling. To evaluate the performance of the aforementioned estimators, we run simulation experiments based on the graphical results. The TIIHLLBE model's adaptability has been demonstrated through applications to real-world datasets.
1. Introduction
In the last years, many different statisticians have attracted the attention of generated families of distributions such as Kumaraswamy-G by Cordeiro and De Castro [1], sine-G by Kumar et al. [2], type II half logistic-G [3], exponentiated M-G by Bantan etb al. [4], and Topp-Leone odd Fréchet by Al-Marzouki et al. [5], among others.
A new family of continuous distributions with a different scale parameter θ > 0 called the type II half logistic-G (TIIHL-G) family was studied by Hassan et al. [3]. The cumulative distribution function (cdf) of TIIHL-G is expressed bywhere is cdf of baseline model with parameter vector and is cdf derived by the T-X generator proposed by Dara and Ahmad [6]. The pdf of the TIIHL-G family is given asrespectively. A random variable has pdf (2) which will be defined as Dara and Ahmad [6] proposed the length-biased exponential (LBE) model (or called moment exponential (ME) model) through assigning weight to the exponential (E) model by following the idea of Fisher [7]. They demonstrated that the LBE distribution is more flexible than the exponential (E) model. The cdf and pdf areandrespectively, where is a scale parameter. In this paper, we suggest an extension of the LBE model; this extension is constructed by using the TIIHL-G family and LBE model, the so-called type II half logistic length-biased exponential (TIIHLLBE) distribution.
A nonnegative random variable x has TIIHLLBE distribution with two parameters β, θ > 0 which is constructed by inserting (3) and (4) in (1) and (2), and we get the cdf and pdf, respectively:and
The remainder of this article is outlined as follows. In Section 2, the linear representation of TIIHLLBE pdf and cdf is presented. The reliability measures of TIIHLLBE model are discussed in Section 3. Basic properties of distribution inclusive moments, moment generating function, the conditional moment, entropy, and probability-weighted moments are calculated in Section 4. In Section 5, its order statistics are derived. Parameter estimation by maximum likelihood (ML), Cramer–von Mises (CVM), and Anderson–Darling (AD) estimation methods is discussed in Section 6. Monte Carlo simulation schemes are performed in Section 7. In Section 8, we applied real datasets to investigate the potentiality of the TIIHLLBE by using some measures of goodness of fit such as the Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC), Consistent Akaike Information Criterion (CAIC), Hannan–Quinn Information Criterion (HQIC), and Kolmogorov–Smirnov (KS) statistic (with its p value (PV). Finally, conclusion is given in Section 9.
2. Linear Representation
In this section, a linear representation of the pdf and cdf is presented to introduce statistical properties of TIIHLLBE distribution. Use the next binomial expansionwhere |z|<1 and b is a positive real noninteger. By applying (7) in the next term, we get
Inserting the previous equation in (6), we have
Again applying the general binomial theorem, we get
Again using the binomial expansion, we getwhere
A linear representation for the cdf can be calculated by using binomial expansion for , where s is an integer:
By applying (7) in the next term, we get
Then,
Again applying the binomial expansion to the next term,
Then,
Again inserting the binomial expansion, we getwhere
3. The Reliability Measures of Length-Biased Exponential Distribution
In this section, we introduce the reliability measures such as survival function, hazard rate function, reversed hazard rate function, residual, and reversed residual lifetime. The reliability function (survival function) of TIIHLLBE distribution is given by
The hazard rate (hrf) function or failure rate and reversed hrf for the TIIHLLBE are obtained byand
The plots of the pdf, cdf, and survival and hazard rate functions of TIIHLLBE with different values of parameters are mentioned in Figures 1–4.
The moment of the residual life, say , is given by
Then,where is the upper incomplete gamma function.
Now, the moment of the reversed residual life is calculated as
Then,where is the lower incomplete gamma function.
4. Properties
In this section, we introduce some properties of TIIHLLBE distribution such as moments, moment generating function, the conditional moment, entropy, and probability-weighted moments (PMWs).
4.1. Moments
The moment of x denoted by can be derived from (6) as follows:
Then,
When r = 1, we get .
The moment generating function of x is given from (6) as
Numerical values for specific values of parameters of the first four ordinary moments: variance skewness (SK), and kurtosis (KU) of the TIIHLLBE model are listed in Tables 1 and 2.
4.2. Conditional Moment
Let X denote a random variable with the pdf given in (6). The upper incomplete moment, say , can be derived aswhere is the upper incomplete gamma function.
Similarly, the lower incomplete moment function is given bywhere is the lower incomplete gamma function.
4.3. Entropy
If x has the pdf given in (6), then the Rényi entropy is defined by
By using (6), then
By using the binomial expansion more times for the last equation, we getwhere Then,
The Rényi entropy of TIIHLLBE distribution is
4.4. Probability-Weighted Moments
The PWMs of X, say , are defined by the formula
The PWMs of TIIHLLBE distribution are obtained by substituting (11) and (17) into (35), and is defined as follows:
Then,
5. Order Statistics
Order statistics have been used in many applications of statistics, such as life testing and reliability. Let be a random sample of size from the pdf given by (6) and the cdf given by (5). Let be the ordered random sample from a TIIHLLBE of size n. Then, the of , is given by
By inserting (11) and (17) in (38) and replacing s by , we getwhere
Now, we have derived an exact expression for the single moments of order statistics from TIIHLLBE. We shall first establish the exact expression for . Using (39), we obtain the moment of order statistics as follows:
Then, we get
For , in (41), we get mean of order statistics.
The joint of and , , is given bywhere
The exact expression for . Using (43), we obtain the and moments of order statistics as follows:
By using the binomial expansion, we getwhere
By using (11) and (17) and replacing s by , we get
Substituting the value of in (45) and simplifying the resulting equation, we get
6. Statistical Inference
In this section, we propose several approaches to estimate the parameters and of TIIHLLBE distribution model using maximum likelihood (ML), Cramer–von Mises (CVM), and Anderson–Darling (AD) methods of estimation.
6.1. Method of Maximum Likelihood
Let be a random sample of size n from TIIHLLBE model with parameters and , and the log-likelihood function is
For calculation of MLE estimation, we need partial derivatives of by parametersandwhere The estimations of the parameters can be found by as solution of the system
6.2. Method of Minimum Distance
In this part, we consider well-known CVM-type minimum distance estimators [8–10], which are based on the next statistic:
Using form of the objective function, we can find thatandwhere The estimations of the parameters can be found by as solution of the system
6.3. Anderson–Darling Method
The method of Anderson–Darling estimation was studied in [11] in the context of statistical tests.where
For minimizing the function, , we need to find partial derivatives by parameters :andwhere . In results estimations of the parameters can be find as solution of the system
7. Monte Carlo Simulation
A Monte Carlo simulation study is studied to evaluate the behavior of the estimates with respect to their bias and root mean square error (RMSE). We calculate next characteristic of the estimations:
Consider results of the Monte Carlo simulation for parameters with generated random sample simulations from TIIHLLBE distribution and sample size .
The bias and RMSE of the estimates of the unknown parameters are evaluated. Monte Carlo simulation results are mentioned in Table 3.
8. Applications
In this section, we compare the TIIHLLBE with some other known competitive models to demonstrate its importance in data modeling. The MLE method is used to estimate the parameters of the competitive models. The AIC, BIC, CAIC, HQIC, KS, and PV model selection criteria and goodness-of-fit tests are used to choose best model.
Data I. The first dataset represents the survival times of 72 guinea pigs (in days) infected with virulent tubercle bacilli, and these data were introduced by Bjerkedal [12].
Data II. The second dataset (called hazard time data) consists of 20 patients referring to the lifetime and relating to relief times (in minutes). Patients who received an analgesic were introduced by Gross and Clark in [13].
Some descriptive statistics of both datasets are mentioned in Table 4, and the boxplots are shown in Figure 5.
The total time test (TTT) plot (see [14]) is an important plot approach to verify whether the data can be applied to a specific model or not. The TTT plots of two real datasets are shown in Figure 6, and this plot shows that the empirical hrf of both datasets is increasing, as shown in Figure 7.
We will compare the fits of the TIIHLLBE model with those of other competitive models, namely: Marshall-Olkin E (MOE), Burr X-E (BrXE), Kumaraswamy E (KuE), beta E (BE), Kumaraswamy Marshall-Olkin E (KuMOE), generalized Marshall-Olkin E (GMOE), Marshall-Olkin Kumaraswamy E (MOKuE), ME, and E models (see [15]).
The MLEs and standard errors (SEs) are calculated via these two datasets, and numerical results are mentioned in Tables 4 and 5. Depending on the numerical values which is obtained in Tables 6 and 7 along with Figures 8 and 9, the TIIHLLBE model is much better than the above-mentioned extensions of the E model, so the TIIHLLBE model is a good alternative to these models in both datasets (Table 8).
The parameters of TIIHLLBE distribution are estimated by using three different estimation methods, ML, CVM, and AD estimation methods. The efficiency of the estimation methods is the same in the two datasets as shown in Tables 9 and 10. We see that the CVM estimation method is the best among the others in the two datasets.
9. Concluding Remarks
In this article, a new model, called TIIHLLBE distribution, is introduced. Some statistical properties of the proposed distribution such as moments, order statistics, conditional moments, probability-weighted moment, and Rényi entropy are derived and discussed. The estimation of the model parameters is discussed through ML, CVM, and AD methods. Monte Carlo simulation study is carried out to compare the performance of the three different estimates. Applications to two real datasets indicate that the new model is superior to the fits when compared to the other well-known distributions.
Data Availability
The numerical dataset used to support the findings of this study is available from the corresponding author upon request.
Conflicts of Interest
The author declares that there are no conflicts of interest to disclose in relation to this work.