Abstract
This article introduces a new probability model based on reflected parameter called the reflected Pareto (RP) distribution. The key properties of the RP model are investigated. A simulation study of the RP model is conducted to evaluate the performances of its estimators. A real-life application is considered to examine the performance of proposed model. The different criteria are discussed numerically as well as graphically to show the flexibility of the RP model. The exponential weighted moving average control charts based on the maximum likelihood and modified maximum likelihood estimators for the shape parameter of the RP distribution are obtained. Detailed simulation results of proposed charts are performed to examine and analyze the performance of these charts with three in-control average run length values and two sample sizes. Finally, the application of the proposed control charts is shown by considering a real-life data set.
1. Introduction
The applications of Pareto distribution have been discussed in various applied fields such as actuarial science, medical science, and engineering. The estimation methods for the parmeters of the Pareto distribution is discussed in [1]. The estimation of the shape and scale parameters of the Pareto model are obtained through the maximum likelihood (ML) and moment estimations approaches [1]. The comparison between estimation methods of the Pareto parameters is addressed in [2]. The classical and Bayesian estimation for Pareto distribution is reported in [3]. Some basic methods for estimation of parameters of the generalized Pareto distribution are addressed by [4]. Evaluation of various generalized Pareto probability distributions for flood frequency analysis are discussed in [5].
Several techniques have been used in the literature to propose new forms of the Pareto model by adding additional parameters for generating new efficient and flexible Pareto models. For example, the Pareto alpha-power Pareto (APP) [6], inverse Pareto (IP) [7], new Kumaraswamy-Pareto (NKP) [8], beta-Pareto (BP) [9], exponentiated Weibull-Pareto [10], among others.
In this paper, a new Pareto distribution is introduced called the reflected Pareto (RP) distribution. The application of RP model is reported on the access of electricity data. We addressed its properties. The proposed model is compared with some competing models based on goodness-of-fit measures including the Kolmogorov–Smirnov (KS) test, its value, Akaike information criteria (AIC), Bayesian information criteria (BIC), consistent Akaike information criteria (CAIC), and Hannan-Quinn information criteria (HQIC) to verify its superiority. The estimation of the shape parameter of the RP distribution is conducted using the ML and modified ML (MML) estimators. Some new modified control charts based on the RP model such as exponential weighted moving average (EWMA) control charts. Visual presentation of the EWMA control charts is presented in both usual and 3D plots.
The new EWMA control chart for monitoring multinomial proportions is suggested in [11]. The Shewhart and EWMA control charts for monitoring the shape parameter of reflected power function distribution are studied in [12]. The are proposed. The comparison of modified Shewhart, EWMA, and HEWMA control charts using Monte Carlo simulation and real‐life applications are presented [13].
This article is arranged in the following sections. In Section 2, the identification of the proposed model is introduced. In Section 3, properties of the RP model are discussed. Estimation of the RP parameters and a simulation study are presented in Section 4. Section 5 provides a real-life data application. The newly proposed control charts are explored in Section 6. The conclusion of this study is discussed in Section 7.
2. Model Identification
The probability density function (PDF) of the Pareto distribution has the formwhere and are shape and scale parameters respectively.
The cumulative distribution function (CDF) of Pareto distribution reduces to
If is a reflected parameter, then by setting in the PDF of Pareto distribution (1), we obtain the PDF of the RP distribution as follows:
The CDF of the RP distribution reduces to
The survival function (SF) and hazard rate function (HRF) of the RP distribution are given by
2.1. Asymptotic Behavior of RP Distribution
In this section, some asymptotic behavior of the RP distribution relevant to PDF, CDF, SF, and HRF are given by(i)(ii)(iii)(iv)(v)(vi)(vii)(viii)
2.2. Special Cases
In this section, some special cases of the RP distribution are addressed.(1)By setting = in the PDF of the RP distribution, we get the Pareto distribution.(2)Setting in the RP density and multiply it by , gives PDF of the power function distribution.
Figure 1 provides the probability distribution loop plot. In Figure 2, the plots show that the RP distribution is reflected image of the Pareto (P) distribution. Figure 3 gives the plots for the CDF and HRF of the RP distribution. The HRF of the RP model is always increasing.
3. Properties
This section presents some properties of RP distribution.
3.1. Moments
The th moments of the RP distribution follows as
3.2. Quantile Function
The quantile function (QF) of the RP distribution has the following form:
Note that can be used to generate random variates from the RP model.
3.3. Order Statistics
Let denotes a random sample of size and denote the order statistics. The PDF of th order statistic, , is obtained as
Then, the PDF of th order statistic of the RP distribution follows as
The PDF of largest order statistic of the RP distribution is obtained as follows:
The PDF of smallest order statistics of the RP distribution reduces to
4. Estimation of Parameters
The parameters of the RP distribution are estimated by two estimation techniques called the maximum likelihood (ML) and modified ML (MML) estimation.
4.1. Maximum Likelihood
The ML estimator (MLE) of the RP parameter follows as
The MLE of the parameter takes the form
As when took partial derivative w.r.t to , it does not exist so likelihood function is maximize by taking
where is maximum value in the data.
4.2. Modified Maximum Likelihood
The MML estimators (MMLEs) of the RP parameters are given by
4.3. Simulation Study
The three combinations of the RP distribution parameters used to compare the two estimation methods, numerically based on simulation results. The results are reported for the three samples sizes 100, 200, and 400. The R software is used to obtain the simulation results based on the following steps.(1)Random values are obtained from the RP distribution for three sample sizes and two methods of estimation.(2)Step 1 is repeated N = 5000 times. The RP parameters are estimated for each parameter combination and each sample, using two estimators including the MML, MMLE.(3)The average values (Avg) and mean square errors (MSE) of estimates are obtained for each parameter combination with each sample.
The results are reported in Table 1. These results show that both estimation methods achieved the consistency property, i.e., the MSE decrease as increases, for all parameter combinations.
5. Real Life Application
In this section, the RP model is fitted using a real-life data set to show its flexibility in modeling real data as compared to other models.
The real data set is available online through the link: . The data refer to “Access to electricity (% of population).” “Access to electricity is the percentage of population with access to electricity. Electrification data are collected from industry, national surveys and international sources.” The data is available for many countries in that file. The data of country “Tunisia” is selected for this study and the data values are: 86.8, 88.7, 90.4, 92, 93.2, 94.2, 94.80000305, 97.30000305, 97.80000305, 98.40000153, 99, 99.30000305, 99.40000153, 99.40000153, 99.40000153, 99.5, 99.5, 99.5, 99.5, 99.69999695, 99.80000305, 99.90000153, 100, 100, 99.80000305, 100, 100.
The proposed model is compared with new Weibull-Pareto (NWP) [14], exponentiated Pareto (EP) [15] and tapered Pareto (TP) [16] distributions. These models are compared with each other on the bases of some goodness-of-fit measures including the Kolmogorov–Smirnov (KS) test, its p-value (KS p-value), Akaike information criteria (AIC), Hannan-Quinn information criteria (HQIC), Bayesian information criteria (BIC), and consistent AIC (CAIC).
Table 2 reports the MLEs of the parameters of the fitted distributions along with the above-mentioned measures for the analyzed real data. The numerical values in this table illustrate that the proposed RP model provides a better fit as compared to competing models.
The box, Q-Q, TTT and KDE plots are given in Figure 4. The estimated PDF, estimated CDF, estimated HRF and PP plots of the RP model are given in Figure 5. The 3D plots of the estimated PDF and HRF for the data are reported in Figure 6. The visual comparison based on the AIC, BIC, CAIC, and HQIC measures are displayed in Figure 7. These plots supports the results in Table 2.
6. EWMA Control Chart
6.1. EWMA Control Chart Based on the MLE
The EWMA statistics at time is given bywhere is a weighting constant, whose values lies between 0 and 1 .
Now, the MLE of shape parameter of the RP distribution used instead of using variable , i.e
Hence, where is the EWMA statistic on current time and is EWMA statistic on previous time.
The EWMA control limits based on the MLE are obtained as
The remaining term after ignoring the term are given by
The random values are generated with from the RP distribution, with parameters as displayed in Figure 8. The obtained are computed from the random values. Repeated this step for 30 times and computed . The control limits of the EWMA control charts based on are computed from the above obtained results and plotted against the sample (subgroup).
6.2. EWMA Control Chart Based on MMLE
The EWMA statistic at time is given bywhere is a weighting constant, whose values lies between 0 and 1 .
Here, the MMLE of shape parameter of the RP distribution used instead of using variable , i.e
where is the EWMA statistics on current time and is the EWMA statistics on previous time.
The EWMA control limits based on MMLE are obtained as follows:
The remaining term after ignoring the term are given as follows:
The random values are generated with from the RP distribution with parameter values as given in Figure 9. Then, is obtained by computing the random values. Repeated this step for 30 times and computed . The control limits of the EWMA control charts based on are computed from the above obtained results and plotted against the sample (subgroup).
6.3. Simultaneous Study
For checking the performance of the EWMA control chart based on , a Monte Carlo simulation study is performed, where = MLE or MMLE. Simulations are performed based on the following steps.(1)The random values are generated with from the RP distribution, with parameters .(2)The estimate are computed from the random values that obtained in step 1.(3)For both types of estimators, the above two steps are repeated 2000 times and is computed.(4)The control limits of the EWMA control charts based on are computed from the results obtained in 3rd step.(5)The is fixed for in-control state of the process and found that value of L at which for in-control state for both type of estimator. Same procedure is repeated for and .(6)The different shifts i.e., are used and the is computed against the values of shifts for both types of estimators.(7)The values of SDRL, are also reported.(8)All above steps are repeated for
In Tables 3–8 and Figures 10–15, the comparison of the and are obtained. By using the it is noted that the provided a smaller as compared to the for all shifts.
6.4. Application of the EWMA Control Charts
A data set is adopted from Tables 9 and 10 in [17] and the proposed EWMA control charts by using the MLE and MMLE. The results are listed in Tables 9 and 10. The results also are displayed visually in Figures 15 and 16.
The simulated results of EWMA control chart based on the MLE from real data are reported in Table 9 and Figure 15.
The simulated results of the EWMA control chart based on the MMLE from real data are reported in Table 10 and Figures 16 and 17.
7. Conclusions
A new probabilistic model is introduced by adding a new parameter to the Pareto distribution called the reflected Pareto (RP) distribution. Some properties of the RP distribution are discussed. Moreover, the RP distribution parameters are estimated by using two estimation methods. The performance of the two methods is explored through simulations. A real-life data set is fitted using the RP model to prove its superiority as compared to some competing models. Additionally, the EWMA control chart based on the maximum likelihood and modified maximum likelihood estimators are discussed in detail. Finally, simulations and real-life data analysis are discussed to validate the results of the EWMA control chart.
Data Availability
The data used to support the findings of this study are included in the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest to report regarding the present study.
Acknowledgments
The authors extend their appreciation to the Deputyship for Research and Innovation, Ministry of Education in Saudi Arabia for funding this research work through project number 445-9-916.