Abstract
The Pareto model corresponds to the power law widely used in physics, biology, and many other fields. In this article, a new generalized Pareto model with a heavy right tail is introduced and studied. It exhibits an upsidedown bathtubshaped failure rate (FR) function. The moments, quantiles, FR function, and mean remaining life function are examined. Then, its parameters are estimated by maximum likelihood, least squared error, and Anderson–Darling (a weighted least squared error) approaches. A simulation study is conducted to verify the efficiency and consistency of the discussed estimators. Analysis of Floyd River flood discharges in James, Iowa, USA, from 1935 to 1973 shows that the proposed model can be quite useful in real applications, especially for extreme value data.
1 Introduction
The heavy righttailed Pareto model which is characterized by the distribution function
and the probability distribution function (PDF)
occurs in a diverse range of physical phenomena. Generally, it is useful when there is an equilibrium in distribution of “small” to “large” values, e.g., the size of transmitted files on a computer network consisting many small files and few large ones, or the size of human settlements consisting of many small and few large cities villages/hamlets. Moreover, the sizes of solar flares, oil reserves in oil fields, earthquakes, corporations, and lunar craters have similar property which is referred to as “power law” property. Newman [1] reviewed some power law forms and theories explaining them. The Pareto model is recognized by its heavy right tail in the literature and shows a decreasing failure rate function. It is useful in biology, reliability engineering, survival analysis, quality control, economics, computer science, geophysics, and many other scientific fields. For detailed information about Pareto and related distributions and their features see Arnold [2], Zhang et al. [3], and Zhang et al. [4].
Bak and Sneppen [5], Sornette [6], and Carlson and Doyle [7] among many others used the Pareto as a power law model in their research. Also, Burroughs and Tebbens [8] fitted the Pareto model to earthquake and wildfire observations, and Schroeder et al. [9] described plate fault data by the Pareto model. Moreover, some researchers defined modified versions of the Pareto model and applied them in their studies. These modified Pareto models are more flexible for data generated in various phenomena. For example, Akinsete et al. [10] introduced a beta Pareto model; Nassar and Nada [11] and Mahmoudi [12] proposed a beta generalization of the Pareto distribution, Alzaatreh et al. [13] used the gamma distribution to propose a modified Pareto model; and Zea et al. [14], Elbatal [15] and Bourguignon et al. [16] defined extensions of the Pareto distribution. Papastathopoulos and Tawn [17] applied one extended Pareto model for tail estimation. Moreover, Mead [18], Elbatal and Aryal [19], Korkmaz et al. [20], Ghitany et al. [21], Tahir et al. [22], Ihtisham et al. [23], Haj Ahmad and Almetwally [24], Jayakumar et al. [25], and recently Jayakumar et al. [26] defined and studied a new model with heavier right tail than Pareto.
In this article, a new flexible generalized Pareto distribution with heavy right tail and upside down bathtubshaped (UBT) FR function is introduced and studied. The novelty of the model is that it gathers the heavy right tail same as the Pareto model and UBT FR form in one model. Thus, the main advantage of the proposed model is that it is useful when the data show a fat right tail and UBT FR function. Such data can be observed in hydrology or other situations with extreme values. The remaining of the article is organized as follows. In Section 2, the new model is defined and its basic attributes like the moments and quantiles are discussed. Then, some important dynamic measures of it like FR and mean residual life (MRL) functions are studied. The aim of Section 3 is to estimate the parameters of the proposed model. In Section 4, the efficiency and consistency of the considered models are investigated by simulations. Then, the proposed model and some alternatives are fitted to consecutive flood discharges of the Floyd river located in James, Iowa, USA, during 1935 to 1973 to show its applicability.
2 The new modified Pareto distribution
The new generalized Pareto,
and the PDF
In Figure 1, the PDF is drawn for some values of the parameters and show a unimodal form for it. It seems that the coefficient
which is a special case of the modified Weibull model defined by Kayid and Djemili [27].
Like the baseline Pareto, the proposed GP model has a heavy right tail. For example, in comparison with the wellknown Weibull distribution we can write
Moreover, in comparison with the baseline Pareto model
which indicates that GP has a heavy right tail like the baseline Pareto model.
Lemma 1
Let
Proof
Since
which gives the result.□
The expectation of a random variable
Proposition 1
The kth moment of
Proof
Let
Take
One good tendency measure which may be applied in place of the moments is the quantile function which at point
where
2.1 Dynamic measures
The FR function of the proposed model is
Proposition 2
The FR function of
Proof
By differentiation of the FR function, we found that the sign of
which is a decreasing function and
In fact, the coefficient
The proof of the following proposition could be trivially obtained by comparing the reliability functions of the assumed random variables and is omitted.
Proposition 3
Let
Figure 1 draws the FR function for some parameters and shows that under the conditions of Proposition 3,
The MRL function of
By Proposition 2, it results that the MRL function has an increasing or bathtub form, see Lai and Xie [31]. Figure 2 shows the MRL for some parameter values.
Another prominent dynamic tendency measure is the
The special case
3 Inference
Assume
3.1 ML method
The loglikelihood function of
The value (
and
The Fisher information matrix can be estimated by replacing parameters by ML estimate in the following Fisher information matrix.
It is a wellknown and very practical technique to approximate the distribution of the MLE by multivariate normal distribution. The random vector
3.2 LSE and AD methods
In the LSE approach for estimating the parameters, we are interested to find parameter values minimizing the following expression:
which causes the distance between estimated and empirical distributions to be the smallest possible value. That is, the LSE estimates are given by
The AD approach is a weighted version of the LSE method with weight
4 Simulation study
To simulate one random variable
In this simulation study, some values for parameters are selected. Then, in every run,
and
and similarly for



80  150  
Method 

B  MSE  B  MSE 
ML  0.1, 0.1, 1  0.0057  0.0125  −0.0008  0.0077 
0.0250  0.0046  0.0103  0.0017  
−0.0259  0.0308  −0.0065  0.0162  
0.2, 0.07, 2  −0.0096  0.0204  −0.0167  0.0122  
0.0333  0.0063  0.0180  0.0028  
−0.0057  0.1242  0.0248  0.0761  
1, 0.2, 0.1  −0.0178  0.0824  −0.0052  0.0431  
0.0079  0.0027  0.0043  0.0013  
0.0052  0.0017  0.0019  0.0008  
LSE  0.1, 0.1, 1  0.0403  0.0296  0.0215  0.0167 
0.0289  0.0097  0.0116  0.0038  
−0.0542  0.0434  −0.0302  0.0220  
0.2, 0.07, 2  0.0314  0.0379  0.0114  0.0218  
0.0362  0.0169  0.0208  0.0086  
−0.0873  0.1649  −0.0311  0.0996  
1, 0.2, 0.1  −0.0749  0.1495  −0.0329  0.0843  
−0.0007  0.0042  0.0004  0.0023  
0.0122  0.0027  0.0064  0.0015  
AD  0.1, 0.1, 1  0.0076  0.0173  −0.0046  0.0087 
−0.0079  0.0038  −0.0070  0.0018  
−0.0008  0.0342  0.0066  0.0165  
0.2, 0.07, 2  −0.0041  0.0242  −0.0072  0.0148  
−0.0083  0.0046  −0.0089  0.0028  
0.0025  0.1366  0.0130  0.0809  
1, 0.2, 0.1  −0.0757  0.0965  −0.0488  0.0513  
−0.0148  0.0026  −0.0086  0.0015  
0.0151  0.0021  0.0092  0.0010 
In every cell, the first, second and third lines are corresponding to
5 Application
Table 2 shows the consecutive flood discharges in terms of
1,460  4,050  3,570  2,060  1,300  1,390  1,720  6,280  1,360  7,440  5,320 
1,400  3,240  2,710  4,520  4,840  8,320  13,900  71,500  6,250  2,260  318 
1,330  970  1,920  15,100  2,870  20,600  3,810  726  7,500  7,170  2,000 
829  17,300  4,740  13,400  2,940  5,660 
Model 



AIC  CVM  AD  KS 





GP  0.4655  910.36  3250.49  758.82  0.0210  0.1732  0.0704 
0.9964  0.9961  0.9829  
Pareto  0.3012  —  4579.41  763.08  0.0946  0.7474  0.1358 
0.6144  0.5195  0.4295  
MOP  0.5346  25.5049  398.83  761.08  0.0295  0.2761  0.0775 
0.9790  0.9576  0.9590  
DAL  0.2710  1.7938  0.0000025  760.19  0.0362  0.2853  0.0815 
0.9533  0.9485  0.9392  
IW  2404.70  1.0144  —  759.97  0.0494  0.3853  0.0861 
0.8828  0.8624  0.9110  
MOIW  1133.76  1.6714  733.82  759.44  0.0266  0.1994  0.0673 
0.9870  0.9906  0.9894  
EP  0.00016  0.00055  2.5330  765.08  0.0946  0.7472  0.1357 
0.6145  0.5197  0.4307  
gamma  0.9171  —  0.000135  769.81  0.2059  1.2343  0.1467 
0.2567  0.2546  0.3369  
MOG  0.9136  0.9120  0.00013  771.27  0.1895  1.1754  0.1437 
0.2894  0.2768  0.3612  
Weibull  0.8715  —  0.0005  0.768.26  0.1495  1.0543  0.1272 
0.3922  0.3295  0.5124  
PECR  17054.5  0.00015  88691.6  772.00  0.2371  1.3342  0.1494 
0.2060  0.2215  0.3163 
The alternative models are Pareto; exponentiated Pareto (EP); Marshal–Olkin Pareto (MOP); Dimitrakopoulou, Adamidis, and Loukas (DAL) modified Weibull model proposed by Dimitrakopoulou et al. [35]; inverse Weibull (IW); Marshal–Olkin inverse Weibull (MOIW); gamma, Marshal–Olkin gamma (MOG); Weibull and Pareto exponential competing risk (PECR).
The parameters of the mentioned models are estimated by the ML method. The R programming language was used for computations, and all optimizations were done by the builtin function “optim” of R. The Akaike information criterion (AIC), Cramer–von Mises (CVM) statistics, AD and Kolmogorov–Smirnov (KS) statistics are reported for every model. Clearly, the proposed GP and MOIW show a closerun. However, the GP outperforms other models and provides a good description of the data. Figure 4 draws the empirical and fitted distribution function for GP and some of the alternatives which show better fits. The estimated FR function is plotted in Figure 5 and confirms a UBT form for the FR function. Also, histogram of the data and estimated PDF are plotted in the right side of Figure 5.
6 Conclusion
One new flexible GP model which preserves the heavy right tail attribute but exhibits an early increasing FR function is introduced. The limiting behavior of the proposed model is similar to the baseline Pareto, but the attributes differ at beginning of the support. The proposed GP model has a UBT FR function. The simulation results show that the ML estimator is efficient and consistent. Applying the model on one flood discharge data of the Floyd river shows that the proposed GP model could be useful in describing many data sets which occur in a wide variety of physical phenomena. There are many future related topics. For example, studying a mixture of the proposed GP model or introducing proper extensions of the GP model based on the underlying physical justifications.
Acknowledgments
The authors thank the two anonymous reviewers for their careful reading of our manuscript and their many insightful comments and suggestions. This work was supported by Researchers Supporting Project number RSP2022R464, King Saud University, Riyadh, Saudi Arabia.

Funding information: This work was supported by Researchers Supporting Project number RSP2022R464, King Saud University, Riyadh, Saudi Arabia.

Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

Conflict of interest: The authors state no conflict of interest.
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© 2022 Mansour Shrahili and Mohamed Kayid, published by De Gruyter
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