1. Introduction

Distribution fitting has been of great importance in different areas of science. It is used to select a model (statistical distribution) which best describes the pattern (fits) of a dataset generated by some random process. The advantage of distribution fitting to data is that it allows for the development of valid models of random process and by so doing gives possibilities for proper predictions of future occurrences which in turn help in making better decisions. The challenge faced in distribution fitting is obtaining the model with the best fit. This includes getting the method of estimation which is most efficient, the appropriate probability distribution, the appropriate goodness of fit criterion to compare the efficiency of the model and nature of the data.

The Pareto distribution is one of such statistical distribution where such challenges are faced. The Pareto distribution is a power law probability distribution that is used in modeling many observable phenomena in description of social, scientific, geophysical, and actuarial science. It has been widely applied in the area of economics, trade, Insurance, Meteorology, Business and social sciences to model some real life phenomena. This has been showcased in the works of Nursamsiah et al. (2018), Ghitany et al. (2018) and Amand et al. (2016). There is a wide range of situation where the choice of the model can be made focusing just on the specific task to be accomplished while inaccuracy in unimportant areas is wittingly accepted. Nevertheless, sometimes the challenge is not on the choice of the model but the method of estimation or fitting the model. Such cases may arise in the Pareto distribution when the data is too heavy-tailed or having outliers or extreme values or when the sample sizes are small. The challenge in such cases is not just to choose a model but to find a robust and most appropriate method of fitting the distribution to the data.

There is a hierarchy of Pareto distributions known as Pareto Type I, II, III, IV, and Feller–Pareto distributions (Barry, 1983; Johnson et al., 1994). Burroughs and Tebbens (2001a) estimated parameters of the truncated Pareto distribution by least squares fitting on a probability plot while Burroughs and Tebbens (2001b) minimized the mean squared error fit on a plot of the tail distribution function). Beg (1981) also provided maximum likelihood estimates of the lower truncation parameter, scale, and probability of exceedance for a truncated Pareto distribution. The maximum likelihood estimator (MLE) of the scale parameter (α) when the lower truncation limit is known was presented by Cohen and Whitten (1988), with some recommendations for the case when the lower truncation limit is not known. vanZyl (2015) showed that random variables of the generalized three-parameter Pareto distribution can be transformed to that of the Pareto distribution. They tested the performance of the estimation of the shape parameter of generalized Pareto distribution using transformed observations and their results indicated that it improved the performance with respect to relative efficiency. Quandt (1966) used the Method of Moments (MOM), while Baxter (1980) and Cook and Mumme (1981) used the method of Maximum Likelihood Estimation (MLE) for the Pareto distribution. vanMontfort and Witter (1986) used the MLE to fit the Generalized Pareto distribution to represent the Dutch POT rainfall series and used an empirical correction formula to reduce bias of the scale and shape parameter estimates. Kim et al. (2017) proposed a new parameter estimator that utilizes a pivotal quantity based on the regression framework, allowing separate estimation of the two parameters in a straightforward manner. They affirmed that the method has the ability of overcoming the limitation of the Maximum Likelihood estimation method. Dalpatadu and Singh (2015) attempted to estimate the parameters of a two-parameter Pareto distribution using a Minimization Technique of a distance function such as the Kolmogorov-Smirnov distance with the help of the Golden Search procedure. They showed that the method was fairly simple and accurate and may be improved using a larger sample; but the work did not compare the method with existing estimation methods. Other authors estimated the parameters of a Generalized Pareto Distribution (GPD) using various methods (see Oztekin, 2005; Hosking et al., 1987). Bhatti et al. (2018a) however proposed three modified percentiles estimators of the parameter of a two-parameter Pareto distribution. They based their modifications on median, geometric mean, and expectation of empirical cumulative distribution function (CDF) of the first-order statistic. The proposed estimators were compared with the traditional percentile estimators through a Monte Carlo simulation. Results showed that the proposed estimators provided efficient and precise parameter estimates that the traditional percentile estrimators considered. Also, Zaher et al (2014) obtained the Fuzzy Least Squares estimator for the two-parameter Pareto distribution and compared it with the different methods of nestimation – the Trimmed Linear moments (TL-moments), Linear moments (L-moments) and Linear Quantile moments (LQ-moments). It was shown that the proposed Fuzzy Least Squares estimator was preferable at all times. Bhatti et al. (2018b) proposed some modifications in the maximum likelihood estimation methods of a Pareto distribution and evaluated their performances. They discovered that the modified maximum likelihood estimators based on expectation of empirical cumulative distribution function (CDF) of first order statistic performed much better than the traditional maximum likelihood estimator and other modified estimators on median and coefficient of variatiion. Hussain et al. (2018) also proposed four modified moment estimators for the parameter of the Pareto distribution. These estimators were compared with the traditional method of moments. The modified moment estimators based on mean of first order statistics and expectation of empirical cumulative distribution function of first order statistics were demonstrated to perform better than the traditional estimator and other modified moment estimators. In the same vein, Zaka et al. (2013) considered modifications of maximum likelihood, moments and percentile estimators of a two-parameter Poareto distribution. They concluded that for some combinations of parameter values, some of the modified traditional maximum likelihood, moments and percentile estimators in terms of bias, mean square error and total deviation. However, this work seeks to estimate the parameters of a two-parameter Pareto distribution to simulated data using some methods of estimation (Method of Moments, Maximum Likelihood method, Least Squares Methods and Ridge Regression Method) at different samples sizes, and to compare the fits by these methods of estimation using some goodness of fit criteria.

2. Methods of Analysis

2.1. Estimating Parameters of a Two-parameter Pareto Distribution

The two-parameter Pareto distribution (also called the Type I Pareto distribution) can be defined in terms of its cumulative distribution function as:

(3.1)

where: β = shape parameter, and α = scale parameter (see Pareto, 1965).

By definition in equation (1.2), to obtain the density function (probability density function), we take the partial derivative of the cumulative density function with respect to x and obtain:

(3.2)

(see Pareto, 1965).

2.1.1. Method of Moments (MOM)

By definition (1.5), the kth moment of the Pareto distribution is given as:

In order to obtain the estimate of from a sample of n observations, we recall that the probability of an observation greater than x is . Thus, the probability that all n sample values are greater than x is . This is, therefore also the probability that the lowest sample value is greater than x.

Thus, the c.d.f of the lowest sample is

(3.3)

Following the procedure by Quandts (1964), we obtain method of moment’s estimates as:

(3.4)

(3.5)

Where is the minimum value and is the mean.

2.1.2. Method of Maximum Likelihood (MLE)

Given that are random variables that follow the Pareto distribution, the likelihood function denoted by for the sample is:

(3.6)

By taking the natural log of the likelihood function:

(3.7)

To obtain the estimate for each parameter, we differentiate equation (3.7) w.r.t. each parameter and equate it to zero.

(3.8)

Simplifying the equation and making the subject formula, we have:

(3.9)

Since the likelihood function (L) is not bounded with respect to α, a maximum likelihood estimate cannot be obtained for α by differentiating L w.r.t. α. According to Richard (1964), since α is the lower bound, we may maximize L subject to the constraint:

(3.10)

Thus, we see that L is maximized w.r.t. α subject to (3.10) when

(3.11)

Therefore, α is estimated by the minimum value of the observation (Quandt, 1964).

2.1.3. Least Squares Method (LSE)

For the estimation of probability distribution parameters, the least squares method (LSM) is extensively used in reliability engineering and mathematics problems.

Given that the cumulative density function of the Pareto distribution is given as:

(3.12)

According to Rasheed and Ahter (2011), the method of least squares estimates is given as

(3.13)

(3.14)

Where

2.1.4. Ridge Regression Method (RR)

Like the method of LSE, the ridge regression estimates of and can be obtained by minimizing the error sum of square for the model:

subject to the single constraint that , where is a finite positive constraint. Using the method of Lagrange's multiplier, we obtain:

(3.15)

(3.16)

The Least Squares Estimates are:

(3.17)

(3.18)

where:

2.2. Goodness of Fit Tests

AL-Fawzan (2000) referred to the use of the procedure of MSE and TD as a goodness of fit test.

2.2.1. Mean Square Error (MSE)

The MSE can be calculated using the formula below:

(3.19)

where is the value of the cumulative distribution function of the two-parameter Pareto distribution using the estimated parameters, and is the empirical cumulative distribution function. (Rasheed and Ahter, 2011).

2.2.2. Total Deviation (TD)

The total deviation TD can be calculated for each method as:

(3.20)

where α and β are the known parameters, and are the estimated parameters by any method.

The method with the minimum total absolute deviation would be selected as the most efficient parametric method for fitting the distribution or estimating the parameters.

3. Results and Discussion

3.1. Results

The data used for evaluating the adequacy of the methods of estimating the parameters of the Pareto distribution (fitting the Pareto distribution) is a simulated data from R. In the simulation, 1000 random numbers that follows a pareto distribution of shape parameter (β=5) and scale parameter (α=10) were generated. Different samples of sizes 20, 40, 60 and 80 were drawn from the simulated population repeated for about 100 times and their respective fits were estimated.

Figure 1 indicates the plot of the two-parameter Pareto density function with shape parameter, β=5 and Location parameter, α =10. From the figure, it is observed that the function tails heavily to the right. This is one of the properties of the Pareto distribution.

Figure 1. Density plot and Cumulative distribution plot for shape = 5, scale = 10

Table 1 shows the estimates of the parameters of the distribution, the performance measures of the parameters using different estimation methods under different sample sizes. When the sample size, n = 80, the estimates of α and β with the least Mean Square Error of 2.0417 x 10^-7 and Total Deviation of 0.0058 were those of Maximum Likelihood Estimation Method (MLE). This is seconded by those of Ordinary Least Squares Method (OLS) with the Mean Square Error of 1.8225 x 10^-7 and Total Deviation of 0.0131. For n = 60, the MLE and OLS methods of parameter estimation give the smallest and smaller Mean Squre Errors and Total Deviations of (2.6667 x 10^-7, 1.5602 x 10^-6) and (0.0173, 0.0349) respectively. The same trend follows for sample sizes of 40 and 20. The rest of the methods performs in the following order: Ridge Regression (RR) and Method of Moment (MME). It is also observed from the Table that the estimates of the parameters tend to be closer to the original values as the sample size increases; hence the methods of estimation have so far shown the properties of consistency.

Table 1. Parameter Estimation of the Two-parameter Pareto distribution

The goodness of fit criteria (Mean Square Error and Total Deviation) select the method of Maximum Likelihood as the best method of estimation, followed by the method of Ordinary Least Squares as they give the least and smaller values respectively.

3.2. Discussion of Results

From Table 1, the Maximum Likelihood Estimator was identified as the most efficient estimator because it gave the least MSE and TD for all sample sizes considered in this work. This agrees with the work of Witter (1986) where MLE proved to be the most efficient method to fit the generalized Pareto Distribution to represent the Dutch POT rainfall series. This was followed by the method of ordinary least squares. The maximum likelihood estimator and the method of ordinary least squares have been known to be consistent estimators. These have been proven mathematically by Quandts (1964).

4. Conclusions

Based on the results obtained from this study, we have sufficient evidence to conclude that the Maximum Likelihood Estimation method is a more adequate method for fitting the two-parameter Pareto distribution when the series is free of outliers. The maximum likelihood estimator has also proven to be the most efficient estimator above the method of moment, ordinary least square estimator and the ridged regression estimator. It is also observed from the analysis that all methods seemed to be consistent in estimating the parameters.