This continuous distribution, belonging to the family of power distributions, is defined by two parameters in its simplest or type I form ( Ahmad & Almetwally, 2020 ). The first parameter, a scale parameter often denoted as x_m (though its notation varies widely), represents both the peak or mode and the minimum value of the distribution ( Chattamvelli & Shanmugam, 2021 ). The second parameter, a shape parameter denoted as α ( Fedotenkov, 2020 ), is commonly referred to as the Pareto tail index ( Andria, 2022 ). The parameter space for both spans the interval (0, ∞), while the support of the distribution is defined over the interval [x_m, ∞).

The profile of its density function depicts a curve that sharply decreases from its maximum value and becomes asymptotic to the horizontal axis, exhibiting positive skewness and leptokurtosis with a long right tail ( Figure 1 ). The value of α that precisely models the 80/20 law is log₄(5) = ln(5)/ln(4) ≈ 1.161, which is considered too extreme for wealth distributions, where α is typically greater than 1.5 ( Yang & Zhou, 2022 ).

$X~\text{Pareto}\left(x_m,\alpha \right)$

$x_m\text{and}\alpha \in \left(0,\infty \right)\text{and}x\in [x_m,\infty )$

The definite integral of the density function allows the calculation of the probability that a value x of the continuous random variable X falls within a given integration interval, which must lie within the domain of the distribution. It is denoted by f_X(x). The analytical expression of this function for the type I Pareto distribution is shown in Equation 1.

$f_X\left(x\right)=\alpha \frac{x_m^{\alpha }}{x^{\alpha +1}},x\ge x_m$ (1)

This function provides the probability that a continuous random variable X takes a value between the lower limit of the distribution and a specified value x, where x belongs to the domain of the distribution. it is denoted by F_X(x). The analytical expression of this function for the type I Pareto distribution is presented in Equation 2.

$F_X\left(x\right)=P\left(X\le x\right)={\displaystyle {\int }_{x_m}^x{f}_X\left(x\right)\text{d}x}=1-{\left(\frac{x_m}{x}\right)}^{\alpha },x\ge x_m$ (2)

Also known as the tail function, it is denoted by ${\bar{F}}_X\left(x\right)$. This function provides the probability that a continuous random variable X takes a value between a specified value x and the upper limit of the distribution, where x belongs to the domain of the distribution. The analytical expression for this function in the case of the type I Pareto distribution is presented in Equation 3.

${\bar{F}}_X\left(x\right)=P\left(X>x\right)={\displaystyle {\int }_{x_m}^{\infty }{f}_X\left(x\right)\text{d}x}={\left(\frac{x_m}{x}\right)}^{\alpha },x\ge x_m$ (3)

It is the inverse of the cumulative distribution function. It is denoted by Q_X(x) or $F_X^{-1}\left(x\right)$. It provides the value of X that accumulates the probability corresponding to the given argument, where the argument takes values in the interval (0, 1). The analytical expression for this function in the case of the type I Pareto distribution is shown in Equation 4.

$Q_X\left(p\right)=\frac{x_m}{\sqrt[\alpha ]{1-p}}=x_m{\left(1-p\right)}^{-\frac{1}{\alpha }},p\in \left(0,1\right)$ (4)

The k-th derivative of this function, evaluated at the point 0, gives the k-th order moments of the distribution, where the k-th order moment is the mathematical expectation of the variable raised to the k-th power. It is denoted by M_X(t). The moment generating function of the type I Pareto distribution is not defined for a non-trivial interval of t around 0, but it is defined for values of t ≤ 0. Its analytical expression is shown in Equation 5.

$M_X\left(t\right)=E\left({\text{e}}^{tx}\right)={\displaystyle {\int }_{x_m}^{+\infty }{\text{e}}^{tx}{f}_X\left(x\right)\text{d}x}=\{\begin{array}{ll}\alpha {\left(-x_{m}t\right)}^{\alpha }\Gamma \left(-\alpha ,-x_{m}t\right)\hfill & t<0\hfill \\ 1\hfill & t=0\hfill \\ \text{Undefined}\hfill & t>0\hfill \end{array}$ (5)

Upper incomplete gamma function

$\Gamma \left(-\alpha ,-x_{m}t\right)={\displaystyle {\int }_{-x_{m}t}^{\infty }{u}^{-\alpha -1}{\text{e}}^{-u}\text{d}u}$

The characteristic function corresponds to the Fourier transform (with a sign inversion) applied to the analytical expression of the density function. It allows for the analysis of the behavior of the moments of a distribution. It is denoted by C_X(t). The analytical expression of this function for the type I Pareto distribution is shown in Equation 6, where $i=\sqrt{-1}$ and t ∊ R.

$\begin{array}{l}{C}_X\left(t\right):ℝ\to ℂ;\\ C_X\left(t\right)=E\left({\text{e}}^{itx}\right)={\displaystyle {\int }_{x_m}^{\infty }{\text{e}}^{itx}{f}_X\left(x\right)\text{d}x}=\alpha {\left(-i{x}_{m}t\right)}^{\alpha }\Gamma \left(-\alpha ,-i{x}_{m}t\right)\end{array}$ (6)

Mathematical expectation or arithmetic mean: This measure corresponds to the first (non-central) moment. For the type I Pareto distribution, it can be calculated using its two parameters, as shown in Equation 7. Its computation requires the shape parameter α to be greater than 1.

$\mu \left(X\right)=E\left(X\right)={\displaystyle {\int }_{x_m}^{\infty }x{f}_X\left(x\right)\text{d}x}=\frac{\alpha }{\alpha -1}{x}_m,\alpha >1$ (7)

Geometric mean: It is the antilogarithm of the mathematical expectation of the logarithm of the values, typically using the natural base. For the type I Pareto distribution, it can be calculated from its two parameters, as shown in Equation 8.

${\mu }_g\left(X\right)=G\left(X\right)={\text{e}}^{E\left(\ln \left(X\right)\right)}=x_m\sqrt[\alpha ]{\text{e}}=x_m{\text{e}}^{1/\alpha }$ (8)

Harmonic Mean: It is the reciprocal of the mathematical expectation of the reciprocals of the values. For the type I Pareto distribution, it can be calculated using its two parameters, as shown in Equation 9.

${\mu }_h\left(X\right)=H\left(X\right)=\frac{1}{E\left(1/X\right)}=x_m\left(1+\frac{1}{\alpha }\right)$ (9)

Median: It is the 0.5 quantile. For the type I Pareto distribution, it can be calculated using its two parameters, as shown in the Equation 10.

$Mdn\left(X\right)=Q_X\left(p=0.5\right)=\frac{x_m}{\sqrt[\alpha ]{1/2}}=x_m\sqrt[\alpha ]{2}$ (10)

Mode: It corresponds to the peak of the distribution. The type I Pareto distribution is unimodal, with its single peak occurring at the scale parameter x_m (Equation 11).

$Mo\left(X\right)=\left\{x|\max \left[f_X\left(x\right)\right]\right\}=x_m$ (11)

Non-central k-th moment: It is the mathematical expectation of the values raised to the k-th power. For the type I Pareto distribution, it can be calculated using its two parameters, as shown in Equation 12. For its calculation, the shape parameter α must be greater than the power k to which the values are raised.

$E\left(X^k\right)={\displaystyle {\int }_{x_m}^{\infty }{x}^k{f}_X\left(x\right)\text{d}x}=\frac{\alpha }{\alpha -k}{x}_m^k,\alpha >k$ (12)

Variance: It corresponds to the second central moment or the expected value of the squared differences from the arithmetic mean. For the type I Pareto distribution, it can be calculated using its two parameters, as shown in in Equation 13. Its computation requires the shape parameter α to be greater than 2.

$\begin{array}{c}Var\left(X\right)={\sigma }^2\left(X\right)=E\left[{\left(X-E\left(X\right)\right)}^2\right]={\displaystyle {\int }_{x_m}^{\infty }{\left(x-E\left(X\right)\right)}^2{f}_X\left(x\right)\text{d}x}\\ =\frac{\alpha x_m^2}{{\left(\alpha -1\right)}^2\left(\alpha -2\right)},\alpha >2\end{array}$ (13)

Standard deviation: It is the square root of the variance (Equation 14).

$DE\left(X\right)=\sigma \left(X\right)=\sqrt{Var\left(X\right)}=\frac{x_m}{\alpha -1}\sqrt{\frac{\alpha }{\alpha -2}},\alpha >2$ (14)

Entropy: It corresponds to the mathematical expectation of the additive inverses of the logarithms of the densities, or the mathematical expectation of the information of the values. When the natural logarithm is used, entropy is expressed in natural units of information (nats), which is the most common. If base 10 is used, it is expressed in decimal units of information (dits); whereas, if base 2 is used, it is expressed in binary units of information (bits). For the type I Pareto distribution, it can be calculated using its two parameters, as shown in Equation 15.

$\begin{array}{c}H\left(X\right)=E\left(-\ln \left[f_X\left(x\right)\right]\right)=-{\displaystyle {\int }_{x_m}^{\infty }{f}_X\left(x\right)\ln \left[f_X\left(x\right)\right]\text{d}x}=\ln \left[\frac{x_m}{\alpha }{\text{e}}^{1+\frac{1}{\alpha }}\right]\\ =\ln \left(x_m\right)-\ln \left(\alpha \right)+1+1/\alpha \end{array}$ (15)

Skewness: It corresponds to the third standardized central moment. Karl Pearson’s original notation is used: $\sqrt{{\beta }_1}$ ( Pearson, 1895 ). For the type I Pareto distribution, it can be calculated using the shape parameter α, as shown in Equation 16. Its computation requires this parameter to be greater than 3.

$\sqrt{{\beta }_1\left(X\right)}=\frac{E\left[{\left(X-E\left(X\right)\right)}^3\right]}{{\left(E\left[{\left(X-E\left(X\right)\right)}^2\right]\right)}^{3/2}}=\frac{2\left(\alpha +1\right)}{\alpha -3}\sqrt{\frac{\alpha -2}{\alpha }},\alpha >3$ (16)

Excess Kurtosis: It corresponds to the standardized fourth central moment minus the kurtosis value of the normal distribution, which is 3. Karl Pearson’s original notation is used: β₂ − 3 ( Pearson, 1905 ). For the type I Pareto distribution, it can be calculated using the shape parameter α, as shown in Equation 17. Its computation requires this parameter to be greater than 4.

${\beta }_2\left(X\right)-3=\frac{E\left[{\left(X-E\left(X\right)\right)}^4\right]}{{\left(E\left[{\left(X-E\left(X\right)\right)}^2\right]\right)}^2}=\frac{6\left({\alpha }^3+{\alpha }^2-6\alpha -2\right)}{\alpha \left(\alpha -3\right)\left(\alpha -4\right)},\alpha >4$ (17)

The estimator of the shape parameter α by the method of moments is obtained from the mathematical expectation or arithmetic mean of X, if the value of the location parameter x_m, which is the peak and minimum value of the distribution, is known. This mathematical expectation corresponds to the quotient of the sample mean (numerator) and the difference between the sample mean and the known parameter x_m (denominator). It is valid when α > 2. For more details, refer to Equation 19.

$\begin{array}{l}E\left(X\right)=\frac{\alpha }{\alpha -1}{x}_m\\ \alpha E\left(X\right)-E\left(X\right)-\alpha x_m=0\\ \alpha \left(E\left(X\right)-x_m\right)=E\left(X\right)\\ \alpha =\frac{E\left(X\right)}{E\left(X\right)-x_m}=\frac{\mu \left(X\right)}{\mu \left(X\right)-x_m}\\ \hat{\alpha }=\frac{\hat{E}\left(X\right)}{\hat{E}\left(X\right)-x_m}=\frac{\hat{\mu }\left(X\right)}{\hat{\mu }\left(X\right)-x_m}=\frac{\bar{x}}{\bar{x}-x_m},\text{where}\hat{\mu }\left(X\right)=\bar{x}=\frac{{{\displaystyle \sum }}_{i=1}^n{x}_i}{n}\end{array}$ (19)

The point estimator converges to a normal distribution, since X has a distribution with finite moments ( Rao, 1973 ), which allows for the computation of an asymptotic interval estimator (Equation 20).

$\begin{array}{l}{\hat{\alpha }}_n=\frac{\bar{x}}{\bar{x}-x_m}\underset{d}{\to }N\left(\alpha ,\frac{\alpha {\left(\alpha -1\right)}^2}{n\left(\alpha -2\right)}\right)\\ P\left(\hat{\alpha }-z_{1-\frac{\alpha }{2}}\times \sqrt{\frac{\hat{\alpha }{\left(\hat{\alpha }-1\right)}^2}{n\left(\hat{\alpha }-2\right)}}\le \alpha \le \hat{\alpha }+z_{1-\frac{\alpha }{2}}\times \sqrt{\frac{\hat{\alpha }{\left(\hat{\alpha }-1\right)}^2}{n\left(\hat{\alpha }-2\right)}}\right)\end{array}$ (20)

The estimator by the maximum likelihood method of the scale parameter x_m is the minimum sample value, and the estimator of the shape parameter α is the inverse of the arithmetic mean of the logarithms of the ratios between each value x_i and the minimum sample value ( Siudem, Nowak, & Gagolewski, 2022 ). See Equation 21 and Equation 22 for these estimators, respectively.

${\hat{x}}_m=\min \left(x_1,x_2,\cdots ,x_n\right)=\min \left({\left\{x_i\right\}}_{i=1}^n\right)$ (21)

$\hat{\alpha }=\frac{n}{{{\displaystyle \sum }}_{i=1}^n\ln \left(\frac{x_i}{{\hat{x}}_m}\right)}=\frac{n}{{{\displaystyle \sum }}_{i=1}^n\ln \left(\frac{x_i}{\min \left({\left\{x_i\right\}}_{i=1}^n\right)}\right)}$ (22)

The Fisher information for n data points for these two parameters is given by the 2 × 2 matrix in Equation 23.

$\begin{array}{l}I\left(x_m,\alpha |x\right)\\ =n\left(\begin{array}{cc}-E\left[\frac{\delta }{\delta x_m\delta x_m}\ln f_X\left(x|x_m,\alpha \right)\right]& -E\left[\frac{\delta }{\delta x_m\delta \alpha }\ln f_X\left(x|x_m,\alpha \right)\right]\\ -E\left[\frac{\delta }{\delta \alpha \delta x_m}\ln f_X\left(x|x_m,\alpha \right)\right]& -E\left[\frac{\delta }{\delta \alpha \delta \alpha }\ln f_X\left(x|x_m,\alpha \right)\right]\end{array}\right)\end{array}$

$=n\left(\begin{array}{cc}\alpha /x_m^2& -1/x_m\\ -1/x_m& 1/{\alpha }^2\end{array}\right)=\left(\begin{array}{cc}n\alpha /x_m^2& -n/x_m\\ -n/x_m& n/{\alpha }^2\end{array}\right)$ (23)

The variance of a parameter θ, Var(θ), is always greater than or equal to its Cramér-Rao lower bound CRLB(θ) or the inverse of its Fisher information for n data, 1/I(θ) ( Xu, Sedory, & Singh, 2022 ). Refer to Equation 24.

$Var\left(\hat{\theta }\right)={\sigma }_{\hat{\theta }}^2\ge CICR\left(\hat{\theta }\right)=1/I\left(\theta \right)$ (24)

The estimators obtained by the maximum likelihood method have asymptotic properties of unbiasedness (Equation 25), consistency (Equation 26), efficiency (Equation 27), and normality (Equation 28), which make them very useful ( Song, Roung-Park, & Yoon, 2022 ).

$\begin{array}{l}{\hat{\theta }}_n\text{is}\text{an}\text{asymptotically}\text{unbiased}\text{estimator}\text{of}\theta \\ \text{if}\underset{n\to \infty }{\lim }E\left({\hat{\theta }}_n\right)=\theta \end{array}$ (25)

$\begin{array}{l}{\hat{\theta }}_n\text{is}\text{an}\text{asymptotically}\text{consistent}\text{estimator}\text{of}\theta \\ \text{if}\underset{n\to \infty }{\lim }P\left(\left|{\hat{\theta }}_n-\theta \right|>\epsilon \right)=0\end{array}$ (26)

$\begin{array}{l}{\hat{\theta }}_n\text{is}\text{an}\text{asymptotically}\text{efficient}\text{estimator}\text{of}\theta \\ \text{if}\underset{n\to \infty }{\lim }\frac{CICR\left(\hat{\theta }\right)}{Var\left({\hat{\theta }}_n\right)}=1\end{array}$ (27)

$\begin{array}{l}{\hat{\theta }}_n\text{as}\text{an}\text{estimator}\theta \text{converges}\text{to}\text{normality}\\ \text{if}\underset{n\to \infty }{\lim }\frac{\sqrt{n}\left({\hat{\theta }}_n-\theta \right)}{DE\left(\hat{\theta }\right)}\underset{d}{\to }N\left(0,1\right)\end{array}$ (28)

Based on these asymptotic properties, asymptotic standard errors (aes) can be defined for the maximum likelihood estimators of x_m (Equation 29) and α (Equation 31), along with asymptotic confidence intervals (Equation 30 for the estimator of x_m and Equation 32 for the estimator of α).

$\begin{array}{l}ase\left(\hat{\theta }\right)={\sigma }_{\hat{\theta }}=\sqrt{1/I\left(\theta \right)}\\ ase\left({\hat{x}}_m\right)={\sigma }_{{\hat{x}}_m}=\sqrt{\frac{1}{n\alpha /x_m^2}}=\sqrt{\frac{x_m^2}{n\alpha }}=\frac{x_m}{\sqrt{n\alpha }};\widehat{EEA}\left({\hat{x}}_m\right)=\frac{{\hat{x}}_m}{\sqrt{n\hat{\alpha }}}\end{array}$ (29)

$\begin{array}{l}P\left(\hat{\theta }-z_{1-\frac{\alpha }{n}}{\sigma }_{\hat{\theta }}\le \theta \le \hat{\theta }+z_{1-\frac{\alpha }{n}}{\sigma }_{\hat{\theta }}\right)=1-\alpha \\ P\left({\hat{x}}_m-z_{1-\frac{\alpha }{n}}\frac{{\hat{x}}_m}{\sqrt{n\hat{\alpha }}}\le x_m\le {\hat{x}}_m+z_{1-\frac{\alpha }{n}}\frac{{\hat{x}}_m}{\sqrt{n\hat{\alpha }}}\right)=1-\alpha \end{array}$ (30)

$\begin{array}{l}ase\left(\hat{\theta }\right)={\sigma }_{\hat{\theta }}=\sqrt{1/I\left(\theta \right)}\\ ase\left(\hat{\alpha }\right)={\sigma }_{\hat{\alpha }}=\sqrt{\frac{1}{n/{\alpha }^2}}=\sqrt{\frac{{\alpha }^2}{n}}=\frac{\alpha }{\sqrt{n}};\widehat{EEA}\left(\hat{\alpha }\right)=\frac{\hat{\alpha }}{\sqrt{n}}\end{array}$ (31)

$\begin{array}{l}P\left(\hat{\theta }-z_{1-\frac{\alpha }{n}}{\sigma }_{\hat{\theta }}\le \theta \le \hat{\theta }+z_{1-\frac{\alpha }{n}}{\sigma }_{\hat{\theta }}\right)=1-\alpha \\ P\left(\hat{\alpha }-z_{1-\frac{\alpha }{n}}\frac{\hat{\alpha }}{\sqrt{n}}\le \alpha \le \hat{\alpha }+z_{1-\frac{\alpha }{n}}\frac{\hat{\alpha }}{\sqrt{n}}\right)=1-\alpha \end{array}$ (32)

The estimator of α by the maximum likelihood method is more efficient than that of the method of moments, but it is biased. Given this limitation, a bias-corrected estimator can be defined that has a lower variance than the biased estimator ( Rytgaard, 1990 ) and can be used to achieve more efficient asymptotic estimation (Equation 33). As with the previous definitions, the use of these asymptotic formulas requires a large sample, specifically one larger than 30 and preferably at least 100 ( Mateus & Caeiro, 2022 ). See the variance, standard error, and asymptotic confidence interval of the bias-corrected estimator in Equations 34, 35, and 36, respectively.

${\hat{\alpha }}_c=\frac{n-2}{{{\displaystyle \sum }}_{i=1}^n\ln \left(\frac{x_i}{\min \left({\left\{x_i\right\}}_{i=1}^n\right)}\right)}$ (33)

${\hat{\sigma }}^2\left({\hat{\alpha }}_c\right)={\hat{\alpha }}_c^2/\left(n-2\right)<{\hat{\sigma }}^2\left(\hat{\alpha }\right)$ (34)

$\begin{array}{c}\hat{\sigma }\left({\hat{\alpha }}_c\right)=\sqrt{{\hat{\alpha }}^2/\left(n-2\right)}\\ ={\hat{\alpha }}_c/\sqrt{n-2}\end{array}$ (35)

$P\left({\hat{\alpha }}_c-z_{1-\frac{\alpha }{n}}\frac{{\hat{\alpha }}_c}{\sqrt{n-2}}\le \alpha \le {\hat{\alpha }}_c+z_{1-\frac{\alpha }{n}}\frac{{\hat{\alpha }}_c}{\sqrt{n-2}}\right)=1-\alpha$ (36)

Let X be a random variable following a type I Pareto distribution with scale parameter x_m and shape parameter α. The exact distribution of the sum of the logarithms of each of the n sample data points of X, divided by the minimum sample value, follows an exponential distribution with a rate parameter equal to α ( Rytgaard, 1990 ). See Equation 37.

$\underset{i=1}{\overset{n}{{\displaystyle \sum }}}\ln \left(\frac{x_i}{\min {\left(\left\{x_i\right\}\right)}_{i=1}^n}\right)=\underset{i=1}{\overset{n}{{\displaystyle \sum }}}\ln \left(\frac{x_i}{{\hat{x}}_m}\right)~\text{Exponential}\left(\lambda =\alpha \right)$ (37)

The exact distribution of the maximum likelihood estimator of the scale parameter x_m is a type I Pareto distribution with a scale parameter x_m and a shape parameter n × α, while the exact distribution of the maximum likelihood estimator of the shape parameter α is an inverse gamma distribution with a shape parameter n - 1 and a scale parameter n × α ( Qian, Chen, & He, 2021 ). See Equation 38.

${\hat{x}}_m=\min \left({\left\{x_i\right\}}_{i=1}^n\right)~\text{Pareto}\left(x_m,n\times \alpha \right)$

$\hat{\alpha }=\frac{n}{{{\displaystyle \sum }}_{i=1}^n\ln \left(\frac{x_i}{\min \left({\left\{x_i\right\}}_{i=1}^n\right)}\right)}~\text{Inv-Gamma}\left(\alpha =n-1,\beta =n\times \alpha \right)$ (38)

Let X be a variable with a type I Pareto distribution characterized by parameters x_m = 500 and α = 4.1, representing the biweekly salary in dollars within a company. The task is to calculate the probability of having an income less than or equal to 592, less than 1000, between 800 and 1400, greater than or equal to 600, and greater than 1500. Additionally, the following measures of central tendency are required: the mathematical expectation or arithmetic mean μ(X), geometric mean μ_g(X), harmonic mean μ_h(X), median Mdn(X), and mode Mo(X); measures of variation such as variance σ²(X), standard deviation σ(X), and entropy H(X); as well as shape measures based on standardized central moments, including skewness $\sqrt{{\beta }_1\left(X\right)}$ and excess kurtosis β₂(X). Furthermore, the Gini concentration index G(X) is to be calculated. Finally, it is considered illustrative to plot its density function f_X(x) and cumulative distribution F_X(x).

$X~\text{Pareto}\left(x_n=500,\alpha =4.1\right)$

Equation 2 is applied for the calculation of the probability of having an income less than or equal to 592.

$P\left(X\le 592\right)={\displaystyle {\int }_{500}^{592}{f}_X\left(x\right)\text{d}x}=1-{\left(x_m/x\right)}^{\alpha }=1-{\left(500/592\right)}^{4.1}=0.4997$

Equation 2 is also used to calculate the probability of having an income less than 1000. Since it is a continuous distribution, including or excluding the specific point does not affect the value of the cumulative probability.

$P\left(X<1000\right)={\displaystyle {\int }_{500}^{1000}{f}_X\left(x\right)\text{d}x}=1-{\left(500/1000\right)}^{4.1}=0.9417$

Equation 2 is used to calculate the probability of having an income between 800 and 1400 by taking the difference between two cumulative probabilities.

$P\left(800\le X\le 1400\right)=P\left(X\le 1400\right)-P\left(X\le 800\right)=0.9853-0.8544=1309$

$P\left(X\le 1400\right)=1-{\left(500/1400\right)}^{4.1}=0.9853$

$P\left(X\le 800\right)=1-{\left(500/800\right)}^{4.1}=0.8544$

Equation 3 is used to calculate the probability of having an income greater than or equal to 600.

$P\left(X\ge 600\right)={\left(500/600\right)}^{4.1}=0.4735$

Equation 3 is also used to calculate the probability of having an income greater than 1500. Since it is a continuous distribution, including or excluding the specific point does not affect the complementary or right-tailed probability.

$P\left(X>1500\right)={\left(500/1500\right)}^{4.1}=0.0111$

Measures of central tendency. The arithmetic mean is calculated using Equation 7, the geometric mean using Equation 8, the harmonic mean using Equation 9, the median using Equation 10, and the mode using Equation 11.

$\mu \left(X\right)=E\left(X\right)=\frac{\alpha }{\alpha -1}{x}_m=\frac{4.1}{3.1}\times 500=661.29$

${\mu }_g\left(X\right)={\text{e}}^{E\left(\ln \left(X\right)\right)}=x_m{\text{e}}^{1/\alpha }=500\times {\text{e}}^{1/4.1}=638.11$

${\mu }_h\left(X\right)=H\left(X\right)=\frac{1}{E\left(1/X\right)}=\left(1+\frac{1}{\alpha }\right)x_m=\left(1+\frac{1}{4.1}\right)\times 500=621.95$

$Mdn\left(X\right)=Q_X\left(p=0.5\right)=\sqrt[\alpha ]{2}\times x_m=2^{1/4.1}\times 500=592.10$

$Mo\left(X\right)=\left\{x|\max \left(f_X\left(x\right)\right)\right\}=x_m=500$

Measures of variation. Variance is calculated using Equation 13, standard deviation using Equation 14, and entropy using Equation 15.

$\begin{array}{c}{\sigma }^2\left(X\right)={\mu }_2\left(X\right)=E\left[{\left(X-E\left(X\right)\right)}^2\right]=x_m^2\frac{\alpha }{{\left(\alpha -1\right)}^2\left(\alpha -2\right)}\\ ={500}^2\frac{4.1}{{3.1}^2\times 2.1}=50790.35\end{array}$

$\sigma \left(X\right)={\mu }_2^{1/2}\left(X\right)=\sqrt{E\left[{\left(X-E\left(X\right)\right)}^2\right]}=\frac{x_m}{\alpha -1}\sqrt{\frac{\alpha }{\alpha -2}}=\frac{500}{3.1}\sqrt{\frac{4.1}{2.1}}=225.37$

$H\left(X\right)=E\left[-\ln \left(f_X\left(x\right)\right)\right]=\ln \left[\frac{x_m}{\alpha }{\text{e}}^{1+\frac{1}{\alpha }}\right]=\ln \left[\frac{500}{4.1}{\text{e}}^{1+\frac{1}{4.1}}\right]=6.05\text{nats}$

Skewness and Excess Kurtosis Based on Central Moments. These two shape statistics are calculated using Equations 16 and 17, respectively.

$\sqrt{{\beta }_1\left(X\right)}=\frac{{\mu }_3\left(X\right)}{{\mu }_2^{3/2}\left(X\right)}=\frac{2\left(\alpha +1\right)}{\alpha -3}\sqrt{\frac{\alpha -2}{\alpha }}=\frac{2\times 5.1}{1.1}\sqrt{\frac{2.1}{4.1}}=6.64$

${\beta }_2\left(X\right)-3=\frac{{\mu }_4\left(X\right)}{{\mu }_2^2\left(X\right)}-3=\frac{6\left({\alpha }^3+{\alpha }^2-6\alpha -2\right)}{\alpha \left(\alpha -3\right)\left(\alpha -4\right)}=\frac{6\left({4.1}^3+{4.1}^2-6\times 4.1-2\right)}{4.1\times 1.1\times 0.1}=786.67$

The Gini Index is obtained using Equation 47.

$G\left(X\right)=1/\left(2\alpha -1\right)=1/\left(2\times 4.1-1\right)=0.14$

The Gini index is close to 0, indicating fairly equal wages within the firm. Figure 2 displays the density function f_X(x) and the cumulative distribution F_X(x) of the random variable X, representing biweekly wages (in dollars). The wages follow a type I Pareto distribution with parameters x_m = 500 and α = 4.1.

The R script to calculate these probabilities, descriptive statistics, and the plot of the density and cumulative distribution functions of the type I Pareto distribution (x_m = 500 and α = 4.1) is shown below:

# Load required libraries (packages)

library(cascsim)

library(modeest)

cat ("Probability calculations (rounding to four decimal places)", "")

cat ("Cumulative probability less than or equal to 592 =", round(ppareto(592, xm = 500, alpha = 4.1), 4), "")

cat ("Cumulative probability less than 1000 =", round(ppareto(1000, xm = 500, alpha = 4.1), 4), "")

cat ("Probability in the interval [800, 1400] =", round(ppareto(1400, xm = 500, alpha = 4.1) - ppareto(800, xm = 500, alpha = 4.1), 4), "")

cat ("Probability greater than or equal to 600 =", round(1 - ppareto(600, xm = 500, alpha = 4.1), 4), "")

cat ("Probability greater than 1500 =", round(1 - ppareto(1500, xm = 500, alpha = 4.1), 4), "")

# Descriptive measures for type I Pareto distribution (rounding to three decimal places).

x_m <- 500 # Scale parameter

alpha <- 4.1 # Shape parameter

cat ("Measures of central tendency", "")

cat ("Mathematical expectation of X: μ(X) =", round(alpha / (alpha - 1) * x_m, 3), "")

cat ("Geometric mean of X: μg(X) =", round(x_m * exp(1 / alpha), 3), "")

cat ("Harmonic mean of X: μh(X) =", round((1 + 1 / alpha) * x_m, 3), "")

cat ("Median de X: Mdn(X) =", round(qpareto(0.5, xm = 500, alpha = 4.1), 3), "")

cat ("Mode of X: Mdn(X) =", round(qpareto(0.5, xm = 500, alpha = 4.1), 3), "")

cat ("Measures of variation ", "")

cat ("Variance of X: σ^2(X) =", round(x_m^2 * alpha /(( alpha -1)^2 * (alpha - 2)), 3), "")

cat ("Measures of variation ", "")

cat ("Standard deviation of X: σ(X) =", round(sqrt(x_m^2 * alpha /(( alpha -1)^2 * (alpha - 2))), 3), "")

cat ("Entropy of X: Η(X) =", round(log(x_m / alpha * exp(1+ 1/ alpha)), 3), "nats", "")

cat ("Shape measures", "")

cat("Skewness of X: √b1(X) =", round(2 * (alpha + 1) / (alpha - 3) * sqrt((alpha - 2) / alpha), 3), "")

cat ("Kurtosis of X: b2(X) =", round(6* (alpha^3 + alpha^2 - 6 * alpha - 2) / (alpha * (alpha - 3) * (alpha - 4)), 3), "")

cat ("Gini Index of X: G(X) =", round(1 / (2 * alpha - 1), 3), "")

# Plot of density and cumulative distribution functions.

x <- seq(0, 2000, length.out = 2000)

dpareto <- function (x, x_m, alpha) {

ifelse (x < x_m, 0, alpha * x_m^alpha / (x^(alpha + 1)))}

ppareto2 <- function (x, x_m, alpha) {ifelse(x < x_m, 0, 1 - (x_m / x)^alpha)}

density <- dpareto(x, x_m, alpha)

cumulative <- ppareto(x, x_m, alpha)

par (mar = c(4, 4, 1, 2) + 0.1)

plot (x, density, type = "l", col = "blue", lwd = 2, xlab = "Biweekly wages (in dollars)",

ylab = "Density", ylim = c(0, max(density)))

par (new = TRUE)

plot (x, cumulative, type = "l", col = "red", lwd = 2, axes = FALSE, xlab = "", ylab = "")

axis (4)

mtext ("Cumulative Probability", side = 4, line = 3, col = "red")

legend ("right", legend = c("Density", "Cumulative"),

col = c ("darkblue", "red"), lwd = 2, bty = "n", y.intersp = 1.5)

In this second example, the aim is to generate a random sample of 40 observations from the variable X ~ Pareto (500, 4.1). Based on this sample, the goal is to estimate the parameters x_m and α using maximum likelihood estimators, and to construct 95% confidence intervals for these parameters based on their asymptotic distributions and errors. Finally, the goodness-of-fit will be assessed through a plot of theoretical versus empirical quantiles, a histogram with an overlaid density curve, and the Kolmogorov-Smirnov and Anderson-Darling tests.

In the first column of Table 2 , a random sample of 40 u_i values is drawn from a standard uniform distribution U ~ U [0, 1], generated by means of the 2021 version of the Excel random number generator. Using the quantile function, the u_i values are transformed into a random sample of size 40 for a variable X that follows a type I Pareto distribution with population parameters: x_m = 500 (scale) and α = 4.1 (shape). This data generation process is known as inverse transform sampling, as illustrated in Equation 56. The first value of X obtained by applying Equation 56, starting at u₁ = 0.226, is 532.239.

$\begin{array}{l}{x}_i=x_m{\left(1-u_i\right)}^{-\frac{1}{\alpha }}=500{\left(1-u_i\right)}^{-\frac{1}{4.1}}\\ x_1=500{\left(1-0.226\right)}^{-\frac{1}{4.1}}=532.239\end{array}$ (56)

Note. Generation of X: u_i = sample data in its random order (i = 1, 2, …, 40) drawn from a standard uniform distribution U [0, 1] and x_i = 500(1 - u_i)-1/4.1 = data transformed to follow a type I Pareto distribution (with scale parameter x_m = 500 and shape parameter α = 4.1) via inverse transform sampling. Logarithmic transformation of X and Kolmogorov- Smirnov test: (i) = the order of data xi within the 40-item sample, x_(i) = data x_i sorted in ascending order (empirical quantiles), y_(i) = ln(x_(i)/502.595) = the logarithmic transformation of x_i normalized by the sample minimum, F_Y(y_(i)) = the cumulative distribution function of the Y variable following an exponential distribution with rate parameter λ = 3.957, D⁺ = (i)/40 - F_Y(y_(i)) = the difference between the empirical and theoretical cumulative distribution functions and D^- = F_Y(y_(i)) - ((i)-1)/40 = the difference between the theoretical distribution function and empirical cumulative distribution function with lap of 1. Quantile-Quantile plot: p_(i) = ((i)-1/3)/(n+1/3) = the order of the theoretical quantile and x_t(i) = Q_X[p_(i)] = theoretical quantiles calculated using the quantile function of a type I Pareto distribution with estimated parameters. Σ = sum per column, max = maximum value per column.

Next, the two parameters are estimated pointwise using maximum likelihood estimation (Equation 21 for thse estimate of x_m and Equation 22 for the estimate of α), and their corresponding standard errors (Equation 29 for ${\hat{x}}_m$ and Equation 31 for $\hat{\alpha }$) and 95% asymptotic confidence intervals (Equation 30 for ${\hat{x}}_m$ and Equation 32 for $\hat{\alpha }$) are computed.

Estimation of the scale parameter (x_m):

${\hat{x}}_m=\min \left({\left\{x_i\right\}}_{i=1}^{40}\right)=502.595$

$ase\left({\hat{x}}_m\right)=\hat{\sigma }\left({\hat{x}}_m\right)=\frac{{\hat{x}}_m}{\sqrt{n\times \hat{\alpha }}}=\frac{502.595}{\sqrt{40\times 3.957}}=39.948$

$P\left({\hat{x}}_m-z_{1-\frac{\alpha }{n}}\frac{{\hat{x}}_m}{\sqrt{n\times \hat{\alpha }}}\le x_m\le {\hat{x}}_m+z_{1-\frac{\alpha }{n}}\frac{{\hat{x}}_m}{\sqrt{n\times \hat{\alpha }}}\right)=1-\alpha$

$P\left(502.595-1.96\times 39.948\le x_m\le 502.595+1.96\times 39.948\right)=0.95$

$P\left(x_m\in \left[424.299,580.891\right]\right)=0.95$

Estimation of the shape parameter (α):

$\begin{array}{c}\hat{\alpha }=\frac{n}{{{\displaystyle \sum }}_{i=1}^n\ln \left(\frac{x_i}{{\hat{x}}_m}\right)}=\frac{n}{{{\displaystyle \sum }}_{i=1}^n\ln \left(\frac{x_i}{\min \left({\left\{x_i\right\}}_{i=1}^n\right)}\right)}\\ =\frac{40}{{{\displaystyle \sum }}_{i=1}^{40}\ln \left(\frac{x_i}{502.595}\right)}=\frac{40}{10.108}=3.957\end{array}$

$ase\left(\hat{\alpha }\right)=\hat{\sigma }\left(\hat{\alpha }\right)=\hat{\alpha }/\sqrt{n}=3.957/\sqrt{40}=0.626$

$P\left(\hat{\alpha }-z_{1-\frac{\alpha }{n}}\frac{\hat{\alpha }}{\sqrt{n}}\le \alpha \le \hat{\alpha }+z_{1-\frac{\alpha }{n}}\frac{\hat{\alpha }}{\sqrt{n}}\right)=1-\alpha$

$P\left(3.957-1.96\times 0.626\le \alpha \le 3.957+1.96\times 0.626\right)=0.95$

$P\left(\alpha \in \left[2.731,5.184\right]\right)=0.95$

The bias-corrected formula is also used to estimate the shape parameter α. (Equation 33). Initially, the point estimate of α appears less accurate than the unbiased one; however, the confidence interval is more efficient and appropriate. The standard error is calculated using Equation 35, and the asymptotic confidence interval is determined using Equation 36.

${\hat{\alpha }}_c=\frac{n-2}{{{\displaystyle \sum }}_{i=1}^n\ln \left(\frac{x_i}{{\hat{x}}_m}\right)}=\frac{38}{{{\displaystyle \sum }}_{i=1}^{40}\ln \left(\frac{x_i}{502.595}\right)}=\frac{38}{10.108}=3.759$

$ase\left({\hat{\alpha }}_c\right)=\hat{\sigma }\left({\hat{\alpha }}_c\right)={\hat{\alpha }}_c/\sqrt{n-2}=3.759/\sqrt{38}=0.610$

$P\left({\hat{\alpha }}_c-z_{1-\frac{\alpha }{n}}\frac{{\hat{\alpha }}_c}{\sqrt{n-2}}\le \alpha \le {\hat{\alpha }}_c+z_{1-\frac{\alpha }{n}}\frac{{\hat{\alpha }}_c}{\sqrt{n-2}}\right)=1-\alpha$