Figure 2. Plots of moments under different parametric values of BL2PFD
Moment Generating Function
Apart from generating functions, the moment generating function can be utilized to describe the characteristic of the random variable.
\begin{equation} M_{o}\left(t\right)=\int_{0}^{\beta}{e^{\text{tx}}f(x)dx}\nonumber \\ \end{equation}
If X follows BL2PFD, the moment generating function may be derived as,
\(M_{o}\left(t\right)=\sum_{r=0}^{\infty}{\frac{t^{r}}{r!}\frac{\ a_{j}\ a_{i}\ a_{l}\ \beta^{r}}{B\left(a,b\right)\ (a+l+j)}}\)
\begin{equation} \text{where\ }a_{j}=\sum_{j=0}^{\infty}\frac{\left(-1\right)^{j}\Gamma(b)}{\Gamma\left(b-j\right)\ j!},\ a_{i}=\sum_{i=0}^{\infty}\frac{\left(-1\right)^{i}\Gamma(\frac{r}{\gamma}+1)}{\Gamma\left(\frac{r}{\gamma}+1-i\right)\ i!}\text{\ and\ }a_{l}=\sum_{l=0}^{\infty}\frac{\left(-1\right)^{l}\Gamma(\frac{i}{\alpha}+1)}{\Gamma\left(\frac{i}{\alpha}+1-l\right)\ l!}\nonumber \\ \end{equation}
Random Number Generator
The random number of BL2PFD may be obtained from
\(F(x)=\frac{B_{\left\{1-\left\{1-\left(\frac{x}{\beta}\right)^{\gamma}\right\}^{\alpha}\right\}}(a,b)}{B(a,b)}\)
After simplifying we get,
\begin{equation} x=\beta\left\{1-\left\{1-rbeta(n,a,b)\right\}^{\frac{1}{\alpha}}\right\}^{\frac{1}{\gamma}}\nonumber \\ \end{equation}
Where “\(rbeta(n,a,b)"\) is the random numbers generated from Beta distribution.
Inverse Moments
By definition Inverse moments may be obtained as
\begin{equation} \mu_{-r}^{{}^{\prime}}=\int_{0}^{\beta}{x^{-r}f(x)}\text{dx\ \ \ \ }\nonumber \\ \end{equation}
We get inverse moments for BL2PFD as
\(\mu_{-r}^{{}^{\prime}}=\frac{\ a_{j}\ a_{i}\ a_{l}\ \beta^{-r}}{B\left(a,b\right)\ (a+l+j)}\)
\begin{equation} \text{where\ }a_{j}=\sum_{j=0}^{\infty}\frac{\left(-1\right)^{j}\Gamma(b)}{\Gamma\left(b-j\right)\ j!},\ a_{i}=\sum_{i=0}^{\infty}\frac{\left(-1\right)^{i}\Gamma(\frac{-r}{\gamma}+1)}{\Gamma\left(\frac{-r}{\gamma}+1-i\right)\ i!}\text{\ and\ }a_{l}=\sum_{l=0}^{\infty}\frac{\left(-1\right)^{l}\Gamma(\frac{i}{\alpha}+1)}{\Gamma\left(\frac{i}{\alpha}+1-l\right)\ l!}\nonumber \\ \end{equation}
Vitality function
The vitality function is obtained for BL2PFD as
\(V\left(x\right)=\frac{1}{S\left(x\right)}\int_{x}^{\beta}{\text{x\ f}\left(x\right)}\text{dx}\)
That may be obtained as
\begin{equation} V\left(x\right)=\frac{\frac{\frac{\beta}{B(a,b)}\ a_{j}\ a_{i}\ a_{l}\left[1-\left\{1-\left\{1-\left(\frac{x}{\beta}\right)^{\gamma}\right\}^{\alpha}\right\}^{\left(a+l+j\right)}\right]}{\left(a+l+j\right)}}{1-\left\{\frac{B_{\left\{1-\left\{1-\left(\frac{x}{\beta}\right)^{\gamma}\right\}^{\alpha}\right\}}(a,b)}{B(a,b)}\right\}}\nonumber \\ \end{equation}\begin{equation} \text{where\ }a_{j}=\sum_{j=0}^{\infty}\frac{\left(-1\right)^{j}\Gamma(b)}{\Gamma\left(b-j\right)\ j!},\ a_{i}=\sum_{i=0}^{\infty}\frac{\left(-1\right)^{i}\Gamma(\frac{1}{\gamma}+1)}{\Gamma\left(\frac{1}{\gamma}+1-i\right)\ i!}\text{\ and\ }a_{l}=\sum_{l=0}^{\infty}\frac{\left(-1\right)^{l}\Gamma(\frac{i}{\alpha}+1)}{\Gamma\left(\frac{i}{\alpha}+1-l\right)\ l!}\nonumber \\ \end{equation}
Information Function
The Information Function is given as
\(IF=\int_{0}^{\beta}{\left\{f(x)\right\}^{s}\text{dx}}\)
For BL2PFD the information function is given as
\(IF=\frac{{(\alpha\gamma)}^{s-1}\beta^{(\gamma-1)(s-1)}a_{j}a_{i}a_{l}}{\beta^{(\gamma s-\gamma)}B(a,b)\left(s(a-1)+j+l\right)}\)
\begin{equation} \text{where\ }a_{j}=\sum_{j=0}^{\infty}\frac{\left(-1\right)^{j}\Gamma\left(\frac{\alpha\left(sb-1\right)-\left(s-1\right)}{\alpha}+1\right)}{\Gamma\left(\frac{\alpha\left(sb-1\right)-\left(s-1\right)}{\alpha}+1-j\right)\ j!},\ a_{i}=\sum_{i=0}^{\infty}\frac{\left(-1\right)^{i}\Gamma\left(\frac{\left(\gamma-1\right)-\left(s-1\right)}{\gamma}+1\right)}{\Gamma\left(\frac{\left(\gamma-1\right)-\left(s-1\right)}{\gamma}+1-i\right)\ i!}\nonumber \\ \end{equation}\begin{equation} \text{\ \ and\ }a_{l}=\sum_{l=0}^{\infty}\frac{\left(-1\right)^{l}\Gamma(\frac{i}{\alpha}+1)}{\Gamma\left(\frac{i}{\alpha}+1-l\right)\ l!}\nonumber \\ \end{equation}
Order Statistics
The pdf of the order statistic may be written as
\(f_{1:n}\left(x\right)=\frac{1}{B\left(1,n\right)}f(x)\left\{1-F(x)\right\}^{n-1}\)
For BL2PFD, we may write the lower and upper order statistics as
\begin{equation} f_{1:n}\left(x\right)=\frac{1}{B\left(1,n\right)}\left\{\frac{\left(1-\left(1-\left(\frac{x}{\beta}\right)^{\gamma}\right)^{\alpha}\right)^{a-1}\left(\left(1-\left(\frac{x}{\beta}\right)^{\gamma}\right)^{\alpha}\right)^{b-1}\alpha\left(1-\left(\frac{x}{\beta}\right)^{\gamma}\right)^{\alpha-1}\frac{\gamma x^{\gamma-1}}{\beta^{\gamma}}}{B\left(a,b\right)}\right\}*\nonumber \\ \end{equation}
\(\left[1-\left\{\frac{B_{\left\{1-\left\{1-\left(\frac{x}{\beta}\right)^{\gamma}\right\}^{\alpha}\right\}}(a,b)}{B(a,b)}\right\}\right]^{n-1}\)
and
\(f_{n:n}\left(x\right)=\frac{1}{B\left(1,n\right)}f(x)\left\{F(x)\right\}^{n-1}\)
\begin{equation} f_{n:n}\left(x\right)=\frac{1}{B\left(1,n\right)}\left\{\frac{\left(1-\left(1-\left(\frac{x}{\beta}\right)^{\gamma}\right)^{\alpha}\right)^{a-1}\left(\left(1-\left(\frac{x}{\beta}\right)^{\gamma}\right)^{\alpha}\right)^{b-1}\alpha\left(1-\left(\frac{x}{\beta}\right)^{\gamma}\right)^{\alpha-1}\frac{\gamma x^{\gamma-1}}{\beta^{\gamma}}}{B\left(a,b\right)}\right\}*\nonumber \\ \end{equation}
\(\left[\frac{B_{\left\{1-\left\{1-\left(\frac{x}{\beta}\right)^{\gamma}\right\}^{\alpha}\right\}}(a,b)}{B(a,b)}\right]^{n-1}\)
Incomplete Moments
The incomplete moments are given as
\(\mu_{X|\left(a,b,\alpha,\beta,\gamma\right);r}(p)=\int_{0}^{P}{x^{r}f\left(x\right)\text{dx}}\)
By simplifying for BL2PFD we get
\(\mu_{X|\left(\alpha,\beta,\gamma,\varphi,\theta\right);r}(p)=\frac{a_{j}\ a_{i}\ a_{l}\ \beta^{r}\left[{1-\left\{1-\left(\frac{p}{\beta}\right)^{\gamma}\right\}}^{\alpha}\right]^{(a+l+j)}}{B\left(a,b\right)(a+l+j)}\)
\begin{equation} \text{where\ }a_{j}=\sum_{j=0}^{\infty}\frac{\left(-1\right)^{j}\Gamma(b)}{\Gamma\left(b-j\right)\ j!},\ a_{i}=\sum_{i=0}^{\infty}\frac{\left(-1\right)^{i}\Gamma(\frac{r}{\gamma}+1)}{\Gamma\left(\frac{r}{\gamma}+1-i\right)\ i!}\text{\ and\ }a_{l}=\sum_{l=0}^{\infty}\frac{\left(-1\right)^{l}\Gamma(\frac{i}{\alpha}+1)}{\Gamma\left(\frac{i}{\alpha}+1-l\right)\ l!}\nonumber \\ \end{equation}
Conditional Moments
The conditional moments may be obtained as
\(E\left[X^{r}|X>t\right]=\frac{1}{\overset{\overline{}}{F}(t)}\int_{t}^{\beta}{x^{r}f(x)}\text{dx}\)
The conditional moments for BL2PFD may be obtained by using above expression as
\(E\left[X^{r}|X>t\right]=\frac{1}{\overset{\overline{}}{F}(t)}\frac{a_{j}\ a_{i}\ a_{l}\ \beta^{r}\left[\left\{1-\left(\frac{t}{\beta}\right)^{\gamma}\right\}^{\alpha}\right]^{(a+l+j)}}{B\left(a,b\right)(a+l+j)}\)
\begin{equation} \text{where\ }a_{j}=\sum_{j=0}^{\infty}\frac{\left(-1\right)^{j}\Gamma(b)}{\Gamma\left(b-j\right)\ j!},\ a_{i}=\sum_{i=0}^{\infty}\frac{\left(-1\right)^{i}\Gamma(\frac{r}{\gamma}+1)}{\Gamma\left(\frac{r}{\gamma}+1-i\right)\ i!}\text{\ and\ }a_{l}=\sum_{l=0}^{\infty}\frac{\left(-1\right)^{l}\Gamma(\frac{i}{\alpha}+1)}{\Gamma\left(\frac{i}{\alpha}+1-l\right)\ l!}\nonumber \\ \end{equation}
Lorenz and Bonferroni Curve
The Lorenz and Bonferroni curve may be obtained as
\begin{equation} L(p)=\frac{1}{\mu}\int_{0}^{q}{x\sum_{l=0}^{\infty}{t_{l}h_{l+1}(x)}}\text{dx}\nonumber \\ \end{equation}
\(L(p)=\frac{1}{\mu}\frac{a_{j}\ a_{i}\ a_{l}\text{\ β}\left[1-\left\{1-\left(\frac{q}{\beta}\right)^{\gamma}\right\}^{\alpha}\right]^{(a+l+j)}}{B\left(a,b\right)(a+l+j)}\)
\begin{equation} \text{where\ }a_{j}=\sum_{j=0}^{\infty}\frac{\left(-1\right)^{j}\Gamma(b)}{\Gamma\left(b-j\right)\ j!},\ a_{i}=\sum_{i=0}^{\infty}\frac{\left(-1\right)^{i}\Gamma(\frac{1}{\gamma}+1)}{\Gamma\left(\frac{1}{\gamma}+1-i\right)\ i!}\text{\ and\ }a_{l}=\sum_{l=0}^{\infty}\frac{\left(-1\right)^{l}\Gamma(\frac{i}{\alpha}+1)}{\Gamma\left(\frac{i}{\alpha}+1-l\right)\ l!}\nonumber \\ \end{equation}
\(B(p)=\frac{1}{\text{Pμ}}\frac{a_{j}\ a_{i}\ a_{l}\text{\ β}\left[1-\left\{1-\left(\frac{q}{\beta}\right)^{\gamma}\right\}^{\alpha}\right]^{(a+l+j)}}{B\left(a,b\right)(a+l+j)}\)
Characterization of BL2PFDLet “X” be Beta-Lehmann2- Power function variable with Probability density function
\begin{equation} f\left(x\right)=\frac{\left(1-\left(1-\left(\frac{x}{\beta}\right)^{\gamma}\right)^{\alpha}\right)^{a-1}\left(\left(1-\left(\frac{x}{\beta}\right)^{\gamma}\right)^{\alpha}\right)^{b-1}\alpha\left(1-\left(\frac{x}{\beta}\right)^{\gamma}\right)^{\alpha-1}\frac{\gamma x^{\gamma-1}}{\beta^{\gamma}}\ }{B(a,b)};\ \ \ \ \ \ 0<x<\beta\nonumber \\ \end{equation}
And let \(\overset{\overline{}}{F}\left(x\right)\) be the survival function respectively. Then the random variable “X” has BL2PFD if and only if
\begin{equation} V\left(X\middle|x\leq t\right)=\frac{a_{j}\ a_{h}\ a_{l}\ \beta^{2}}{F\left(t\right)B(a,b)}\left[\frac{1-\left\{1-\left(\frac{t}{\beta}\right)^{\gamma}\right\}}{a+j+l}^{\alpha}\right]^{a+j+l}-\left[\frac{a_{j}\ a_{i}\ a_{l}\text{\ β}}{F\left(t\right)B(a,b)}\left[\frac{1-\left\{1-\left(\frac{t}{\beta}\right)^{\gamma}\right\}}{a+j+l}^{\alpha}\right]^{a+j+l}\right]^{2}\nonumber \\ \end{equation}\begin{equation} \text{where\ \ \ \ \ }V\left(X\middle|x\leq t\right):\ Conditional\ variance\nonumber \\ \end{equation}\begin{equation} \text{Also\ }a_{j}=\sum_{j=0}^{\infty}\frac{\left(-1\right)^{j}\Gamma\left(b\right)}{\Gamma\left(b-j\right)\ j!},\ a_{i}=\sum_{i=0}^{\infty}\frac{\left(-1\right)^{i}\Gamma\left(\frac{1}{\gamma}+1\right)}{\Gamma\left(\frac{1}{\gamma}+1-i\right)\ i!},\ a_{l}=\sum_{l=0}^{\infty}\frac{\left(-1\right)^{l}\Gamma\left(\frac{i}{\alpha}+1\right)}{\Gamma\left(\frac{i}{\alpha}+1-l\right)\ l!}\text{\ and}\nonumber \\ \end{equation}\begin{equation} \ a_{h}=\sum_{i=0}^{\infty}\frac{\left(-1\right)^{i}\Gamma(\frac{2}{\gamma}+1)}{\Gamma\left(\frac{2}{\gamma}+1-i\right)\ i!}\nonumber \\ \end{equation}
Proof:
Necessary part:
\begin{equation} E\left(X^{r}\middle|x\leq t\right)=\frac{1}{F(t)}\int_{0}^{t}x^{r}\frac{\left(1-\left(1-\left(\frac{x}{\beta}\right)^{\gamma}\right)^{\alpha}\right)^{a-1}\left(\left(1-\left(\frac{x}{\beta}\right)^{\gamma}\right)^{\alpha}\right)^{b-1}\alpha\left(1-\left(\frac{x}{\beta}\right)^{\gamma}\right)^{\alpha-1}\frac{\gamma x^{\gamma-1}}{\beta^{\gamma}}\ }{B(a,b)}\text{dx}\nonumber \\ \end{equation}\begin{equation} \text{Put}\ 1-\left(1-\left(\frac{x}{\beta}\right)^{\gamma}\right)^{\alpha}=z\nonumber \\ \end{equation}\begin{equation} E\left(X^{r}\middle|x\leq t\right)=\frac{1}{F(t)B(a,b)}\left[\int_{0}^{1-\left(1-\left(\frac{t}{\beta}\right)^{\gamma}\right)^{\alpha}}{\beta^{r}\left\{1-\left(1-z\right)^{1/\alpha}\right\}^{r/\gamma}\left(z\right)^{a-1}\left(1-z\right)^{b-1}}\text{dz}\right]\nonumber \\ \end{equation}\begin{equation} E\left(X^{r}\middle|x\leq t\right)=\frac{a_{j}\ a_{i}\ a_{l}\ \beta^{r}}{F\left(t\right)B\left(a,b\right)}\left[\frac{1-\left\{1-\left(\frac{t}{\beta}\right)^{\gamma}\right\}}{a+j+l}^{\alpha}\right]^{a+j+l}\text{\ \ \ \ \ \ \ \ \ \ }\nonumber \\ \end{equation}\begin{equation} E\left(X\middle|x\leq t\right)=\frac{a_{j}\ a_{i}\ a_{l}\text{\ β}}{F\left(t\right)B\left(a,b\right)}\left[\frac{1-\left\{1-\left(\frac{t}{\beta}\right)^{\gamma}\right\}}{a+j+l}^{\alpha}\right]^{a+j+l}\nonumber \\ \end{equation}\begin{equation} \text{Where\ }a_{j}=\sum_{j=0}^{\infty}\frac{\left(-1\right)^{j}\Gamma(b)}{\Gamma\left(b-j\right)\ j!},\ a_{i}=\sum_{i=0}^{\infty}\frac{\left(-1\right)^{i}\Gamma(\frac{1}{\gamma}+1)}{\Gamma\left(\frac{1}{\gamma}+1-i\right)\ i!}\text{\ and\ }a_{l}=\sum_{l=0}^{\infty}\frac{\left(-1\right)^{l}\Gamma(\frac{i}{\alpha}+1)}{\Gamma\left(\frac{i}{\alpha}+1-l\right)\ l!}\nonumber \\ \end{equation}
Put r=2
\begin{equation} E\left(X^{2}\middle|x\leq t\right)=\frac{a_{j}\ a_{h}\ a_{l}\ \beta^{2}}{F\left(t\right)B\left(a,b\right)}\left[\frac{1-\left\{1-\left(\frac{t}{\beta}\right)^{\gamma}\right\}}{a+j+l}^{\alpha}\right]^{a+j+l}\text{\ \ \ \ \ \ \ \ \ \ }\nonumber \\ \end{equation}\begin{equation} \text{Where\ }a_{j}=\sum_{j=0}^{\infty}\frac{\left(-1\right)^{j}\Gamma(b)}{\Gamma\left(b-j\right)\ j!},\ a_{h}=\sum_{i=0}^{\infty}\frac{\left(-1\right)^{i}\Gamma(\frac{2}{\gamma}+1)}{\Gamma\left(\frac{2}{\gamma}+1-i\right)\ i!}\text{\ and\ }a_{l}=\sum_{l=0}^{\infty}\frac{\left(-1\right)^{l}\Gamma(\frac{i}{\alpha}+1)}{\Gamma\left(\frac{i}{\alpha}+1-l\right)\ l!}\nonumber \\ \end{equation}\begin{equation} V\left(X\middle|x\leq t\right)=\frac{a_{j}\ a_{h}\ a_{l}\ \beta^{2}}{F\left(t\right)B(a,b)}\left[\frac{1-\left\{1-\left(\frac{t}{\beta}\right)^{\gamma}\right\}}{a+j+l}^{\alpha}\right]^{a+j+l}-\left[\frac{a_{j}\ a_{i}\ a_{l}\text{\ β}}{F\left(t\right)B\left(a,b\right)}\left[\frac{1-\left\{1-\left(\frac{t}{\beta}\right)^{\gamma}\right\}}{a+j+l}^{\alpha}\right]^{a+j+l}\right]^{2}\ \ \ (7)\nonumber \\ \end{equation}
Also Sufficient part
\begin{equation} V\left(X\middle|x\leq t\right)=\frac{1}{F\left(t\right)}\int_{0}^{t}x^{2}dx-\ \left\{\frac{1}{F\left(t\right)}\int_{0}^{t}x\text{dx}\right\}^{2}\ \nonumber \\ \end{equation}\begin{equation} V\left(X\middle|x\leq t\right)=t^{2}-2\int_{0}^{t}\frac{\text{xF}\left(x\right)}{F\left(t\right)}dx-\ \left\{t-\int_{0}^{t}\frac{F\left(x\right)}{F\left(t\right)}\text{dx}\right\}^{2}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (8)\nonumber \\ \end{equation}
Equate (7) and (8), we get
\begin{equation} t^{2}-2\int_{0}^{t}\frac{\text{xF}\left(x\right)}{F\left(t\right)}dx-\ \left\{t-\int_{0}^{t}\frac{F\left(x\right)}{F\left(t\right)}\text{dx}\right\}^{2}=\frac{a_{j}\ a_{h}\ a_{l}\ \beta^{2}}{F\left(t\right)B(a,b)}\left[\frac{1-\left\{1-\left(\frac{t}{\beta}\right)^{\gamma}\right\}}{a+j+l}^{\alpha}\right]^{a+j+l}-\left[\frac{a_{j}\ a_{i}\ a_{l}\text{\ β}}{F\left(t\right)B(a,b)}\left[\frac{1-\left\{1-\left(\frac{t}{\beta}\right)^{\gamma}\right\}}{a+j+l}^{\alpha}\right]^{a+j+l}\right]^{2}\nonumber \\ \end{equation}\begin{equation} t-\int_{0}^{t}\frac{F\left(x\right)}{F\left(t\right)}dx=\frac{a_{j}\ a_{i}\ a_{l}\text{\ β}}{F\left(t\right)B(a,b)}\left[\frac{1-\left\{1-\left(\frac{t}{\beta}\right)^{\gamma}\right\}}{a+j+l}^{\alpha}\right]^{a+j+l}\nonumber \\ \end{equation}
Therefore
\(t^{2}-2\int_{0}^{t}\frac{\text{xF}\left(x\right)}{F\left(t\right)}\text{dx}\)=\(\frac{a_{j}\ a_{h}\ a_{l}\ \beta^{2}}{F\left(t\right)B(a,b)}\left[\frac{1-\left\{1-\left(\frac{t}{\beta}\right)^{\gamma}\right\}}{a+j+l}^{\alpha}\right]^{a+j+l}\)
Differentiate w.r.t “t”
\(t^{2}f\left(t\right)=\)\(\frac{a_{j}\ a_{h}\ a_{l}\ \beta^{2}}{B\left(a,b\right)}\left[{1-\left\{1-\left(\frac{t}{\beta}\right)^{\gamma}\right\}}^{\alpha}\right]^{a+j+l-1}\alpha\left\{1-\left(\frac{t}{\beta}\right)^{\gamma}\right\}^{\alpha-1}\frac{\gamma t^{\gamma-1}}{\beta^{\gamma}}\)
As
\begin{equation} {a_{j}\ a_{h}\ a_{l}\left[{1-\left\{1-\left(\frac{t}{\beta}\right)^{\gamma}\right\}}^{\alpha}\right]}^{a+j+l-1}=\left(1-\left(1-\left(\frac{t}{\beta}\right)^{\gamma}\right)^{\alpha}\right)^{a-1}\left(\left(1-\left(\frac{t}{\beta}\right)^{\gamma}\right)^{\alpha}\right)^{b-1}\left(\frac{t}{\beta}\right)^{2}\nonumber \\ \end{equation}
Therefore
\begin{equation} t^{2}f\left(t\right)=\frac{\beta^{2}}{B\left(a,b\right)}\left(1-\left(1-\left(\frac{t}{\beta}\right)^{\gamma}\right)^{\alpha}\right)^{a-1}\left(\left(1-\left(\frac{t}{\beta}\right)^{\gamma}\right)^{\alpha}\right)^{b-1}\left(\frac{t}{\beta}\right)^{2}\text{\ α}\left\{1-\left(\frac{t}{\beta}\right)^{\gamma}\right\}^{\alpha-1}\frac{\gamma t^{\gamma-1}}{\beta^{\gamma}}\nonumber \\ \end{equation}\begin{equation} f\left(t\right)=\frac{1}{B\left(a,b\right)}\left(1-\left(1-\left(\frac{t}{\beta}\right)^{\gamma}\right)^{\alpha}\right)^{a-1}\left(\left(1-\left(\frac{t}{\beta}\right)^{\gamma}\right)^{\alpha}\right)^{b-1}\text{\ α}\left\{1-\left(\frac{t}{\beta}\right)^{\gamma}\right\}^{\alpha-1}\frac{\gamma t^{\gamma-1}}{\beta^{\gamma}}\nonumber \\ \end{equation}
The pdf of BL2PFD
  1. Results
  2. Maximum Likelihood Method (MLM)
Let x1, x2 ,…, xn be a random sample of size “n” from the BL2PFD. The log-likelihood function for the BL2PFD is given by
\begin{equation} L\left(a,b,\alpha,\ \beta,\gamma\right)=nln\left(\frac{\text{αγ}}{\beta^{\gamma}}\right)+n\left(a-1\right)\ln\left\{1-\left(1-\left(\frac{x_{i}}{\beta}\right)^{\gamma}\right)^{\alpha}\right\}+n\left(\alpha b-1\right)\ln\left(1-\left(\frac{x_{i}}{\beta}\right)^{\gamma}\right)+n\left(\gamma-1\right)\ln x_{i}\nonumber \\ \end{equation}
The score vector is
\(U_{a}\left(a,b,\alpha,\ \beta,\gamma\right)=\frac{\partial}{\partial a}L\left(a,b,\alpha,\ \beta,\gamma\right)\)
\(U_{b}\left(a,b,\alpha,\ \beta,\gamma\right)=\frac{\partial}{\partial b}L\left(a,b,\alpha,\ \beta,\gamma\right)\)
\begin{equation} U_{\alpha}\left(a,b,\alpha,\ \beta,\gamma\right)=\frac{\partial}{\partial\alpha}L\left(a,b,\alpha,\ \beta,\gamma\right)\nonumber \\ \end{equation}\begin{equation} U_{\beta}\left(a,b,\alpha,\ \beta,\gamma\right)=\frac{\partial}{\partial\beta}L\left(a,b,\alpha,\ \beta,\gamma\right)\nonumber \\ \end{equation}\begin{equation} U_{\gamma}\left(a,b,\alpha,\ \beta,\gamma\right)=\frac{\partial}{\partial\gamma}L\left(a,b,\alpha,\ \beta,\gamma\right)\nonumber \\ \end{equation}
The parameters of BL2PFD can be obtained by solving the above equations resulting from setting the five partial derivatives of\(L\left(a,b,\alpha,\beta,\gamma\right)\) equals to zero.
Estimation of BL2PFD Parameters from “common percentiles” (P.E)
In [21] proposed a percentile estimator of the shape parameter, based on any two sample percentiles. After [21], [22] also discussed it, in which he estimated the parameters of Weibull distribution with the help of percentiles.
Let \(x_{1\ },x_{2},x_{3},\ldots,\ x_{n}\) be a random sample of size n drawn from Probability density function of BL2PFD. The cumulative distribution function of BL2PFD with shape and scale parameters\(\ \text{γ\ and\ β\ }\), respectively
\begin{equation} F\left(x\right)=\left\{\frac{B_{\left\{1-\left\{1-\left(\frac{x}{\beta}\right)^{\gamma}\right\}^{\alpha}\right\}}(a,b)}{B(a,b)}\right\}\nonumber \\ \end{equation}
By solving we get
\begin{equation} x=\beta\left\{1-\left\{1-rbeta(n,a,b)\right\}^{\frac{1}{\alpha}}\right\}^{\frac{1}{\gamma}}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }(9)\nonumber \\ \end{equation}
Where “\(rbeta(n,a,b)"\) is the random numbers generated from Beta distribution.
Let P75 and P25 are the 75th and 25th Percentiles,\(\text{therefore\ }\left(9\right)\text{\ becomes\ }\)
\begin{equation} P_{75}=\beta\left\{1-\left\{1-0.75\right\}^{\frac{1}{\alpha}}\right\}^{\frac{1}{\gamma}}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (10)\nonumber \\ \end{equation}\begin{equation} P_{25}=\ \beta\left\{1-\left\{1-0.25\right\}^{\frac{1}{\alpha}}\right\}^{\frac{1}{\gamma}}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (11)\nonumber \\ \end{equation}
Solving the above equations, we get
\begin{equation} \left(\frac{P_{75}}{P_{25}}\right)^{\gamma}=\ \left\{\frac{1-\left\{1-0.75\right\}^{\frac{1}{\alpha}}}{\left\{1-\left\{1-0.25\right\}^{\frac{1}{\alpha}}\right\}}\right\}\nonumber \\ \end{equation}\begin{equation} \gamma\ln{\left(\frac{P_{75}}{P_{25}}\right)=\ \ln{\ \left\{\frac{1-\left\{1-0.75\right\}^{\frac{1}{\alpha}}}{\left\{1-\left\{1-0.25\right\}^{\frac{1}{\alpha}}\right\}}\right\}}}\nonumber \\ \end{equation}\begin{equation} \hat{\gamma}=\ \frac{\ln{\ \left\{\frac{1-\left\{1-0.75\right\}^{\frac{1}{\alpha}}}{1-\left\{1-0.25\right\}^{\frac{1}{\alpha}}}\right\}}}{\ln\left(\frac{P_{75}}{P_{25}}\right)}\nonumber \\ \end{equation}\begin{equation} \text{and\ \ \ \ \ \ \ \ \ \ \ \ \ \ }\hat{\beta}=\ \frac{P_{75}}{\left[1-\left\{1-0.75\right\}^{\frac{1}{\alpha}}\right]^{\frac{1}{\hat{\gamma}}}}\nonumber \\ \end{equation}\begin{equation} \text{generally\ \ \ \ }\hat{\gamma}=\ \frac{\ln{\ \left\{\frac{1-\left\{1-H\right\}^{\frac{1}{\alpha}}}{1-\left\{1-L\right\}^{\frac{1}{\alpha}}}\right\}}}{\ln\left(\frac{P_{H}}{P_{L}}\right)}\nonumber \\ \end{equation}\begin{equation} \text{and\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\hat{\beta}=\ \frac{P_{H}}{\left[1-\left\{1-H\right\}^{\frac{1}{\alpha}}\right]^{\frac{1}{\hat{\gamma}}}}\nonumber \\ \end{equation}
Where H= Maximum Percentage, L= Minimum Percentage and P = Percentile
A simulation study is used in order to compare the performance of the proposed estimation methods. We carry out this comparison taking the samples of sizes as n = 40 and 150 with pairs of (\(\beta\), γ) = {(1, 2), (2, 1) and (1.5, 1.5)}. We generated random samples of different sizes by observing that if\(R_{i}\) is random number taking (0, 1), then\(x_{i}=\ \beta\left\{1-\left\{1-rbeta(n,a,b)\right\}^{\frac{1}{\alpha}}\right\}^{\frac{1}{\gamma}}\)is the random number generation from BL2PFD with (\(a,\ b,\alpha,\ \beta\ and\ \gamma\)) parameters. All results are based on 5000 replications.
Such generated data have been used to obtain estimates of the unknown parameters. The results obtained from parameters estimation of the 2-parameters (shape and scale parameters) of BL2PFD using different sample sizes and different values of parameters with mean square error MSE.
\begin{equation} \text{M.S.E\ }\left(\hat{\beta}\right)=\ E\left[\left(\hat{\beta}\mathbf{\ \ }\beta\right)^{2}\right],\ M.S.E\ \left(\hat{\gamma}\right)=\ E\left[\left(\hat{\gamma}\mathbf{\ }\gamma\right)^{2}\right]\ \nonumber \\ \end{equation}
Table 2. Estimates for the parameters of BL2PFD with different estimation methods under the sample size 40 when\(a=1,\ b=2\ and\ \alpha=3\)