Median Based Unit Weibull Distribution (MBUW): Does the Higher Order Probability Weighted Moments (PWM) Add More Information over the Lower Order PWM in Parameter Estimation

2.1. Calculating ${\mathit{M}}_{1,0,1}$

Using binomial expansion in equation (13):

Now exchange integration with summation in equations (13) & (14):

Where the integral is:

The same steps follow for:

Substitute s=1 in equation (17) and (18):

For

Substitute equation (19) and (20) into equation (14)

$$M_{\mathrm{1,0},1}={\displaystyle \frac{360+108{\alpha }^{\beta }}{1044{\alpha }^{\beta }+580{\alpha }^{2\beta }+155{\alpha }^{3\beta }+20{\alpha }^{4\beta }+{\alpha }^{5\beta }+720}}$$

2.2. Calculate $M_{\mathrm{1,1},0}$

Using binomial expansion:

Exchange the integral and the sum

The same steps are followed and give:

$$\underset{0}{\overset{1}{\int }}D\left(y\right)dy={\displaystyle \frac{6}{{\alpha }^{\beta }}}\sum _{i=0}^r{3^r\left(-1\right)}^i\left({\displaystyle \genfrac{}{}{0pt}{}{r}{i}}\right){\left({\displaystyle \frac{2}{3}}\right)}^i\left({\displaystyle \frac{{\alpha }^{\beta }}{3+2r+i+{\alpha }^{\beta }}}\right)\dots \dots \left(25\right)$$

(25)

Substitute Equation (24) and Equation (25) into Equation (22) and let r=1

To sum up, PWMs method for estimating the parameters using $M_{\mathrm{1,0},1}$ and $M_{\mathrm{1,1},0}$ :

Step 1: Calculate the population PWMs for the order p=1

Step 2: Calculate the estimated sample PWMs whether the unbiased or biased estimators and equate these estimators with the corresponding population PWMs.

Step 3: The above equations construct system of equations to be solved numerically. In this paper, the author used Levenberg-Marquardt (LM) algorithm. The objective functions to be minimized are equations (27) and (28).

Differentiate the previous equations (27-28) with respect to alpha and beta

The Jacobian matrix is $\left[\begin{array}{cc}{\displaystyle \frac{\partial }{\partial \alpha }}M\_101& {\displaystyle \frac{\partial }{\partial \beta }}M\_101\\ {\displaystyle \frac{\partial }{\partial \alpha }}M\_110& {\displaystyle \frac{\partial }{\partial \beta }}M\_110\end{array}\right]$

$$\left(y-f\left({\mathit{\theta }}^{\left(\mathit{a}\right)}\right)\right)=\left[\begin{array}{c}{\widehat{\alpha }}_r-{\displaystyle \frac{360+108{\alpha }^{\beta }}{1044{\alpha }^{\beta }+580{\alpha }^{2\beta }+155{\alpha }^{3\beta }+20{\alpha }^{4\beta }+{\alpha }^{5\beta }+720}}\\ {\widehat{\beta }}_s-{\displaystyle \frac{60+6{\alpha }^{\beta }}{74{\alpha }^{\beta }+15{\alpha }^{2\beta }+{\alpha }^{3\beta }+120}}\end{array}\right],$$

$where{\mathit{\theta }}^{\left(\mathit{a}\right)}=\left[\begin{array}{c}{\alpha }^{\left(a\right)}\\ {\beta }^{\left(a\right)}\end{array}\right]$

Apply LM algorithm:

$${\theta }^{\left(a+1\right)}={\theta }^{\left(a\right)}+{\left[J^{\text{'}}\left({\theta }^{\left(a\right)}\right)J\left({\theta }^{\left(a\right)}\right)+{\lambda }^{\left(a\right)}{I}^{\left(a\right)}\right]}^{-1}\left[J\text{'}\left({\theta }^{\left(a\right)}\right)\right]\left(y-f\left({\theta }^{\left(a\right)}\right)\right)$$

Where the parameters used in the first iteration are the initial guess, then they are updated according to the sum of squares of errors.

LM algorithm is an iterative algorithm.

$J^{\text{'}}\left({\theta }^{\left(a\right)}\right)$ : this is the Jacobian function which is the first derivative of the objective function evaluated at the initial guess ${\theta }^{\left(a\right)}$ .

${\lambda }^{\left(a\right)}$ is a damping factor that adjusts the step size in each iteration direction, the starting value usually is 0.001 and according to the sum square of errors (SSE) in each iteration this damping factor is adjusted:

${SSE}^{\left(a+1\right)}\ge {SSE}^{\left(a\right)}soupdate:{\lambda }_{updated}=10*{\lambda }_{old}$

${SSE}^{\left(a+1\right)}<{SSE}^{\left(a\right)}soupdate:{\lambda }_{updated}={\displaystyle \frac{1}{10}}*{\lambda }_{old}$

$f\left({\theta }^{\left(a\right)}\right)$ : is the objective function (population PWMs, $M_{\mathrm{1,0},1}$ & $M_{\mathrm{1,1},0}$) evaluated at the initial guess.

$y$ : is the sample estimates of population PWMs , the $M_{\mathrm{1,0},1}$ and $M_{\mathrm{1,1},0}$.

Steps of the LM algorithm:

1-

Start with the initial guess of parameters (alpha and beta).

2-

Substitute these values in the objective function and the Jacobian.

3-

Choose the damping factor, say lambda=0.001

4-

Substitute in equation (LM equation) to get the new parameters.

5-

Calculate the SSE at these parameters and compare this SSE value with the previous one when using initial parameters to adjust for the damping factor.

6-

Update the damping factor accordingly as previously explained.

7-

Start new iteration with the new parameters and the new updated damping factor, i.e., apply the previous steps many times till convergence is achieved or a pre-specified number of iterations is accomplished.

The value of this quantity: ${\left[J^{\text{'}}\left({\theta }^{\left(a\right)}\right)J\left({\theta }^{\left(a\right)}\right)+{\lambda }^{\left(a\right)}{I}^{\left(a\right)}\right]}^{-1}$ can be considered a good approximation to the variance—covariance matrix of the estimated parameters. Standard errors for the estimated parameters are the square root of the diagonal of the elements in this matrix divided by sample size.

2.3. Calculate $M_{\mathrm{1,0},2}$

Substitute for s=2 in equation (13) and solve to get:

2.4. Calculate $M_{\mathrm{1,2},0}$

Substitute for r=2 in Equation (21) and solve to get:

To sum up PWMs method for estimating the parameters using $M_{\mathrm{1,0},2}$ and $M_{\mathrm{1,2},0}$ :

Step 1: Calculate the population PWMs for the order p=1. See equation (29) & (30)

Step 2: Calculate the estimated sample PWMs whether the unbiased or biased estimators and equate these estimators with the corresponding population PWMs.

Step 3: The above equations construct system of equations to be solved numerically. In this paper, the author used Levenberg-Marquardt (LM) algorithm.

The objective functions to be minimized are Equations (29) and (30).

Differentiate the previous Equations (29) and (30) with respect to alpha and beta

The Jacobian matrix is $\left[\begin{array}{cc}{\displaystyle \frac{\partial }{\partial \alpha }}M\_102& {\displaystyle \frac{\partial }{\partial \beta }}M\_102\\ {\displaystyle \frac{\partial }{\partial \alpha }}M\_120& {\displaystyle \frac{\partial }{\partial \beta }}M\_120\end{array}\right]$

Apply LM algorithm

$${\theta }^{\left(a+1\right)}={\theta }^{\left(a\right)}+{\left[J^{\text{'}}\left({\theta }^{\left(a\right)}\right)J\left({\theta }^{\left(a\right)}\right)+{\lambda }^{\left(a\right)}{I}^{\left(a\right)}\right]}^{-1}\left[J\text{'}\left({\theta }^{\left(a\right)}\right)\right]\left(y-f\left({\theta }^{\left(a\right)}\right)\right)$$