Abstract
Probability distributions have great use in reliability engineering where the researchers try to find the distribution of the different processes. To meet the needs of the reliability engineers, we have proposed a simple probability distribution named as Beta Lehman-2 which may be proved more useful as compare to already existing models of the probability distributions. The aim of the study is to show the performance of the proposed distribution over already existing distributions. In this study, a new Beta Lehmann-2 Power function distribution (BL2PFD) is proposed. We suggest a new generator that will modify the Power function distribution called Beta Lehmann-2 generator (BL2-G). The various properties of the new distribution have been discussed in detail such as moments, vitality function, conditional moments and order statistics etc. We have also characterized the BL2PFD based on conditional variance. This distribution can be used for approximately symmetric data (normal data), positive and negative skewed data. The application of this distribution is illustrated by using data sets from medical and engineering sources. The shape of the new distribution has been studied for applied sciences. After analyzing data, we conclude that the proposed model BL2PFD perform better in all the data sets while compared to different competitor models.
Keywords: Beta Lehmann-2 Power function distribution, Characterization of truncated distribution, Lehmann alternatives, Percentile estimator, Power function distribution.
Introduction
The researchers in Engineering sciences mostly study the reliability of different components by taking the help from probability distributions that are simple in mathematical expression instead of using mathematically complex probability distributions. In [1] introduced the power function as the inverse of Pareto distribution. [2] showed that power function distribution is better to fit for failure data over exponential, lognormal and Weibull because it provides a better fit.
More studies about the application of this distribution and its applications can be found in [3, 4 and 5]. For modeling heterogeneous population, [6] talked about the two component mixture of one-parameter Power function distribution. Estimation of the parameters of the two-parameter Power function distribution was studied by [7] through the methods of least squares, relative least squares and ridge regression. According to its applicability in real life situations for modeling survival data, [8] proposed the modification of the Power function distribution as Weibull-Power function distribution. By using the Bayesian inference, [9] estimated the parameter of the one-parameter Power function distribution. In [10] introduced the Transmuted Power function distribution by following Shaw and Buckley [11]. In [12] proposed the modification of the Power function distribution by using Marshall and Olkin [13] technique. In [14] proposed the McDonald Power function distribution and [15] proposed the Kumaraswamy Power function distribution. In [16] discussed the parameters estimation for continuous uniform distribution using modified percentile estimators. Further [17] introduced the exponentiated generalized class of Power function distribution.
Materials and Methods
Lehmann alternatives were introduced by [18] in the two-sample hypothesis testing context and are useful in survival analysis.
\(\varnothing\left(x\right)=1-\left\{1-G\left(x\right)\right\}^{\alpha}\text{\ \ \ \ \ \ }\)(Lehmann2 relationship)
In [19] proposed the Beta generator (Beta-G).
\begin{equation} F\left(x\right)=\frac{B_{\varnothing\left(x\right)}(a,b)}{B(a,b)}\nonumber \\ \end{equation}
Then the mixture of these two techniques is known as Beta Lehmann-2 generator (BL2-G). The probability density function (pdf) and cumulative distribution function (cdf) of the BL2-G are given as
\(F\left(x\right)=\frac{B_{1-\left\{1-G\left(x\right)\right\}^{\alpha}}(a,b)}{B(a,b)}\)(1)
And
\begin{equation} f\left(x\right)=\frac{\left(1-\left(1-G\left(x\right)\right)^{\alpha}\right)^{a-1}\left(\left(1-G\left(x\right)\right)^{\alpha}\right)^{b-1}\alpha\left(1-G\left(x\right)\right)^{\alpha-1}g\left(x\right)\ }{B(a,b)}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2)\nonumber \\ \end{equation}\begin{equation} \text{Where\ G}\left(x\right):cdf\ and\ g\left(x\right):pdf\text{\ of\ any\ probability\ distribution}\nonumber \\ \end{equation}
In this work, we suggest a new distribution that will generalize the Power function distribution (PFD) by using the above mentioned technique. We have derived some of the main structural properties of this distribution. We have also characterized the distribution by conditional moments (Right and Left Truncated mean), doubly truncated mean (DTM) and conditional variance. Maximum likelihood method (MLM) and Percentile estimation (P.E) method are used to estimate the shape and scale parameters of BL2PFD. The application of this distribution is illustrated by using data sets from medical and engineering sources.
Model Identification For Beta Lehmann-2 Power function distribution (BL2PFD)
The pdf and cdf of Power function distribution are given as follows
\(g\left(x\right)=\frac{\gamma x^{\gamma-1}}{\beta^{\gamma}};\ \ \ \ 0<x<\beta,\ \ \ \ \ \gamma>0\)(3)
and
\(G\left(x\right)=\left(\frac{x}{\beta}\right)^{\gamma}\) (4)
Where γ and β  are the shape and scale parameters.
Following the generator (1), the BL2PFD is obtained by putting (3) and (4) in (2) and simplifying, we get
\(f\left(x\right)=\frac{\left(1-\left(1-\left(\frac{x}{\beta}\right)^{\gamma}\right)^{\alpha}\right)^{a-1}\left(\left(1-\left(\frac{x}{\beta}\right)^{\gamma}\right)^{\alpha}\right)^{b-1}\alpha\left(1-\left(\frac{x}{\beta}\right)^{\gamma}\right)^{\alpha-1}\frac{\gamma x^{\gamma-1}}{\beta^{\gamma}}\ }{B(a,b)}\ \ ;\ \ \ 0<x<\beta\ \ \ \ \ \)(5)
and associated cdf is obtained by putting (4) in (1) as
\(F\left(x\right)=\frac{B_{\left\{1-\left\{1-\left(\frac{x}{\beta}\right)^{\gamma}\right\}^{\alpha}\right\}}(a,b)}{B(a,b)}\)(6)
We may observe α, a and b are the tuning parameters. γ as the shape andβ as scale parameter.
By definition, the survival function is
\begin{equation} S\left(x\right)=1-F\left(x\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}
And the Hazard Rate Function (HRF) of probability distribution is given as
\begin{equation} H\left(x\right)=\frac{f(x)}{S(x)}=\frac{\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)}}{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}
Asymptotic Behavior
The behavior of the pdf, cdf, hazard and survival functions of BL2PFD are being investigated as x → 0 and x → ∞.
  1. \(\operatorname{}{f\left(x\right)}=0\ ;\forall\ possible\ values\ of\ \alpha,\ a,\ b,\ \gamma\ and\ \beta.\)
  2. \(\operatorname{}{f\left(x\right)}=\infty\ ;\ \forall\ possible\ values\ of\ \alpha,\ a,\ b,\ \gamma\ and\ \beta.\)
  3. \(\operatorname{}{F\left(x\right)}=0\ ;\forall\ possible\ values\ of\ \alpha,\ a,\ b,\ \gamma\ and\ \beta.\)
  4. \(\operatorname{}{F\left(x\right)}=1\ ;if\ x=\text{β\ and\ }\forall\ possible\ values\ of\ \alpha,\ a,\ b,\ \gamma\ and\ \beta.\)
  5. \(\operatorname{}{F\left(x\right)}=0\ ;if\ x\neq\text{β\ if\ γ}=0\ and\ \alpha\neq 0\ .\)
  6. \(\operatorname{}{F\left(x\right)}=1\ ;if\ x\neq\text{β\ if\ γ}>0\ and\ \alpha=0\ .\)
  7. \(\operatorname{}{S\left(x\right)}=1\ ;if\ x\neq\text{β\ if\ γ}=0\ and\ \alpha\neq 0\ .\)
  8. \(\operatorname{}{S\left(x\right)}=0\ ;if\ x\neq\text{β\ if\ γ}>0\ and\ \alpha=0.\)
  9. \(\operatorname{}{H\left(x\right)}=0\ ;\forall\ possible\ values\ of\ \alpha,\beta,\ \gamma,\ \varphi\ and\ \theta.\)
  10. \(\operatorname{}{H\left(x\right)}=\infty\ ;\forall\ possible\ values\ of\ \alpha,\beta,\ \gamma,\ \varphi\ and\ \theta.\)
  11. Characteristics of Hazard function using Glaser method
In [20] had defined the conditions of increasing, decreasing, and upside-down bathtub-shaped failure rate. We use these conditions in our proposed distribution.
\begin{equation} \eta\left(x\right)=-\frac{\overset{\acute{}}{f}\left(x\right)}{f\left(x\right)}\nonumber \\ \end{equation}\begin{equation} \eta\left(x\right)=-\beta^{\gamma}\left[\frac{(\gamma-1)}{x}-\left(\alpha b-1\right)\left\{\frac{\frac{\gamma x^{\gamma-1}}{\beta^{\gamma}}}{\left(1-\left(\frac{x}{\beta}\right)^{\gamma}\right)}\right\}+\alpha(a-1)\left\{\frac{\left(1-\left(\frac{x}{\beta}\right)^{\gamma}\right)^{\alpha-1}}{\left\{1-\left(1-\left(\frac{x}{\beta}\right)^{\gamma}\right)^{\alpha}\right\}}\right\}\frac{\gamma x^{\gamma-1}}{\beta^{\gamma}}\right]\nonumber \\ \end{equation}
If x > 0, then the values of\(\overset{\acute{}}{\eta}\left(x\right)\) under the following conditions are given in Table 1.Table 1. Values of \(\overset{\acute{}}{\eta}\left(x\right)\)under the following conditions