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 → ∞.
- \(\operatorname{}{f\left(x\right)}=0\ ;\forall\ possible\ values\ of\ \alpha,\ a,\ b,\ \gamma\ and\ \beta.\)
- \(\operatorname{}{f\left(x\right)}=\infty\ ;\ \forall\ possible\ values\ of\ \alpha,\ a,\ b,\ \gamma\ and\ \beta.\)
- \(\operatorname{}{F\left(x\right)}=0\ ;\forall\ possible\ values\ of\ \alpha,\ a,\ b,\ \gamma\ and\ \beta.\)
- \(\operatorname{}{F\left(x\right)}=1\ ;if\ x=\text{β\ and\ }\forall\ possible\ values\ of\ \alpha,\ a,\ b,\ \gamma\ and\ \beta.\)
- \(\operatorname{}{F\left(x\right)}=0\ ;if\ x\neq\text{β\ if\ γ}=0\ and\ \alpha\neq 0\ .\)
- \(\operatorname{}{F\left(x\right)}=1\ ;if\ x\neq\text{β\ if\ γ}>0\ and\ \alpha=0\ .\)
- \(\operatorname{}{S\left(x\right)}=1\ ;if\ x\neq\text{β\ if\ γ}=0\ and\ \alpha\neq 0\ .\)
- \(\operatorname{}{S\left(x\right)}=0\ ;if\ x\neq\text{β\ if\ γ}>0\ and\ \alpha=0.\)
- \(\operatorname{}{H\left(x\right)}=0\ ;\forall\ possible\ values\ of\ \alpha,\beta,\ \gamma,\ \varphi\ and\ \theta.\)
- \(\operatorname{}{H\left(x\right)}=\infty\ ;\forall\ possible\ values\ of\ \alpha,\beta,\ \gamma,\ \varphi\ and\ \theta.\)
- 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