In this paper, a new one parameter life lime distribution has been proposed and named Ram Awadh distribution. Its moments and moments based measures have been derived. Statistical properties including stochastic ordering, mean deviations, Bonferroni and Lorenz curves, order statistics, Renyi entropy and stress–strength measure have been discussed. Simulation study of proposed distribution has also been discussed. For estimating its parameter method of moments and method of maximum likelihood have been discussed. Goodness of fit of Ram Awadh distribution has been presented and compared with other lifetime distributions of one parameter. It was found superior than other one parameter life time distributions.

Keywords: moments, reliability measures, stochastic ordering, mean deviation, bonferroni and lorenz curves, order statistics, renyi entropy measure, estimation of parameters, goodness of fit

One parameter new life time distribution having parameters $\lambda$ is defined by its pdf

$f(x;\lambda )=\frac{{\lambda }^6}{{\lambda }^6+120}(\lambda +x^5)e^{-\lambda x};x>0,\lambda >0$  (1.1)

$f_2(x;\lambda )=p{g}_1(x;\lambda )+(1-p)g_2(x;\lambda ,6)$

The corresponding cumulative distribution function (cdf) of (1.1) is given by

$F(x;\lambda )=1-[1+\frac{\lambda x({\lambda }^4{x}^4+5{\lambda }^3{x}^3+20{\lambda }^2{x}^2+60\lambda x+120)}{{\lambda }^6+120}]e^{-\lambda x};x>0,\lambda >0$  (1.2)

In this paper, new one parameter life time distribution has been proposed and named Ram Awadh distribution. Its raw moments and central moments have been obtained and coefficients of variation, skewness, kurtosis and index of dispersion have been discussed. Its hazard rate function, mean residual life function, stochastic ordering, mean deviations, Bonferroni and Lorenz curves, order statistics , Renyi entropy measure and stress – strength have been discussed. Both the method of moments and the method of maximum likelihood have been discussed for estimating the parameter of Ram Awadh distribution. A simulation study of distribution has also been carried out. The goodness of fit of the proposed distribution has been presented and compared with other lifetime distributions of one parameter.

${\mu }_r=\frac{r!\left[{\lambda }^6+(r+1)(r+2)(r+3)(r+4)(r+5)\right]}{{\lambda }^r({\lambda }^6+120)};r=1,2,3,\mathrm{...}$  (2.1)

${\mu }_1=\frac{{\lambda }^6+720}{\lambda ({\lambda }^6+120)}$,${\mu }_2=\frac{2({\lambda }^6+2520)}{{\lambda }^2({\lambda }^6+120)}$

${\mu }_3=\frac{6({\lambda }^6+6720)}{{\lambda }^3({\lambda }^6+120)}$, ${\mu }_4=\frac{24({\lambda }^6+15120)}{{\lambda }^4({\lambda }^6+120)}$

${\mu }_2=\frac{({\lambda }^{12}+3840{\lambda }^6+86400)}{{\lambda }^2{({\lambda }^6+120)}^2}$

${\mu }_3=\frac{2({\lambda }^{18}+12960{\lambda }^{12}-172800{\lambda }^6+1036800)}{{\lambda }^3{({\lambda }^6+120)}^3}$

${\mu }_4=\frac{9({\lambda }^{24}+25280{\lambda }^{18}+2054400{\lambda }^{12}+271872000{\lambda }^6+3317760000)}{{\lambda }^4{({\lambda }^6+120)}^4}$

$C.V=\frac{\sigma }{{\mu }_1}=\frac{\sqrt{({\lambda }^{12}+3840{\lambda }^6+86400)}}{({\lambda }^6+720)}$

${\beta }_2=\frac{{\mu }_4}{{\mu }_2}=\frac{9({\lambda }^{24}+25280{\lambda }^{18}+2054400{\lambda }^{12}+271872000{\lambda }^6+3317760000)}{{({\lambda }^{12}+3840{\lambda }^6+86400)}^2}$

$\gamma =\frac{{\sigma }^2}{{\mu }_1}=\frac{({\lambda }^{12}+3840{\lambda }^6+86400)}{\lambda ({\lambda }^6+120)({\lambda }^6+720)}$

The value of index of dispersion will be one at . To study the nature of C.V, $\sqrt{{\beta }_1}$ , ${\beta }_2$ , and of Ram Awadh distribution, graphs of C.V, $\sqrt{{\beta }_1}$ , ${\beta }_2$ , and of Ram Awadh distribution have been drawn for varying values of the parameter and presented in Figure 3.

There are two important reliability measures namely hazard rate function and mean residual life function. Let $X$ be a continuous random variable with pdf $f\left(x\right)$ and cdf $F\left(x\right)$ . The hazard rate function and the mean residual life function of are respectively defined as

$h\left(x\right)=\underset{\Delta x\to 0}{\lim }\frac{P(X<x+\Delta x|X>x)}{\Delta x}=\frac{f\left(x\right)}{1-F\left(x\right)}$   (3.1)

The corresponding $h\left(x\right)$ and $m\left(x\right)$ of Ram Awadh distribution (1.1) are as follows:

$h\left(x\right)=\frac{{\lambda }^6(\lambda +x^5)}{({\lambda }^5{x}^5+5{\lambda }^4{x}^4+20{\lambda }^3{x}^3+60{\lambda }^2{x}^2+120\lambda x+{\lambda }^6+120)}$   (3.1)

and $m\left(x\right)=\frac{1}{\left(\begin{array}{l}{\lambda }^5{x}^5+5{\lambda }^4{x}^4+20{\lambda }^3{x}^3\\ +60{\lambda }^2{x}^2+120\lambda x+{\lambda }^6+120\end{array}\right)}\underset{x}{\overset{\infty }{\int }}\left(\begin{array}{l}{\lambda }^5{t}^5+5{\lambda }^4{t}^4+20{\lambda }^3{t}^3+60{\lambda }^2{t}^2\\ +120\lambda t+{\lambda }^6+120\end{array}\right)e^{-\lambda t}dt$

$=\frac{({\lambda }^5{x}^5+10{\lambda }^4{x}^4+60{\lambda }^3{x}^3+240{\lambda }^2{x}^2+600\lambda x+{\lambda }^6+720)}{\lambda ({\lambda }^5{x}^5+5{\lambda }^4{x}^4+20{\lambda }^3{x}^3+60{\lambda }^2{x}^2+120\lambda x+{\lambda }^6+120)}$   (3.4)

The graphs of $h\left(x\right)$ and $m\left(x\right)$ of Ram Awadh distribution for varying values of parameter are presented in Figure 4 & 5.

Figure 4 $h\left(x\right)$ Plots of Ram Awadh distribution for varying values of $\lambda$.

Figure 5 $m\left(x\right)$ Plots of Ram Awadh distribution for varying values of $\lambda$.

For judging the comparative behavior of continuous distribution, it is important tool.

 A random variable $X$ is said to be smaller than a random variable in the

• Stochastic order if for all $\left(X{\le }_{st}Y\right) if F_X\left(x\right)\ge F_Y\left(x\right) for all x$
• Hazard rate order if for all $\left(X{\le }_{st}Y\right) if h_X\left(x\right)\ge h_Y\left(x\right) for all x$
• Mean residual life order if for all $\left(X{\le }_{mrl}Y\right) if m_X\left(x\right)\le m_Y\left(x\right) for all x$
• Likelihood ratio order if decreases in $\left(X{\le }_{mrl}Y\right) if \frac{f_X\left(x\right)}{f_Y\left(x\right)} for all x$.

The following results due to The following results due to Shaked & Shanthikumar⁸ are well known for establishing stochastic ordering of distributions.

$$\begin{gathered} X{\le }_{lr}Y\Rightarrow X{\le }_{hr}Y\Rightarrow X{\le }_{mrl}Y \\ \underset{X{\le }_{st}Y}{\Downarrow } \end{gathered}$$

The Ram Awadh distribution is ordered with respect to the strongest "likelihood ratio ordering" as established in the following theorem:

Theorem: Let and Ram Awadh distribution and respectively. If then and hence , and .

Proof: We have

$\frac{f_X(x;{\lambda }_1)}{f_Y(x;{\lambda }_2)}=\frac{{\lambda }_1({\lambda }_2+120)}{{\lambda }_2({\lambda }_1+120)}{e}^{-({\lambda }_1-{\lambda }_2)x};x>0$

Now $\ln \frac{f_X(x;{\lambda }_1)}{f_Y(x;{\lambda }_2)}=\ln \left[\frac{{\lambda }_1({\lambda }_2+120)}{{\lambda }_2({\lambda }_1+120)}\right]+\ln \left(\frac{{\lambda }_1+x^5}{{\lambda }_2+x^5}\right)-({\lambda }_1-{\lambda }_2)x$

Thus if ${\theta }_1>{\theta }_2$, $\frac{d}{dx}\ln \frac{f_X(x;{\theta }_1)}{f_Y(x;{\theta }_2)}<0$. This means that $X{\le }_{lr}Y$ and hence $X{\le }_{hr}Y, X{\le }_{mrl}Y and X{\le }_{st}Y$.

The mean deviation about mean and median defined by

${\delta }_1\left(X\right)=\underset{0}{\overset{\infty }{\int }}|x-\mu |f\left(x\right)dx$ and ${\delta }_2\left(X\right)=\underset{0}{\overset{\infty }{\int }}|x-M|f\left(x\right)dx$ respectively, where $\mu =E\left(X\right)$ and $M=\text{Median }\left(X\right)$. The measures ${\delta }_1\left(X\right)$ and ${\delta }_2\left(X\right)$ can be calculated using the following simplified relationships

${\delta }_1\left(X\right)=\underset{0}{\overset{\mu }{\int }}(\mu -x)f\left(x\right)dx+\underset{\mu }{\overset{\infty }{\int }}(x-\mu )f\left(x\right)dx$

$=\mu F\left(\mu \right)-\underset{0}{\overset{\mu }{\int }}xf\left(x\right)dx-\mu [1-F\left(\mu \right)]+\underset{\mu }{\overset{\infty }{\int }}xf\left(x\right)dx$

$=2\mu F\left(\mu \right)-2\mu +2\underset{\mu }{\overset{\infty }{\int }}xf\left(x\right)dx$

$=2\mu F\left(\mu \right)-2\underset{0}{\overset{\mu }{\int }}xf\left(x\right)dx$

and ${\delta }_2\left(X\right)=\underset{0}{\overset{M}{\int }}(M-x)f\left(x\right)dx+\underset{M}{\overset{\infty }{\int }}(x-M)f\left(x\right)dx$

$=MF\left(M\right)-\underset{0}{\overset{M}{\int }}xf\left(x\right)dx-M[1-F\left(M\right)]+\underset{M}{\overset{\infty }{\int }}xf\left(x\right)dx$

$=-\mu +2\underset{M}{\overset{\infty }{\int }}xf\left(x\right)dx$

$=\mu -2\underset{0}{\overset{M}{\int }}xf\left(x\right)dx$   (5.2)

$\underset{0}{\overset{\mu }{\int }}xf(x;\lambda )dx=\mu -\frac{\left\{\begin{array}{l}{\lambda }^7\mu +{\lambda }^6({\mu }^6+1)+6{\lambda }^4{\mu }^4(\lambda \mu +5)+120{\mu }^{}{\lambda }^2(\lambda \mu +5)+\\ 120{\lambda }^2{\mu }^2(\lambda \mu +3)+720(\lambda \mu +1)\end{array}\right\}{e}^{-\lambda \mu }}{\lambda ({\lambda }^6+120)}$   (5.3)

$\underset{0}{\overset{M}{\int }}xf(x;\theta )dx=\mu -\frac{\left\{\begin{array}{l}{\lambda }^{7}M+{\lambda }^6(M^6+1)+6{\lambda }^4{M}^4(\lambda M+5)+120M^{}{\lambda }^2(\lambda M+5)+\\ 120{\lambda }^2{M}^2(\lambda M+3)+720(\lambda M+1)\end{array}\right\}{e}^{-\lambda M}}{\lambda ({\lambda }^6+120)}$   (5.4)

${\delta }_1\left(X\right)=\frac{2\left\{{\lambda }^5{\mu }^5+10{\lambda }^3{\mu }^3(\lambda \mu +6)+120\lambda \mu (2\lambda \mu +5)+({\lambda }^6+720)\right\}{e}^{-\lambda \mu }}{\lambda ({\lambda }^6+120)}$   (5.5)

${\delta }_2\left(X\right)=\frac{2\left\{\begin{array}{l}{\lambda }^{7}M+{\lambda }^6(M^6+1)+6{\lambda }^4{M}^4(\lambda M+5)+120M^{}{\lambda }^2(\lambda M+5)+\\ 120{\lambda }^2{M}^2(\lambda M+3)+720(\lambda M+1)\end{array}\right\}{e}^{-\lambda M}}{\lambda ({\lambda }^6+120)}-\mu$   (5.6)

It was given by Bonferron,9 and Bonferroni and Gini indices have applications not only in economics to study income and poverty, but it has also in many applications in different fields, such as demography, insurance and medicine. It can be defined as

$B\left(p\right)=\frac{1}{p\mu }\underset{0}{\overset{q}{\int }}xf\left(x\right)dx=\frac{1}{p\mu }[\underset{0}{\overset{\infty }{\int }}xf\left(x\right)dx-\underset{q}{\overset{\infty }{\int }}xf\left(x\right)dx]=\frac{1}{p\mu }[\mu -\underset{q}{\overset{\infty }{\int }}xf\left(x\right)dx]$ (6.1)

$L\left(p\right)=\frac{1}{\mu }\underset{0}{\overset{q}{\int }}xf\left(x\right)dx=\frac{1}{\mu }[\underset{0}{\overset{\infty }{\int }}xf\left(x\right)dx-\underset{q}{\overset{\infty }{\int }}xf\left(x\right)dx]=\frac{1}{\mu }[\mu -\underset{q}{\overset{\infty }{\int }}xf\left(x\right)dx]$  (6.2)

$B\left(p\right)=\frac{1}{p\mu }\underset{0}{\overset{p}{\int }}{F}^{-1}\left(x\right)dx$ (6.3)

$L\left(p\right)=\frac{1}{\mu }\underset{0}{\overset{p}{\int }}{F}^{-1}\left(x\right)dx$ (6.4)

The Bonferroni and Gini indices are thus defined as

$B=1-\underset{0}{\overset{1}{\int }}B\left(p\right)dp$ (6.5)

and $G=1-2\underset{0}{\overset{1}{\int }}L\left(p\right)dp$ (6.6)

Using pdf of Ram Awadh distribution (1.1), it can be written

$\underset{q}{\overset{\infty }{\int }}xf(x;\lambda )dx=\frac{\left\{\begin{array}{l}{\lambda }^{7}q+{\lambda }^6(q^6+1)+6{\lambda }^4{q}^4(\lambda q+5)+120q^{}{\lambda }^2(\lambda q+5)\\ +120{\lambda }^2{q}^2(\lambda q+3)+720(\lambda q+1)\end{array}\right\}{e}^{-\lambda q}}{\lambda ({\lambda }^6+120)}$ (6.7)

$B\left(p\right)=\frac{1}{p}[1-\frac{\left\{\begin{array}{l}{\lambda }^{7}q+{\lambda }^6(q^6+1)+6{\lambda }^4{q}^4(\lambda q+5)+120q^{}{\lambda }^2(\lambda q+5)\\ +120{\lambda }^2{q}^2(\lambda q+3)+720(\lambda q+1)\end{array}\right\}{e}^{-\lambda q}}{({\lambda }^6+720)}]$ (6.8)

and $L\left(p\right)=1-\frac{\left\{\begin{array}{l}{\lambda }^{7}q+{\lambda }^6(q^6+1)+6{\lambda }^4{q}^4(\lambda q+5)+120q^{}{\lambda }^2(\lambda q+5)\\ +120{\lambda }^2{q}^2(\lambda q+3)+720(\lambda q+1)\end{array}\right\}{e}^{-\lambda q}}{({\lambda }^6+720)}$ (6.9)

Now using equations (6.8) and (6.9) in (6.5) and (6.6), the Bonferroni and Gini indices of Ram Awadh distribution are thus given as

$B=1-\frac{\left\{\begin{array}{l}{\lambda }^{7}q+{\lambda }^6(q^6+1)+6{\lambda }^4{q}^4(\lambda q+5)+120q^{}{\lambda }^2(\lambda q+5)\\ +120{\lambda }^2{q}^2(\lambda q+3)+720(\lambda q+1)\end{array}\right\}{e}^{-\lambda q}}{({\lambda }^6+720)}$ (6.10)

$G=\frac{2\left\{\begin{array}{l}{\lambda }^{7}q+{\lambda }^6(q^6+1)+6{\lambda }^4{q}^4(\lambda q+5)+120q^{}{\lambda }^2(\lambda q+5)\\ +120{\lambda }^2{q}^2(\lambda q+3)+720(\lambda q+1)\end{array}\right\}{e}^{-\lambda q}}{({\lambda }^6+720)}-1$ (6.11)

Order statistics

Let $X_1,X_2,\mathrm{...},X_n$ be a random sample of size $n$ from Ram Awadh distribution (1.1). Let $X_{\left(1\right)}<X_{\left(2\right)}<\mathrm{...}<X_{\left(n\right)}$ denote the corresponding order statistics. The pdf and the cdf of the th order statistic, say $Y=X_{\left(k\right)}$ are given by

$f_Y\left(y\right)=\frac{n!}{(k-1)!(n-k)!}{F}^{k-1}\left(y\right){\{1-F\left(y\right)\}}^{n-k}f\left(y\right)$

$=\frac{n!}{(k-1)!(n-k)!}\sum _{l=0}^{n-k}\left(\begin{array}{c}n-k\\ l\end{array}\right){(-1)}^l{F}^{k+l-1}\left(y\right)f\left(y\right)$

$=\sum _{j=k}^n\sum _{l=0}^{n-j}\left(\begin{array}{c}n\\ j\end{array}\right)\left(\begin{array}{c}n-j\\ l\end{array}\right){(-1)}^l{F}^{j+l}\left(y\right)$

Thus, the pdf and the cdf of k^th order statistic of Ram Awadh distribution (1.1) are obtained as

$f_Y\left(y\right)=\frac{n!{\lambda }^6(1+x^5)e^{-\lambda x}}{({\lambda }^6+120)(k-1)!(n-k)!}\sum _{l=0}^{n-k}\left(\begin{array}{c}n-k\\ l\end{array}\right){(-1)}^l$

$\times {[1-\frac{\left\{\lambda x({\lambda }^4{x}^4+5{\lambda }^3{x}^3+20{\lambda }^2{x}^2+60\lambda x+120)\right\}{e}^{-\lambda x}}{{\lambda }^6+120}]}^{k+l-1}$

and $F_Y\left(y\right)=\sum _{j=k}^n\sum _{l=0}^{n-j}\left(\begin{array}{c}n\\ j\end{array}\right)\left(\begin{array}{c}n-j\\ l\end{array}\right){(-1)}^l{[1-\frac{\left\{\lambda x({\lambda }^4{x}^4+5{\lambda }^3{x}^3+20{\lambda }^2{x}^2+60\lambda x+120)\right\}{e}^{-\lambda x}}{{\lambda }^6+120}]}^{j+l}$

Entropy measure

Entropy of a random variable $X$ is a measure of variation of uncertainty. A popular entropy measure is Renyi entropy. If $X$ is a continuous random variable having probability density function $f(.)$, then Renyi entropy is defined as

$T_R\left(\gamma \right)=\frac{1}{1-\gamma }\log \{\int f^{\gamma }\left(x\right)dx\}$

Thus, the Renyi entropy for Ram Awadh (1.1) can be obtained as

$=\frac{1}{1-\gamma }\log \left[\underset{0}{\overset{\infty }{\int }}\frac{{\lambda }^{7\gamma }}{{({\lambda }^6+120)}^{\gamma }}{\left(x^5\right)}^{\gamma }\sum _{j=0}^{\infty }\left(\begin{array}{c}\gamma \\ j\end{array}\right){\left(\frac{x^5}{\lambda }\right)}^j{e}^{-\lambda \gamma x}dx\right]$

$=\frac{1}{1-\gamma }\log \left[\sum _{j=0}^{\infty }\left(\begin{array}{c}\gamma \\ j\end{array}\right)\frac{{\lambda }^{7\gamma }}{{({\lambda }^6+120)}^{\gamma }}\underset{0}{\overset{\infty }{\int }}{e}^{-\lambda \gamma x}{x}^{5j+1-1}dx\right]$

$=\frac{1}{1-\gamma }\log \left[\sum _{j=0}^{\infty }\left(\begin{array}{c}\gamma \\ j\end{array}\right)\frac{{\lambda }^{7\gamma }}{{({\lambda }^6+120)}^{\gamma }}\frac{\Gamma (5j+1)}{{\left(\lambda \gamma \right)}^{5j+1}}\right]$

$=\frac{1}{1-\gamma }\log \left[\frac{{\lambda }^{7\gamma -5j-1}}{{({\lambda }^6+120)}^{\gamma }}\sum _{j=0}^{\infty }\left(\begin{array}{c}\gamma \\ j\end{array}\right)\frac{\Gamma (5j+1)}{{\left(\gamma \right)}^{5j+1}}\right]$

This process consists in generating N=10,000 pseudo–random samples of sizes 20, 40, 60, 80 and 100 from Ram Awadh distribution. Acceptance and rejection method has been used for this study. Average bias and mean square error of the MLEs of the parameter $\lambda$ are estimated using the following formulae
Average Bias = $\frac{1}{N}\sum _{j=1}^n({\hat{\lambda }}_j-\lambda )$  , MSE= $\frac{1}{N}\sum _{j=1}^n({\hat{\lambda }}_j-\lambda {)}^2$

The following algorithm can be used to generate random sample from Ram Awadh distribution.

Algorithm

Rejection method: To simulate from the density $f_X$, it is assumed that envelope density h from which it can simulate, and that have some $k<\infty$ such that $\sup {}_x\frac{f_X\left(x\right)}{h\left(x\right)}\le k$ Simulate X from h.

1. Generate Y~U(0,kh(X) , where $k=\frac{{\lambda }^6}{({\lambda }^6+120)}$
2. If $Y<f_X\left(x\right)$ then return X, otherwise go back to step 1.

The average bias (mean square error) of simulated estimate of parameter $\lambda$ for different values of n and $\lambda$ are presented in Table 1.

Figure 6 Estimated mean squared error of the MLEs for different values of $\lambda$ and $n.$

The graphs of estimated mean square error of the maximum likelihood estimate (MLE) for different values of parameter $\lambda$ and $n$ have been shown in Figure 6.

It explains the life of a component which has random strength X that is subjected to a random stress Y. When the stress applied to it exceeds the strength, the component fails instantly and the component will function adequately till $X>Y$. Therefore, $R=P(Y<X)$ is a measure of component reliability.

 Let X and Y be independent strength and stress random variables having Ram Awadh (1.1) with parameter ${\lambda }_1$  and ${\lambda }_2$  respectively. Then the stress–strength reliability $R$ can be obtained as

$R=P(Y<X)=\underset{0}{\overset{\infty }{\int }}P\left(Y<X|X=x\right)f_X\left(x\right)dx$

$=\underset{0}{\overset{\infty }{\int }}f(x;{\lambda }_1)F(x;{\lambda }_2)dx$

$=1-\frac{{\lambda }_1^6\left[\begin{array}{l}{\lambda }_2+10{\lambda }_2^{15}{\lambda }_1+45{\lambda }_2^{14}{\lambda }_1^2+120{\lambda }_2^{13}{\lambda }_1^3+210{\lambda }_2^{12}{\lambda }_1^4\\ +(252{\lambda }_1^5+120){\lambda }_2^{11}+(210{\lambda }_1^6+600{\lambda }_1+720){\lambda }_2^{10}\\ +(120{\lambda }_1^7+1200{\lambda }_1^2+5400{\lambda }_1){\lambda }_2^9+(1200{\lambda }_1^3+18600{\lambda }_1^2+45{\lambda }_1^8){\lambda }_2^8\\ +10{\lambda }_1^3({\lambda }_1^6+60{\lambda }_1+3900){\lambda }_2^7+{\lambda }_1^4({\lambda }_1^6+120{\lambda }_1+55320){\lambda }_2^6\\ +(55440{\lambda }_1^5+6652800){\lambda }_2^5+(39600{\lambda }_1^6+4752000{\lambda }_1){\lambda }_2^4\\ +(19800{\lambda }_1^7+2376000{\lambda }_1^2){\lambda }_2^3+(66600{\lambda }_1^8+792000{\lambda }_1^3){\lambda }_2^2\\ +(1320{\lambda }_1^9+158400{\lambda }_1^4){\lambda }_2+120{\lambda }_1^{10}+14400{\lambda }_1^5\end{array}\right]}{({\lambda }_1+120)({\lambda }_2+120){({\lambda }_1+{\lambda }_2)}^{11}}$

Method of moments estimates (MOME) of parameters

Equating population mean of Ram Awadh distribution to the corresponding sample mean, MOME $\hat{\lambda }$ of $\lambda$ is the solution of following non–linear equation ${\lambda }^7\overline{x}+120\lambda \overline{x}-({\lambda }^6+720)=0$

${\lambda }^7\overline{x}+120\lambda \overline{x}-({\lambda }^6+720)=0$ (10.1)

Maximum likelihood estimates (MLE) of parameters

Let $(x_1,x_2,x_3,\mathrm{...},x_n)$ be a random sample of size $n$ from Ram Awadh (1.1)). The likelihood function, of Ram Awadh distribution is given by

$L={\left(\frac{{\lambda }^6}{{\lambda }^6+120}\right)}^n\prod _{i=1}^n(\lambda +x_i)e^{-n\lambda \overline{x}}$

$\ln L=n\ln \left(\frac{{\lambda }^6}{{\lambda }^6+120}\right)+\sum _{i=1}^n\ln (\lambda +x_i)-n\lambda \overline{x}$

$\frac{\partial \ln L}{\partial \lambda }=\frac{6n}{\lambda }-\frac{6n{\lambda }^5}{({\lambda }^6+120)}+\sum \frac{1}{(\lambda +x^5)}-n\overline{x}=0$ (10.2)

Data set 1:

This data is related with behavioral sciences, collected by N. Balakrishnan, Victor Leiva & Antonio Sanhueza,¹¹ the detailed about the data are given in Balkrishnan et al.,¹² The scale “General Rating of Affective Symptoms for Preschoolers (GRASP)” measures, which are:

19(16) 20(15) 21(14) 22(9) 23(12) 24(10) 25(6) 26(9) 27(8) 28(5) 29(6) 30(4) 31(3) 32(4) 33 34 35(4) 36(2) 37(2) 39 42 44

Data set 2: This data set is the strength data of glass of the aircraft window reported by Fuller et al.,¹² 18.83 20.8 21.657 23.03 23.23 24.05 24.321 25.5 25.52 25.8 26.69 26.77 26.78 27.05 27.67 29.9 31.11 33.2 33.73 33.76 33.89 34.76 35.75 35.91 36.98 37.08 37.09 39.58 44.045 45.29 45.381

Data Set 3: The following data represent the tensile strength, measured in GPa, of 69 carbon fibers tested under tension at gauge lengths of 20mm (Bader and Priest)¹³:

1.312 1.314 1.479 1.552 1.700 1.803 1.861 1.865 1.944 1.958 1.966 1.997 2.006 2.021 2.027 2.055 2.063 2.098 2.140 2.179 2.224 2.240 2.253 2.270 2.272 2.274 2.301 2.301 2.359 2.382 2.382 2.426 2.434 2.435 2.478 2.490 2.511 2.514 2.535 2.554 2.566 2.570 2.586 2.629 2.633 2.642 2.648 2.684 2.697 2.726 2.770 2.773 2.800 2.809 2.818 2.821 2.848 2.880 2.954 3.012 3.067 3.084 3.090 3.096 3.128 3.233 3.433 3.585 3.858

For the above three data sets, Ram Awadh distribution has been fitted along with one parameter exponential, Lindley and Akash, Shanker, Sujatha, Ishita and Pranav distribution. The pdf and cdf of one parameter fitted distributions are presented in Table 2. The ML estimates, values of and K–S statistics of the fitted distributions are presented in Table 3. As we know that the best distribution corresponds to the lower values of and K–S.

Profile plot of parameter and fitted plot for dataset–1, 2 and 3 are presented in Figures 7–9 respectively. From the graph, it is observed that Ram Awadh distribution is closer to observed dataset in comparison to other distributions of one parameter.

Figure 7 Profile of parameter and fitted probability plots for data set-1.

Figure 8 Profile of parameter and fitted probability plots for data set-2.

Figure 9 Profile of parameter and fitted probability plots for data set-3.

In this paper, a new one parameter lifetime distribution named Ram Awadh distribution has been proposed. Its mathematical properties including moments, measure of dispersion, hazard rate function, mean residual life function, stochastic ordering, mean deviations, order statistics, Bonferroni and Lorenz curves, and stress–strength reliability have been discussed. Simulation study of Ram Awadh distribution has also been discussed. The method of moments and the method of maximum likelihood estimation have been derived for estimating the parameter. In the last, three numerical examples of real lifetime data sets have been illustrated to test the goodness of fit of the Ram Awadh distribution. Its fit was found satisfactory over exponential, Lindley, Sujatha, Ishita, Akash , Shanker and Pranav distribution.

Note: The paper is named Ram Awadh distribution in the name of my Father Shri Ram Awadh Shukla.

None.

Author declares that there is no conflict of interest.

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