The Marshall-Olkin extended inverted Kumaraswamy distribution: Theory and applications

2.1 Limiting behavior for the MOEIK density function

2.2 Shapes of the density and hazard rate function

The Fig. 1(a) and (b) represent the behavior of the density function and explains the tractability and flexibility of the model graphically with its sub-families. The pdf plot shows that for $\beta <1$ , the newly developed model has exponentially decreasing behavior. For $\beta =1$ , the curves also have exponentially decreasing behavior but starts from y-axis while for $\beta >1$ , the distribution has unimodal positively skewed behavior. Fig. 1(c) and (d) represents behavior of hazard rate function (hrf) which is unimodal at various parameter combinations and also have bathtub and decreasing behavior for small $\alpha \mathit{and}\beta$ , while $\lambda$ approaches to zero. Moreover, the observed model can have upside down bathtub behavior as the x-axis increases.

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Fig. 1

Density function 1 (a) and 1 (b) and hazard rate function 1 (c) and 1 (d) at different parameter values.

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Table: Some special models of MOEIK distribution.

2.3 Quantile function

Let random variable $X\sim \mathit{MOEIK}(\lambda ,\alpha ,\beta )$ having pdf in Eq. (5). The computation of the quantile function, say $x=Q(u)$ of the MOEIK distribution can be attained by inverting Eq. (6)

3.1 Moments

This subsection provides the unambiguous expressions of the rth non-central moment and cumulants. Numerous important features and characteristics such as tendencies, coefficient of skewness, coefficient of kurtosis and dispersions can be studied from moments. The given theorem provide moments.

The rth raw moments of the MOEIK distribution can be obtained by following the definition of the moments as $${\mu }_r^{\prime }={\int }_0^{\infty }{x}^{r}f(x\text{;}\alpha ,\beta ,\lambda )\mathit{dx}=\lambda \alpha \beta {\displaystyle \sum _{i,j=0}^{\infty }}{w}_{i,j}{\int }_0^{\infty }{x}^r{(1+x)}^{-\alpha -1}{(1-{(1+x)}^{-\alpha })}^{\beta j+\beta -1}\mathit{dx}$$

The cumulants $(k_s)$ of the MOEIK distribution are attained from above expression as $$k_s={\mu }_s^{\prime }-{\displaystyle \sum _{k=1}^{s-1}}\left(\begin{array}{c}\hfill s-1\hfill \\ \hfill k-1\hfill \\ \hfill \hfill \end{array}\right)k_k{\mu }_{s-k}^{\prime },{\mu }_s={\displaystyle \sum _{k=0}^s}\left(\begin{array}{c}\hfill s\hfill \\ \hfill k\hfill \\ \hfill \hfill \end{array}\right){(-1)}^k{\mu }_1^{\prime k}{\mu }_{s-k}^{\prime }$$ where ${\mu }_1^{\prime }=k$ . Thus, $$k_2={\mu }_2^{\prime }-{({\mu }_1^{\prime })}^2,$$ $$k_3={\mu }_3^{\prime }-3{\mu }_2^{\prime }{\mu }_1^{\prime }+2{({\mu }_1^{\prime })}^3,$$ $$k_4={\mu }_4^{\prime }-4{\mu }_3^{\prime }{\mu }_1^{\prime }-3{({\mu }_2^{\prime })}^2+12{\mu }_2^{\prime }{({\mu }_1^{\prime })}^2-6{({\mu }_1^{\prime })}^4$$

The coefficient of skewness and the coefficient of kurtosis for the MOEIK distribution can also obtained from the third and the fourth standardized cumulants by using formulae ${\gamma }_1=\frac{k_3}{k_4^{3/2}}\mathit{and}{\gamma }_2=\frac{k_4}{k_2^2}$ respectively. Table 1 represents the behavior of raw moments, skewness and kurtosis for MOEIK distribution.

3.2 Incomplete moments

Researchers have general interest to compute the rth incomplete moments of life-time models. The incomplete moments are mostly utilized to explain the Bonferroni and Lorenz curves while it also helps to evaluate the mean deviation. Let X follows the MOEIK distribution and its rth incomplete moment could be evaluated from $$\phi (\text{x})={\int }_0^{\text{x}}{\text{v}}^{\text{r}}\text{f}(\text{v})\text{dv}=\lambda \alpha \beta {\displaystyle \sum _{i,j=0}^{\infty }}{w}_{i,j}{\int }_0^x{v}^r{(1+v)}^{-\alpha -1}{(1-{(1+v)}^{-\alpha })}^{\beta j+\beta -1}\mathit{dv}$$ $$\mathit{Let}{(1+v)}^{-\alpha }=z,v=z^{-\frac{1}{\alpha }}-1\mathit{and}\mathit{dv}=-\frac{1}{\alpha }{z}^{-\frac{1}{\alpha }-1}\mathit{dz}=\mathit{ck}{\beta }^r{\displaystyle \sum _{i,j,k=0}^{\infty }}{w}_{i,j}{(-1)}^{r+k}\left(\begin{array}{c}\hfill r\hfill \\ \hfill k\hfill \\ \hfill \hfill \end{array}\right){\int }_{{(1+x)}^{-\alpha }}^1{z}^{1-\frac{r}{\alpha }-1}{(1-z)}^{\beta j+\beta -1}\mathit{dz}$$

3.3 Mean deviation

A most appropriate tool used to evaluate the average absolute deviation of the observation from their mean is mean deviation. We can find the mean deviation about the mean and median from the following expressions given respectively.

Mean deviation is used in numerous important fields such as economics, demography, insurance, medicines and reliability.

3.4 Mode

For symmetrical distributions such as the standard normal, t-distribution and Cauchy distribution, the mean, median and mode are same say 0. But for the asymmetrical distribution we have to follow the general method with necessary and sufficient conditions to find the mode. The mode of the MOEIK distribution can be obtained by taking the first derivative and equate it to zero.

$f^{\prime }(x)=0$ .

The density function of the MOEIK in Eq. (6) is given as $$f(x)=\frac{\lambda \alpha \beta {(1+x)}^{-\alpha -1}{(1-{(1+x)}^{-\alpha })}^{\beta -1}}{{[1-(1-\lambda )(1-{(1-{(1+x)}^{-\alpha })}^{\beta })]}^2}$$

Following the definition of mode $$f^{\prime }(x)=\lambda {\alpha }^2\beta {(1+x)}^{-2\alpha -2}{(1-{(1+x)}^{-\alpha })}^{\beta -2}\left[\frac{(\beta -1)(1-(1-\lambda )(1-{(1-{(1+x)}^{-\alpha })}^{\beta }))+2\beta {(1-{(1+x)}^{-\alpha })}^{\beta }}{{\{1-(1-\lambda )(1-{(1-{(1+x)}^{-\alpha })}^{\beta })\}}^3}\right]=0$$

Since ${(1+x)}^{-2\alpha -2}>0,{(1-{(1+x)}^{-\alpha })}^{\beta -2}>0\mathit{and}{[1-(1-\lambda )(1-{(1-{(1+x)}^{-\alpha })}^{\beta })]}^3>0,\alpha ,\beta ,\theta >0\mathit{for}\mathit{all}x>0$ . Therefore, $$(\beta -1)(1-(1-\lambda )(1-{(1-{(1+x)}^{-\alpha })}^{\beta }))+2\beta {(1-{(1+x)}^{-\alpha })}^{\beta }=0$$

3.5 Rényi entropy

Entropy is generally belongs to statistical mechanics, more specifically microscopic variables. Therefore, more precisely we can say that Entropy is the measure of disorder in a microscopic system. A common measure of entropy is the Rényi entropy and has much importance in many fields such as statistical inference, classification, problems identification in statistics, econometrics and pattern recognition in computer sciences. The given theorem provides expression for Rényi entropy.

3.6 Order statistics

Let the random variables and its ordered values are denoted as $X_{(1)},X_{(2)},X_{(3)},\dots ,X_{(n)}$ . The probability density function (pdf) of order statistics is obtained using the function $$f_{s:n}(x)=\frac{n!}{(s-1)!(n-s)!}{[F(x)]}^{s-1}{[1-F(x)]}^{n-s}f(x)$$

The density of the nth ordered statistics follows the MOEIK distribution is $$f_{s:n}(x)=\frac{n!}{(s-1)!(n-s)!}{\left[\frac{{(1-{(1+x)}^{-\alpha })}^{\beta }}{1-(1-\lambda )(1-{(1-{(1+x)}^{-\alpha })}^{\beta })}\right]}^{s-1}{\left[1-\frac{{(1-{(1+x)}^{-\alpha })}^{\beta }}{1-(1-\lambda )(1-{(1-{(1+x)}^{-\alpha })}^{\beta })}\right]}^{n-s}\frac{\lambda \alpha \beta {(1+x)}^{-\alpha -1}{(1-{(1+x)}^{-\alpha })}^{\beta -1}}{{[1-(1-\lambda )(1-{(1-{(1+x)}^{-\alpha })}^{\beta })]}^2}$$

The rth moment of order statistic is obtained as $$E(X_{s:n}^r)=\frac{n!\lambda \alpha \beta }{(s-1)!(n-s)!}{\displaystyle \sum _{i,j,k=0}^{\infty }}{\varphi }_{i,j,k}{\int }_0^{\infty }{x}^r{(1+x)}^{-\alpha -1}{(1-{(1+x)}^{-\alpha })}^{\beta (s+i+k)-1}\mathit{dx}$$

On simplifications, we get $$=\frac{n!\lambda \beta }{(s-1)!(n-s)!}{\displaystyle \sum _{i,j,k,m=0}^{\infty }}{\psi }_{i,j,k,m}{\int }_0^1{z}^{1-\frac{m}{\alpha }-1}{(1-z)}^{\beta (s+i+k)-1}\mathit{dz}$$ $$\mathit{where}{\psi }_{i,j,k,m}={\varphi }_{i,j,k}{\displaystyle \sum _{m=0}^{\infty }}{(-1)}^{m+r}\left(\begin{array}{c}\hfill r\hfill \\ \hfill m\hfill \\ \hfill \hfill \end{array}\right)$$

The expressions for rth moment of order statistic becomes

3.7 Maximum likelihood estimation

This study adopts the maximum likelihood estimation method so that it is mostly used and provides the maximum information about the properties of the estimated parameters. Moreover, the normal approximation of these estimators can frankly be managed systematically and mathematically for large sample theory. Consequently, the ML estimation has adopted to estimate the unknown parameters $(\alpha ,\beta \text{and}\lambda )$ of the MOEIK distribution.

Let random variable X belong to the observed distribution and have vector of the parameters ${(\alpha ,\beta ,\lambda )}^T$ with size n. The sample likelihood function is achieved as $${\displaystyle \prod _{i=1}^n}f(x\text{;}\alpha ,\beta ,\lambda )={\alpha }^n{\beta }^n{\lambda }^n{\displaystyle \prod _{i=1}^n}\frac{{(1+x)}^{-\alpha -1}{(1-{(1+x)}^{-\alpha })}^{\beta -1}}{{[1-(1-\lambda )(1-{(1-{(1+x)}^{-\alpha })}^{\beta })]}^2}$$

The log-likelihood function is

Now we have to maximize the log-likelihood function given in Eq. (20) to get the ML estimates of unknown parameters of the Marshall-Olkin extended Kumaraswamy distribution. For this purpose, we take the first derivative of the log-likelihood equation with respect to parameters and equate to zero respectively.

The exact solution of the derived ML estimator for unknown parameters in Eqs. (21)–(23) is genuinely not possible. Therefore, it is more appropriate to use the nonlinear optimization algorithms such as a Newton-Raphson algorithm for maximizing the likelihood function numerically. We can use R (optimal function or maxBFGS function), or MATHEMATICA (Maximize function). After application of large sample property of the ML Estimates, MLE $\widehat{\theta }$ can be treated as being approximately normal with the mean θ and the variance-covariance matrix equal to the inverse of the expected information matrix, i.e. $\sqrt{n(\widehat{\theta }-\theta )\to N(0,{\mathit{nI}}^{-1}(\theta ))},I(\theta )$ is the information matrix then its inverse of matrix is $I^{-1}(\theta )$ provide the variances and covariance’s. The $I(\widehat{\theta })$ variance-covariance matrix is actually equal to the inverse of the expected information matrix $I^{-1}(\widehat{\theta })$ is given as

The elements of the variance-covariance information matrix are given in Appendix.