A new probability model for modeling of strength of carbon fiber data: properties and applications

Abstract

The procedures to discover proper new models in probability theory for different data collections are highly prevalent these days among the researchers of this area whenever existing literature models are not appropriate. Before delivering a product, manufacturers of raw materials or finished materials must follow some compliance standards in various engineering disciplines to avoid severe losses. Materials of high strength are necessary to ensure the safety of human lives along with infrastructures to elude the significant obligations linked with the provisions of non-compliant products. Using probability theory, we introduce the weighted version of inverted Kumaraswamy Distribution, which could be considered a better model than some other sub-models used to model Carbon fiber’s strength data. We derive various statistical properties of this distribution such as cumulative distribution, moments, mean residual life, reversed residual life functions, moment generating function, characteristic function, harmonic mean, and geometric mean. Parameters are estimated through the maximum likelihood method and ordinary moments. Simulation studies are carried out to illustrate the theoretical results of these two approaches. Furthermore, two real data sets of Carbon fibers strength are utilized to contrast the proposed model and its sub-models like inverted Kumaraswamy distribution and Kumaraswamy Sushila distribution through different goodness of fit criteria such as Akaike Information Criterion (AIC), corrected Akaike information criterion, and the Bayesian Information Criterion (BIC). Results reveal the outperformance of the proposed model compared to other models, which render it a proper interchange of the current sub-models.

A Moments and related measures

It is well known that the moments of a r.v. are fundamental characteristics. They allow studying skewness and kurtosis of a variable following a certain distribution and permits to construct an estimator of the parameters. Here, we focus on the different moments of the WIKD.

1.1 A.0.1 The central and non-central moments

1.2 A.1 The central and non-central moments

Let X be a random variable has WIK distribution with parameters $\alpha$, $\beta$ and k . The non central moment of order r,

$$\begin{aligned} E[X^r] =&\int _0^{\infty }x^{r}f_{WIK}(x,\alpha ,\beta ) dx \\ =&\frac{\alpha }{\sum _{i=0}^{k}\left( {\begin{array}{c}k\\ i\end{array}}\right) (-1)^i B\left( \frac{\alpha -k+i}{\alpha },\beta \right) }\int _0^{\infty } x^{r+k}(1+x)^{-(\alpha +1)}\left[ 1-(1+x)^{-\alpha }\right] ^{\beta -1}dx\\&\displaystyle = \frac{\sum _{i=0}^{k+r}\left( {\begin{array}{c}k+r\\ i\end{array}}\right) (-1)^i B\left( \frac{\alpha -k-r+i}{\alpha },\beta \right) }{\sum _{i=0}^k \left( {\begin{array}{c}k\\ i\end{array}}\right) (-1)^{i} B\left( \frac{\alpha -k+i}{\alpha },\beta \right) } \end{aligned}$$

Finally, the non-central moment, of order r, $E[X^r]$ is defined by

$$\begin{aligned} \mu _r=\displaystyle \frac{ W(\alpha ,\beta , k+r) }{ W(\alpha ,\beta , k)} \end{aligned}$$

Hence, the variance is

$$\begin{aligned} \sigma ^2=\frac{\left( W(\alpha ,\beta , k)*W(\alpha ,\beta , k+2)\right) -\left( W(\alpha ,\beta , k+1)\right) ^2 }{\left( W(\alpha ,\beta , k))\right) ^2} \end{aligned}$$

(9)

Similarly, for the central moments $E[X-\mu ]^r$, we have

$$\begin{aligned} E[X-\mu ]^r =\sum _{j=0}^r\left( {\begin{array}{c}r\\ j\end{array}}\right) (-\mu )^{r-j}E[X^j] \end{aligned}$$

Then,

$$\begin{aligned} \mu _r^{Central} =\displaystyle \frac{\sum _{j=0}^r\left( {\begin{array}{c}r\\ j\end{array}}\right) (-1)^{r-j} W(\alpha ,\beta , k+1))^{r-j} W(\alpha ,\beta , k+j) }{(W(\alpha ,\beta , k+2))^r} \end{aligned}$$

(10)

Using the moment definition of WIKD, the harmonic mean of the r.v. X is defined by

$$\begin{aligned} H_m(X)\displaystyle = \frac{1}{E[X^{-1}]}=\frac{ W(\alpha ,\beta , k) }{ W(\alpha ,\beta , k-1)}. \end{aligned}$$

1.3 A.1.1 Moment generating function and characteristic function

Assume that the X follows WIKD. Then, The moment generating function and characteristic function are respectively defined by

$$\begin{aligned} \psi (t)=\frac{1}{\sum _{i=0}^{k}\left( {\begin{array}{c}k\\ i\end{array}}\right) (-1)^i B\left( \frac{\alpha -k+i}{\alpha },\beta \right) }\left( \sum _{n=0}^\infty \sum _{i=0}^{k+n}\left( {\begin{array}{c}k+n\\ i\end{array}}\right) (-1)^iB\left( \frac{\alpha -n-k+i}{\alpha },\beta \right) \frac{t^n}{n!}\right) \end{aligned}$$

and

$$\begin{aligned} \phi (t)= \frac{1}{\sum _{i=0}^{k}\left( {\begin{array}{c}k\\ i\end{array}}\right) (-1)^i B\left( \frac{\alpha -k+i}{\alpha },\beta \right) }\left( \sum _{n=0}^\infty \sum _{i=0}^{k}\left( {\begin{array}{c}k+n\\ i\end{array}}\right) (-1)^iB\left( \frac{\alpha -n-k+i}{\alpha },\beta \right) \frac{(it)^n}{n!}\right) . \end{aligned}$$

We conclude that

$$\begin{aligned} \psi (t)=\frac{1}{W(\alpha ,\beta , k)}\left( \sum _{n=0}^\infty W(\alpha ,\beta , k+n) \frac{t^n}{n!}\right) \end{aligned}$$

and

$$\begin{aligned} \phi (t)= \frac{1}{W(\alpha ,\beta , k)}\left( \sum _{n=0}^\infty W(\alpha ,\beta , k+n) \frac{(it)^n}{n!}\right) \end{aligned}$$

1.4 A.2 Measures of skewness and kurtosis

The skewness and kurtosis coefficients are usually used to judge, respectively, the asymmetry and flatness of a distribution. The skewness coefficient is determined by

$$\begin{aligned} \gamma _1=E\left[ \frac{X-\mu }{\sigma }\right] ^3. \end{aligned}$$

Using (9)and (10), it follows, for the WIKD, that

$$\begin{aligned} \gamma _1=\frac{(W(\alpha ,\beta , k))^2 W(\alpha ,\beta , k+3)-3W(\alpha ,\beta , k) W(\alpha ,\beta , k+1)W(\alpha ,\beta , k+2) +2 (W(\alpha ,\beta , k+1))^3}{\left( W(\alpha ,\beta , k+2)- (W(\alpha ,\beta , k+1)^2)\right) ^3} \end{aligned}$$

Also, the Kurtosis coefficient is

$$\begin{aligned} \gamma _2=E\left[ \frac{X-\mu }{\sigma }\right] ^4. \end{aligned}$$

Thus, for WIKD and using (9)and (10), it is

$$\begin{aligned} \gamma _2&=\frac{ (W(\alpha ,\beta , k))^3W(\alpha ,\beta , k+4)-4(W(\alpha ,\beta , k))^2W(\alpha ,\beta , k+3)W(\alpha ,\beta , k+1)}{\left( W(\alpha ,\beta , k+2)- (W(\alpha ,\beta , k+1)^2)\right) ^2}\\&\quad +\frac{6(W(\alpha ,\beta , k))W(\alpha ,\beta , k+2)(W(\alpha ,\beta , k+1))^2-4(W(\alpha ,\beta , k+1)^4)}{\left( W(\alpha ,\beta , k+2)- (W(\alpha ,\beta , k+1)^2)\right) ^2} \end{aligned}$$