STATISTICS IN TRANSITION new series, March 2018 135 STATISTICS IN TRANSITION new series, March 2018 Vol. 19, No. 1, pp. 135–148, DOI 10.21307/stattrans-2018-008 POWER ISHITA DISTRIBUTION AND ITS APPLICATION TO MODEL LIFETIME DATA Kamlesh Kumar Shukla 1, Rama Shanker 2 ABSTRACT A study on two-parameter power Ishita distribution (PID), of which Ishita distribution introduced by Shanker and Shukla (2017 a) is a special case, has been carried out and its important statistical properties including shapes of the density, moments, skewness and kurtosis measures, hazard rate function, and stochastic ordering have been discussed. The maximum likelihood estimation has been discussed for estimating its parameters. An application of the distribution has been explained with a real lifetime data from engineering, and its goodness of fit shows better fit over two-parameter power Akash distribution (PAD), two- parameter power Lindley distribution (PLD) and one-parameter Ishita, Akash, Lindley and exponential distributions. Key words: Ishita distribution, moments, hazard rate function, stochastic ordering, maximum likelihood estimation, goodness of fit. 1. Introduction The probability density function (pdf) of Ishita distribution introduced by Shanker and Shukla (2017 a) is given by 3 f y;  y2ey ;y 0,0 (1.1) 1 32  pg y;1 pg y; (1.2) 1 2 3 where p 32 g y;ey ;y 0,0 1 1 First author: Department of Statistics, College of Science, Eritrea Institute of Technology, Asmara, Eritrea. E-mail: [email protected]. 2 Corresponding author: Department of Statistics, College of Science, Eritrea Institute of Technology, Asmara, Eritrea. E-mail: [email protected]. 136 K. K. Shukla, R. Shanker: Power Ishita distribution… 3 g y; eyy31 ;y 0,0 2 3 The pdf in (1.1) reveals that the Ishita distribution is a two-component mixture of an exponential distribution (with scale parameter) and a gamma distribution (with shape parameter 2 and scale parameter), with mixing proportion 3 p . Shanker and Shukla (2017 a) have discussed some of its 32 mathematical and statistical properties including its shapes for varying values of the parameter, moments, skewness, kurtosis, hazard rate function, mean residual life function, stochastic ordering, mean deviations, order statistics, Bonferroni and Lorenz curves, Renyi entropy measure, stress-strength reliability, and the applications of the distribution for modelling lifetime data from engineering and medical science. However, there are some situations where the Ishita distribution may not be suitable from either theoretical or applied point of view. Shukla and Shanker (2017) have also obtained a Poisson mixture of Ishita distribution and named it Poisson-Ishita distribution, and studied its various statistical properties, estimation of parameter and the goodness of fit with some real count data sets. The corresponding cumulative distribution function (cdf) of (1.1) is given by  yy2 F y;11 ey;y0,0 (1.3) 1 32   Recall that the pdf and the cdf of two-parameter power Akash distribution (PAD) introduced by Shanker and Shukla (2017 b) and two-parameter power Lindley distribution (PLD) introduced by Ghitany et al. (2013) are respectively given by 3 f x;, 1x2x1ex ;x0,0,0 (1.4) 2 2 2  xx2 F x;,11 ex;x0,0,0 (1.5) 2  2 2    2 f x;, 1xx1ex ;x0,0,0 (1.6) 3 1  x F x;,11 ex;x0,0,0 (1.7) 3  1 A detailed study regarding various properties, estimation of parameters and applications of PAD and PLD can be seen from Shanker and Shukla (2017 b) and STATISTICS IN TRANSITION new series, March 2018 137 Ghitany et al. (2013) respectively. At 1, PAD reduces to Akash distribution introduced by Shanker (2015) having pdf and cdf given by 3 f x; 1x2ex ;x0,0 (1.8) 4 2 2  xx2 F x;11 ex;x0,0 (1.9) 4 2 2   Shanker (2015) has a detailed study about various statistical and mathematical properties of Akash distribution, estimation of parameter and applications for modelling lifetime data from engineering and medical science and showed that Akash distribution gives better fit than both exponential and Lindley distributions. Shanker (2017) has also obtained a Poisson mixture of Akash distribution and named Poisson-Akash distribution and discussed important statistical properties, estimation of parameter using both the method of moments and the method of maximum likelihood and the application for modelling count data. Similarly, at1, PLD reduces to Lindley distribution introduced by Lindley (1958) having pdf and cdf given by 2 f x; 1xex;x0,0 (1.10) 5 1  x  F x;1 1 ex;x0,0 (1.11) 5  1 Ghitany et al. (2008) have a detailed study about various properties of Lindley distribution, estimation of parameter and application for modelling waiting time data from a bank and it has been shown that it gives better fit than exponential distribution. Shanker et al. (2016) have a detailed and critical comparative study of modelling real lifetime data from engineering and biomedical sciences using Akash, Lindley and exponential distribution and observed that each of these one- parameter distribution has some advantage over the other but none is perfect for modelling all real lifetime data. Since Ishita distribution gives better fit than Akash, Lindley and exponential distribution, it is expected and hoped that the two- parameter power Ishita distribution (PID) will provide a better model over two- parameter power Akash distribution (PAD) and power Lindley distribution (PLD) and one-parameter Ishita, Akash, Lindley and exponential distributions. In this paper, a two-parameter power Ishita distribution (PID), which includes one-parameter Ishita distribution, has been introduced and its various properties including shapes for varying values of the parameters, survival function, hazard rate function, moments, stochastic ordering have been studied. The maximum likelihood estimation has been discussed for estimating its parameters. Finally, applications and goodness of fit of PID has been illustrated with a real life time data and fit has been found better over two-parameter power Akash distribution 138 K. K. Shukla, R. Shanker: Power Ishita distribution… (PAD) of Shanker and Shukla (2017 b), two-parameter power Lindley distribution (PLD) of Ghitany et al. (2013), and one-parameter Ishita, Akash, Lindley and exponential distributions. 2. Power Ishita distribution Taking the power transformation X Y1 in (1.1), pdf of the random variable X can be obtained as 3 f x;, x2x1ex ;x0,0,0 (2.1) 6 32  pg x;,1 pg x;, (2.2) 3 4 3 where p 32 g x;,x1ex ;x0,0,0 3 3x31ex g x;, ;x0,0,0 4 2 We would call the density in (2.1) “Power Ishita distribution (PID)” and denote it as PID,. It is obvious that the PID is also a two-component mixture of Weibull distribution (with shape parameter and scale parameter ), and a generalized gamma distribution (with shape parameters 3,  and scale 3 parameter) introduced by Stacy (1962) with their mixing proportion p . 32 The corresponding cumulative distribution function (cdf) of (2.1) can be obtained as  xx2 F x;,11 ex;x0,0,0 (2.3) 6  32    Graphs of the pdf and the cdf of PID for varying values of the parameters have been drawn and presented in Figures 1 and 2 respectively. If 1, the pdf of PID is monotonically decreasing for increasing values of the parameter . But for 1 and increasing values of the parameter, the shapes of the pdf of PID become negatively skewed, positively skewed, symmetrical, platykurtic and mesokurtic; and this means that PID can be used for modelling lifetime data of various nature. STATISTICS IN TRANSITION new series, March 2018 139 Figure.1. Graphs of pdf of PID for varying values of parameters  and  140 K. K. Shukla, R. Shanker: Power Ishita distribution… Figure 2. Graphs of cdf of PID for varying values of parameters  and  3. Survival and hazard rate functions The survival function, Sxand hazard rate function, hx of the PID can be obtained as xx232 Sx;,1F x;, ex;x0,0,0 6  32    (3.1) f x;, 31x2x1 hx;, 6  ; x0, 0, 0 Sx;, xx232 (3.2) The nature and behaviour of hx of the PID for varying values of the parameters  and are shown graphically in Figure 3. It is obvious from the graphs of hxthat it is monotonically decreasing and increasing for increased values of the parameters  and . STATISTICS IN TRANSITION new series, March 2018 141 Figure 3. Graphs of hx of PID for varying values of the parameters  and  4. Moments and related measures Using the mixture representation (2.2), the rth moment about origin of the PID can be obtained as    EXr pxrg x;,dx1 pxrg x;,dx r 3 4 0 0  r  r 23rr2      ;r 1,2,3,.... (4.1) 3r32 142 K. K. Shukla, R. Shanker: Power Ishita distribution… It should be noted that at1, the above expression will reduce to the rth moment about origin of Ishita distribution and is given by r!3r1r2    ;r 1,2,3,... r r32 Therefore, the mean and the variance of the PID are obtained as  1   23121       1 3132 2 2 2232123321 231212    2  62322 The coefficient of skewness and the coefficient of kurtosis of PID, upon substituting for the raw moments and standard deviation, can be obtained using following expressions  3 32  3 2 1 1 Coefficient of Skewness = 3  2  4 46  3  4 3 1 2 1 1 and Coefficient of Kurtosis  . 4 5. Stochastic ordering Stochastic ordering of positive continuous random variables is an important tool for judging their comparative behaviour. A random variable X is said to be smaller than a random variable Y in the (i) stochastic order X  Yif F x F xfor all x st X Y (ii) hazard rate order X  Yif h xh x for all x hr X Y (iii) mean residual life order X  Yif m xm xfor all x mrl X Y f x (iv) likelihood ratio order X  Yif X decreases inx. lr f x Y STATISTICS IN TRANSITION new series, March 2018 143 The following important interrelationships due to Shaked and Shanthikumar (1994) are well known for establishing stochastic ordering of distributions X  Y X  Y X  Y lr hr mrl  X Y st The PID is ordered with respect to the strongest ‘likelihood ratio ordering’ as shown in the following theorem: Theorem: Let X  PID, and Y  PID, . If and   1 1 2 2 1 2 1 2 (or   and ) then X  Y and henceX  Y , X  Y and 1 2 1 2 lr hr mrl X  Y. st Proof: From the pdf of PID (2.1), we have fXx,1,1 1132321x21 x12e1x12x2 ; x0 fYx;2,2 332 x22   2 2 1  2 Now ln fXx,1,1 ln113232ln1x21 lnxx1  x2 fYx;2,2 332  x22  1 2 1 2  2 2 1  2 This gives d ln fXx,1,1 212x21121x221212x2121 12 dx fYx;2,2 x21 x22 x 1 2 x11x21 1 1 2 2 Clearly for   and  (or   and ), d ln fXx,1,10. 1 2 1 2 1 2 1 2 dx fYx;2,2 This means that X  Y and henceX  Y , X  Y andX  Y. Thus PID lr hr mrl st follows the strongest likelihood ratio ordering. 6. Maximum likelihood estimation Let x ,x ,x ,...,x  be a random sample of size n from PID,. Then, 1 2 3 n the log-likelihood function is given by n lnLln f x;, 6 i i1 144 K. K. Shukla, R. Shanker: Power Ishita distribution… n n n nln3lnln32lnx21lnx x.   i i i i1 i1 i1 The maximum likelihood estimates (MLE) ˆ,ˆ of ,of PID (2.1) are the solutions of the following log likelihood equations lnL 3n 3n2 n 1 n    x0   3 2 x2 i i1 i i1 lnL n n x2lnx  n n  2 i i lnx xlnx 0   x2 i i i x1 i i1 i1 These two log likelihood equations do not seem to be solved directly because these cannot be expressed in closed form. However, Fisher’s scoring method can be applied to solve these equations iteratively. Thus, we have 2lnL 3n 3n34 n 1    2 2 322 i1 x22 i 2lnL n x2lnx  n 2lnL 2 i i xlnx   i1 x22 i1 i i  i 2lnL n n x lnx 2 n  4 i i xlnx 2 2 2 i1 x22 i1 i i i The MLE ˆ,ˆ of ,of PID (2.1) are the solution of the following equations 2lnL 2lnL lnL    2  ˆ0    2lnL 2lnL ˆ  lnL   0  2 ˆ   ˆ0 0 ˆ ˆ 0 0 where and are initial values ofand. These equations are solved 0 0 iteratively until sufficiently close estimates of ˆ andˆ are obtained. In this paper, R-software has been used to estimate the parameters and for the considered dataset.