From the log-exponential-power (LEP) distribution are given as F ( x, , ) = e

From the log-exponential-power (LEP) distribution are given as F ( x, , ) = e and (- log x) 1-exp (- log x ) e , x (0, 1) (4) (- log x ) -1 e x respectively, where 0 and 0 are the model parameters. This new unit model is called as LEP distribution and just after here, a random variable X is denoted as X LEP(, ). The connected hrf is given by f ( x, , ) = h( x, , ) = x eexp (- log x )1-exp (- log x ),x (0, 1)(three)-e(- log x) (- log x ) -1 ,x (0, 1).(5)-If the parameter is equal to one, then we have Compound 48/80 Cancer following simple cdf and pdf F ( x, , 1) = – – e1- x and f ( x, , 1) = x –1 e1- x for x (0, 1) respectively. The attainable shapes of the pdf and hrf have already been sketched by Figure 1. In accordance with this Figure 1, the shapes in the pdf can be noticed as numerous shapes such as U-shaped, growing, decreasing and unimodal too as its hrf shapes may be bathtub, growing and N-shaped.LEP(0.2,three) LEP(1,1) LEP(0.25,0.75) LEP(0.05,5) LEP(two,0.five) LEP(0.5,0.five)LEP(0.02,3.12) LEP(1,1) LEP(0.25,0.75) LEP(0.05,five) LEP(2,0.five) LEP(0.5,0.5)hazard rate0.0 0.2 0.4 x 0.6 0.eight 1.density0.0.0.4 x0.0.1.Figure 1. The doable shapes of your pdf (left) and hrf (suitable).Other parts of the study are as follows. Statistical C2 Ceramide supplier Properties in the LEP distribution are provided in Section two. Parameter estimation process is presented in Section 3. Section 4 is devoted towards the LEP quantile regression model. Section five contains two simulation research for LEP distribution and the LEP quantile regression model. Empirical benefits with the study are offered in Section six. The study is concluded with Section 7. two. Some Distributional Properties on the LEP Distribution The moments, order statistics, entropy and quantile function on the LEP distribution are studied.Mathematics 2021, 9,3 of2.1. Moments The n-th non-central moment from the LEP distribution is denoted by E( X n ) which is defined as E( X n )= nx n-1 [1 – F ( x )]dx = 1 – n1x n-1 e1-exp((- log( x)) ) dxBy altering – log( x ) = u transform we obtain E( X n )= 1nee-n u e- exp( u ) du = 1 n ee-n u 1 (-1)i exp(i u ) du i! i =1 (-1)i = 1ne n i=1 i!e-n u exp(i u )du= 1ene = 1e e(-1)i ( i ) j i!j! i =1 j =u j e-n u du(-1)i ( i ) j – j n ( j 1) i!j! i =1 j =Based on the very first 4 non-central moments on the LEP distribution, we calculate the skewness and kurtosis values in the LEP distributions. These measures are plotted in Figure two against the parameters and .ness Kurto sis15000Skew505000 0 0 1 2 3 alpha two 3 a bet 1 0 0 1 2 three alpha 4 5 five four 1 4 five 52 3 a betFigure 2. The skewness (left) and kurtosis (appropriate) plots of LEP distribution.2.2. Order Statistics The cdf of i-th order statistics on the LEP distribution is given by Fi:n ( x ) = Thenr E( Xi:n )k =nn n-k n n F ( x )k (1 – F ( x ))n-k = (-1) j k k k =0 j =n-k F ( x )k j j= rxr-1 [1 – Fi:n ( x )]dx= 1-rk =0 j =(-1) jn n-kn kn-k j1xr-1 e(k j)[1-exp((- log( x)) )] dxBy altering – log( x ) = u transform we obtainMathematics 2021, 9,4 ofr E( Xi:n ) = 1 r n n-kk =0 j =(-1) jn k n k n kn n-kn kn – k k j e je-r u e-(k j) exp( u ) du= 1r = 1r = 1rk =0 j =(-1) jn n-kn – k k j e je -r u 1 (-1)l (k j)l exp(l u ) du l! l =k =0 j =(-1) j (-1) jn n-k(-1)l (k j)l (l )s n – k k j 1 e r l =1 s =0 l!s! je-r u u s duk =0 j =n – k k j 1 (-1)l (k j)l (l )s ( s 1) e j r l =1 s =0 l!s! r s 2.three. Quantile Function and Quantile LEP Distribution Inverting Equation (3), the quantile function on the LEP distribution is offered, we receive x (, ) = e-log(1-log ) 1/,(six)where (0, 1). For the spe.