In the log-exponential-power (LEP) PF-06873600 Epigenetics distribution are offered as F ( x, , ) = e and (- log x) 1-exp (- log x ) e , x (0, 1) (4) (- log x ) -1 e x respectively, exactly where 0 and 0 are the model parameters. This new unit model is named as LEP distribution and immediately after here, a random variable X is denoted as X LEP(, ). The related hrf is offered by f ( x, , ) = h( x, , ) = x eexp (- log x )1-exp (- log x ),x (0, 1)(3)-e(- log x) (- log x ) -1 ,x (0, 1).(5)-If the parameter is equal to a single, then we’ve got following straightforward cdf and pdf F ( x, , 1) = – – e1- x and f ( x, , 1) = x –1 e1- x for x (0, 1) respectively. The achievable shapes from the pdf and hrf have been sketched by Figure 1. In line with this Figure 1, the shapes of your pdf can be noticed as various shapes for example U-shaped, increasing, decreasing and unimodal also as its hrf shapes is often bathtub, growing and N-shaped.LEP(0.2,3) LEP(1,1) LEP(0.25,0.75) LEP(0.05,five) LEP(two,0.5) LEP(0.five,0.five)LEP(0.02,3.12) LEP(1,1) LEP(0.25,0.75) LEP(0.05,five) LEP(2,0.5) LEP(0.five,0.five)hazard rate0.0 0.two 0.4 x 0.six 0.8 1.density0.0.0.four x0.0.1.Figure 1. The possible shapes with the pdf (left) and hrf (ideal).Other parts in the study are as follows. Statistical properties of the LEP distribution are C2 Ceramide Apoptosis provided in Section two. Parameter estimation system is presented in Section three. Section four is devoted towards the LEP quantile regression model. Section 5 consists of two simulation studies for LEP distribution as well as the LEP quantile regression model. Empirical outcomes on the study are offered in Section six. The study is concluded with Section 7. two. Some Distributional Properties of 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 of your LEP distribution is denoted by E( X n ) which can be 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 receive 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 around the 1st 4 non-central moments of the LEP distribution, we calculate the skewness and kurtosis values with 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 five 5 4 1 four five 52 three a betFigure two. The skewness (left) and kurtosis (ideal) plots of LEP distribution.two.2. Order Statistics The cdf of i-th order statistics with 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 changing – 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 (three), the quantile function from the LEP distribution is given, we acquire x (, ) = e-log(1-log ) 1/,(six)exactly where (0, 1). For the spe.