On the log-exponential-power (LEP) Charybdotoxin Cancer Distribution are provided 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 will be the model parameters. This new unit model is known as as LEP distribution and following right here, a random variable X is denoted as X LEP(, ). The associated 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 a single, then we’ve following basic cdf and pdf F ( x, , 1) = – – e1- x and f ( x, , 1) = x –1 e1- x for x (0, 1) respectively. The possible shapes on the pdf and hrf have already been sketched by Figure 1. In line with this Figure 1, the shapes of your pdf is usually noticed as several shapes like U-shaped, increasing, decreasing and unimodal as well as its hrf shapes may be bathtub, escalating and N-shaped.LEP(0.two,3) LEP(1,1) LEP(0.25,0.75) LEP(0.05,5) LEP(2,0.five) LEP(0.five,0.5)LEP(0.02,3.12) LEP(1,1) LEP(0.25,0.75) LEP(0.05,five) LEP(2,0.5) LEP(0.5,0.5)hazard rate0.0 0.two 0.4 x 0.6 0.8 1.density0.0.0.4 x0.0.1.Figure 1. The attainable shapes with the pdf (left) and hrf (suitable).Other parts on the study are as follows. Statistical properties on the LEP distribution are provided in Section 2. Parameter estimation method is presented in Section three. Section 4 is devoted for the LEP quantile regression model. Section five contains two simulation research for LEP distribution plus the LEP quantile regression model. Empirical outcomes of your study are provided in Section 6. The study is concluded with Section 7. 2. Some Distributional Properties of your LEP Distribution The moments, order statistics, entropy and quantile function of the LEP distribution are studied.Mathematics 2021, 9,three of2.1. Moments The n-th non-central moment on 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 changing – log( x ) = u Etiocholanolone Membrane Transporter/Ion Channel transform we get 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 four non-central moments on the LEP distribution, we calculate the skewness and kurtosis values of your LEP distributions. These measures are plotted in Figure 2 against the parameters and .ness Kurto sis15000Skew505000 0 0 1 two 3 alpha two 3 a bet 1 0 0 1 two 3 alpha 4 5 5 4 1 four five 52 3 a betFigure 2. The skewness (left) and kurtosis (proper) plots of LEP distribution.two.2. Order Statistics The cdf of i-th order statistics of your LEP distribution is offered 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 (three), the quantile function on the LEP distribution is given, we obtain x (, ) = e-log(1-log ) 1/,(6)where (0, 1). For the spe.