On the log-exponential-power (LEP) distribution are offered as F ( x, , ) = e and (- log x) 1-exp (- log x ) e , x (0, 1) (four) (- log x ) -1 e x respectively, where 0 and 0 will be the model parameters. This new unit model is known as as LEP distribution and soon after right here, a random variable X is denoted as X LEP(, ). The Charybdotoxin Autophagy associated hrf is offered 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).(five)-If the parameter is equal to one particular, then we have following simple cdf and pdf F ( x, , 1) = – – e1- x and f ( x, , 1) = x –1 e1- x for x (0, 1) respectively. The doable shapes with the pdf and hrf happen to be sketched by Figure 1. In line with this Figure 1, the shapes with the pdf is usually observed as different shapes for example U-shaped, escalating, decreasing and unimodal too as its hrf shapes might be bathtub, increasing and N-shaped.LEP(0.2,3) LEP(1,1) LEP(0.25,0.75) LEP(0.05,five) LEP(2,0.5) LEP(0.5,0.five)LEP(0.02,three.12) LEP(1,1) LEP(0.25,0.75) LEP(0.05,5) LEP(two,0.5) LEP(0.5,0.five)hazard rate0.0 0.2 0.4 x 0.six 0.8 1.density0.0.0.four x0.0.1.Figure 1. The doable shapes with the pdf (left) and hrf (proper).Other components of the study are as follows. Statistical properties of the LEP distribution are offered in ML-SA1 Agonist Section two. Parameter estimation method is presented in Section three. Section four is devoted towards the LEP quantile regression model. Section 5 consists of two simulation research for LEP distribution and also the LEP quantile regression model. Empirical final results in the study are provided 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 of your LEP distribution are studied.Mathematics 2021, 9,three of2.1. Moments The n-th non-central moment of your 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 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 around the initially four 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 three alpha two 3 a bet 1 0 0 1 2 3 alpha four five five four 1 four 5 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 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 two.three. Quantile Function and Quantile LEP Distribution Inverting Equation (3), the quantile function of the LEP distribution is provided, we get x (, ) = e-log(1-log ) 1/,(six)exactly where (0, 1). For the spe.
