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, exactly where 0 and 0 will be the model parameters. This new unit model is known as as LEP distribution and immediately after here, a random variable X is denoted as X LEP(, ). The D-Fructose-6-phosphate disodium salt medchemexpress related hrf is provided 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, then we have following easy cdf and pdf F ( x, , 1) = – – e1- x and f ( x, , 1) = x –1 e1- x for x (0, 1) respectively. The feasible shapes in the pdf and hrf have been sketched by Figure 1. As outlined by this Figure 1, the shapes of the pdf might be observed as many shapes like U-shaped, growing, decreasing and unimodal also as its hrf shapes may be bathtub, escalating and N-shaped.LEP(0.two,three) LEP(1,1) LEP(0.25,0.75) LEP(0.05,5) LEP(2,0.5) LEP(0.five,0.5)LEP(0.02,3.12) LEP(1,1) LEP(0.25,0.75) LEP(0.05,5) LEP(two,0.5) LEP(0.5,0.5)hazard rate0.0 0.two 0.4 x 0.six 0.eight 1.density0.0.0.4 x0.0.1.Figure 1. The DNQX disodium salt Epigenetic Reader Domain probable shapes from the pdf (left) and hrf (right).Other parts on the study are as follows. Statistical properties of the LEP distribution are given in Section 2. Parameter estimation strategy is presented in Section 3. Section four is devoted towards the LEP quantile regression model. Section five includes two simulation research for LEP distribution plus the LEP quantile regression model. Empirical results in the study are provided in Section six. The study is concluded with Section 7. two. Some Distributional Properties from the 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 with the LEP distribution is denoted by E( X n ) that 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 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 on the initially four non-central moments on the LEP distribution, we calculate the skewness and kurtosis values on the LEP distributions. These measures are plotted in Figure two against the parameters and .ness Kurto sis15000Skew505000 0 0 1 two 3 alpha 2 3 a bet 1 0 0 1 two 3 alpha 4 five five four 1 4 five 52 3 a betFigure 2. The skewness (left) and kurtosis (suitable) plots of LEP distribution.2.two. 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,four 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.3. Quantile Function and Quantile LEP Distribution Inverting Equation (three), the quantile function of the LEP distribution is given, we get x (, ) = e-log(1-log ) 1/,(six)where (0, 1). For the spe.
