On min- and max-Kies families: distributional properties and saturation in Hausdorff sense

Abstract

The purpose of this paper is to explore two probability distributions originating from the Kies distribution defined on an arbitrary domain. The first one describes the minimum of several Kies random variables whereas the second one is for their maximum – they are named min- and max-Kies, respectively. The properties of the min-Kies distribution are studied in details, and later some duality arguments are used to examine the max variant. Also the saturations in the Hausdorff sense are investigated. Some numerical experiments are provided.

On the Maximum Likelihood Estimators’ Uniqueness and Existence for Two Unitary Distributions: Analytically and Graphically, with Application

“…According to these results, the graphical method achieves similar outcomes in a shorter period compared to traditional methods. In a future study, we will investigate the efficiency of this graphical method to some recent bounded distributions, including Kies families [23][24][25]. abb2=c(mean(abs(alphamle-alpha))) mre1=c(mean(abs(betamle-beta)/beta)) mre2=c(mean(abs(alphamle-alpha)/alpha)) fullb= rbind(fullb,n,b1,b2) fullm= rbind(fullm,n,m1,m2) fullabb= rbind(fullabb,n,abb1,abb2) fullmre= rbind(fullmre,n,mre1,mre2) print(fark) fullb fullm fullabb fullmre "Maximum Likelihood Method" alphamle=betamle=NULL fullb=fullm=fullabb=fullmre=NULL loglk=function(par){ t=0 beta=par [1] alpha=par [2] f=(1/x)*alpha*beta*(-log(x))^(beta-1)*exp(-alpha*(log(x))^beta) t=t+sum(log(f)) return(-t) } for (j in 1:ds) { cat("\14",j) dev=F while(dev==F) { for(i in 1:n) { u=runif (1) x[i]=1/exp(exp(log(-log(u)/alpha)/beta)) } aa=try(optim(c(beta,alpha),loglk, method = "CG", hessian = T),silent = T) if (!is.character(aa))if (aa$par [1] >0& aa$par[2] >0) dev=T } betamle[j]=aa$par [1] alphamle[j]=aa$par [2] } bitis <-Sys.time() fark <-difftime(bitis, baslangic, units = "secs") print(fark) b1=mean(betamle) b2=mean(alphamle) m1=c(mean((betamle-beta)^2)) m2=c(mean((alphamle-alpha)^2)) abb1=c(mean(abs(betamle-beta))) abb2=c(mean(abs(alphamle-alpha))) mre1=c(mean(abs(betamle-beta)/beta)) mre2=c(mean(abs(alphamle-alpha)/alpha)) fullb= rbind(fullb,n,b1,b2) fullm= rbind(fullm,n,m1,m2) fullabb= rbind(fullabb,n,abb1,abb2) fullmre= rbind(fullmre,n,mre1,mre2) print(fark) fullb fullm) fullabb fullmre…”