Study On The Problem Of Maximum Likelihood Estimation Of The Parameters Of The Kotz Ratio Distribution Under Order Restrictions

Using the normal distribution to deal with the problem of small samples, the unreliabilityof small samples needed to be considered. Statisticians have put forward somenonconventional distributions to solve the problems. The kotz-type distribution is one of thenonconventional distributions. Compared with the normal distributions, it has thecharacteristics of peak and heavy tail. According to the good properties, this thesis willcontinue to discuss the kotz-type distribution on the basis of the S. Nadarajah’s studies.Based on the above studies and the theory of order restrictions, firstly, we study thekotz-type distribution under the order restrictions. The thesis mainly studies the basicproperties of the multiple kotz-type ratio distributions. The maximum likelihood estimationabout the parameters of the multiple kotz-type ratio distributions is given based on thecharacteristics of distribution. Based on the iterative methods, when parameter N is knownand parameter r is unknown, the maximum likelihood estimation of r is given; whenparameter r is known and parameter N is unknown, the maximum likelihood estimationof N is given. In addition, when N and r are both unknown, the maximum likelihoodestimations of N and r are given.Secondly, the maximum likelihood estimation about the parameters of the multiplekotz-type ratio distributions under order restrictions which contain simple order, simple treeorder, umbrella semi-order and simple ring order is studied. According to the PAVAalgorithm, when parameter N is known and parameter r is unknown, the maximumlikelihood estimation of r is given; when parameter r is known and parameter N isunknown, the maximum likelihood estimation of N is given. In addition, when N and rare both unknown, the maximum likelihood estimations of N and r are given. Especially,the simple algorithms are used in simple tree order, umbrella semi-order and simple ringorder.Finally, the proposed methods are applied in the main stock indices closing prices of theEurope: Germany DAX(Ibis), Switzerland SMI, France CAC and UK FTSE, then we find outthe relationship between any two stock indices.