Theory And Methods For Statistical Inference Of Generalized Normal Distribution Based On Order Statistics

In this dissertation,an attempt has been made for statistical inference of Generalized Normal Distribution（GND）,a probability distribution proposed by Nadarajah（2005）.It is obtained from normal distribution by adding a shape parameter into model,it’s also showed some special and classical distribution cases in it such as normal distribution,Laplace distribution.Due to the flexible parametric form of its probability density function,it’s developed for modeling in a wide range of applications:physics,engineering,medicine,biology.For example,synthetic aperture radar（SAR）images modeling for both military and civilian applications,Gabor filters in face recognition and computer vision,texture retrieval modeling using wavelets,radio signal processing,ARCH modeling,etc.However,there is few literatures in theories and applications for GND,especially in order statistics.The aim of this work is to discuss the more related properties of GND and find more suitable methods to infer the location,scale and shape parameters estimation for GND.Roughly,this work is associated with three parts as follow:Firstly,it’s showed that numerical characteristics（moments,skewness,kurtosis）and the monotonicity of kurtosis for GND.In discussing above theories,a new method to compute moments of GND is proposed base on the property that random variable of GND can be expressed as a function of Gamma distribution,which can simplify the deduction process of numerical characteristics and provide a strong theoretical basis for statistical inference of GND.Secondly,its k-order moments based on order statistics has been showed in Section 2.There are few analytical properties of GND are known about its k-order moments which cases are both independent identical distribution（IID）and independent but not identical distribution（INID）based on order statistics.In this dissertation,an explicit closed form expression is derived for the moments of GND based on order statistics,the expression for the lower records and upper records statistics are also being discussed.In particular,it is shown that both of them involve in terms of Gaussian hypergeometric function-Lauricella function,which transformed the multiple integration in the form of incomplete gamma functions into finite summation forms for special functions.Finally,four comparative methods:Method of Moments Estimation（MoM）,L-Moments Estimation（LME）,Probability Weighted Moments Estimation（PWMs）,Maximum Likelihood Estimation（MLE）,have been employed to estimate the unknown parameters of GND.The performance of these four methods are measured by simulation which provide evidence that LME and PWMs are performed better with good precision to estimate scale and shape parameter,especially in case that sample size is small.It’s showed that the estimation differences among MoM,LME,PWMs are gradually decreasing when sample size is larger.It’s also seen that MoM performed good for estimating scale parameter when sample size is small as well as true shape parameter is bigger.LME is recommend to estimate scale and shape parameter when true shape parameter is smaller than 2.And when true shape parameter is bigger than 2,PWMs is more suitable for estimating shape parameter.