Statistical Diagnostics For Skew-normal Data Models Based On The Pena Distance

Normal distribution is one of the most commonly used symmetric distributions in statistical analysis.However,in practical fields such as economy,finance and environmental engineering,the problem that response variables do not satisfy symmetry is also very common.In order to study this kind of problems,statisticians have proposed many statistical models with asymmetry,among which the most typical one is the skew-normal model.In addition,we know that statistical diagnosis is the first step of data analysis,and the main purpose is to diagnose and identify abnormal points or strong influence points in sample data.There are Cook distance,likelihood distance,W-K statistic,AP statistic,etc.In this paper,a new method,Pena distance,is introduced.The traditional way is to delete a set of sample points,the impact on the estimate,or a certain(set)sample points of small disturbance effect on the estimate,whereas Pena distance is one sample points in the sample is affected by the rest of the each sample points,in simple terms,is a sample of each point,after deleting of a particular point of forecasts.In daily life,most of the data are from different populations.If we look at the same population data for simple analysis,we may not find the differences of various categories in the data.In order to better analyze the hidden statistical laws of mixture data,we need to cluster analysis for the mixture data,classify the mixture data according to different indicators or attributes,and then study the data with the same attributes or indicators in detail.For mixed data,mixture of regression models is an important data analysis tool.This article mainly studies the following three aspects:Firstly,the Pena distance is extended from the linear regression model under normal distribution to the linear regression model under skew-normal distribution,so that it can diagnose not only normal data,but also skewed data.Secondly,the Pena distance is extended from the linear regression model under skew-normal distribution to the nonlinear regression model.In this way,the applicable scope of diagnosis is extended again.Not only the linear model can be diagnosed,but also the nonlinear model can be applied.Secondly,for skew-normal data,the statistical diagnosis of location regression model under skew-normal data is established.The likelihood distance,Cook distanceand Pena distance were compared.Finally,the model and method are is illustrated by simulation studies and a real example analysis.Thirdly,on the basis of the first part,considering the possible heteroscedasticity of the data,the scale parameters and skewness parameters are further modeling.The statistical diagnosis of the joint location,scale and skewness models of skew-normal data is proposed.The likelihood distance,Cook distance and Pena distance were also used for comparison.Finally,the model and method are illustrated by simulation studies and a real example analysis.Fourthly,under the skew-normal data,considering this kind of mixture data,nonlinear and asymmetric,still contain abnormal points or strong influence points,if simply the population data diagnosis,the results may not be accurate.So we propose statistical diagnosis for mixture of nonlinear location regression models of under the skewed normal data.The diagnosis of mixture data is not classified as a whole,compared with the diagnosis after classification,then the likelihood distance,Cook distance,Pena distance,three diagnostic statistic to distinguish abnormal points or strong influence.Finally,the model and method are proved to be scientific and reasonable is illustrated by simulation studies and a real example analysis.