Confidence Sets Of Parameters In The P-Norm Distribution

In statistics,it is a persistent research topic to study the confidence intervals of parameters in various distributions.P-norm distribution is a kind of distribution with single-peak and symmetric error,including degenerate distribution,normal distribution,Laplacian distribution and so on.It is more suitable to describe the error distribution than the normal distribution,so it is of great significance to study the confidence set of parameters in the p-norm distribution.In Chapter 2,appropriate pivot quantities are determined by the tools of moment estimation,maximum likelihood estimation(MLE)and central limit theorem,and(approximate)confidence intervals for single and double populations are obtained respectively.In addition,the approximate confidence intervals of the parameters are given by some simulation examples.In Chapter 3,the pivots are constructed in four ways:construct the pivot quantity of approximate Chi-square distribution according to the maximum likelihood estimation and strong consistent;construct the pivot quantity with the strong consistent estimation of parameters instead of the original parameters;construct the pivot quantity with the F distribution instead of the Chi-square distribution;and construct the pivot quantity according to the likelihood ratio test.The approximate joint confidence sets for the parameter(μ,σ~p)in the p-norm distribution are obtained by using these four pivot quantities.At the end of this chapter,some simulation examples are given to illustrate the effectiveness of the above methods.The sizes of the confidence regions obtained by various methods are compared by the graphs of the confidence regions.The four confidence regions become smaller and more consistent with the increase of n.In addition,our conclusions are applied to the GPS surveying residual data,and the four confidence regions all contain the estimated values of the parameters.In Chapter 4,two methods of joint confidence sets are explored,which are maximum Lq-likelihood estimator(MLqE)and Lq-likelihood-ratio-type(LqRT).Using the MLqE of parameter(μ,σ~p)and its approximate distribution to construct the appropriate pivot quantity and obtain the approximate confidence region of(μ,σ~p).The approximate confidence region of(μ,σ~p)is obtained by constructing the appropriate pivot quantity from the Lq-likelihood ratio statistics of p-norm distribution and its approximate distribution.The correctness of these methods is illustrated by the simulation examples,and each confidence region is compared by graphs.The following conclusions are obtained:the approximate confidence region obtained by LqRT method is better than that obtained by MLqE method;the approximate confidence region increases with the increase of q,decreases with the increase of sample size n,and finally approaches the true value.The approximate confidence region obtained by MLqE method decreases with the increase of p,while the relationship between the approximate confidence region obtained by LqRT method and p is not clear.