Asymptotic Statistical Properties Of Gini Regression Under Binary Gaussian Model

Regression analysis was born in the period of modern statistics,and its origin can be traced back to more than one hundred years ago.As a mathematical model,regression analysis is essentially to establish a virtual regression curve to make the curve as close as possible to the sample point of the study.That is,it is a technique for studying the quantitative correlation between multiple sample points.In the field of statistics,there are many kinds of regression analysis methods.But undoubtedly the most classic one should be the ordinary least squares.The ordinary least squares method establishes a regression curve and uses the residuals between the actual sample data and the virtual curve to find the minimum value of the sum of squares of the residuals to obtain the best matching function of the data.Therefore,this method is suitable for obtaining some unknown data,or performing linear fitting and optimizing some parameters.The least squares method has efficient applications in many fields.However,the rationality of this method must satisfy certain classical assumptions.When the ordinary least squares method cannot satisfy several classical assumptions,it will not be able to obtain a reliable estimate.This paper attempts to use the Gini regression method based on Gini mean difference to make up for the shortcomings of the ordinary least squares method.There are two Gini regression methods based on Gini mean difference.The first method is based on the minimization of the residual of the Gini mean difference.The second is to express the Gini covariance between the dependent variable and the independent variable as the weighted sum of the slopes of the regression curve,also known as the Gini semiparametric method.The Gini semiparametric method is similar in structure to the ordinary least square method.The regression coefficients of these two methods have an equivalent.The equivalent is constructed by using the Gini coefficient covariance and the Gini coefficient to replace the covariance and variance in the ordinary least squares method,respectively.The above replacement will generate additional parameters that enable researchers to adjust the statistical analysis according to the needs of the research field.Therefore,we will focus on the second Gini regression method.The second Gini regression method provides a wealth of tools for statistical analysis.In order to make full use of this analysis tool,this paper has carried out the following theoretical work.Firstly,on the basis of the standard binary Gaussian model,we derive the Gini regression coefficient definition formula containing rank information according to the original definition formula of the second Gini regression coefficient.Secondly,we skillfully use the first-order and second-order Delta formulas to derive the asymptotic expressions of the mathematical expectation and variance of the Gini regression coefficient for the first time.Then,in order to prove the reliability and correctness of the theoretical derivation,this paper uses Matlab to simulate Monte Carlo experiment to verify the theoretical derivation results.The asymptotic statistical characteristics of the Gini regression coefficient are analyzed in detail by combining the theoretical derivation results with the actual simulation data.Finally,we introduce the index of asymptotic relative efficiency,and use the ordinary least squares method as a reference method to discuss the asymptotic relative efficiency of the Gini regression method,which further proves that the Gini regression method has excellent asymptotic statistical characteristics in the binary Gaussian environment.At the same time,in order to prove that the second Gini regression method has application value in real work,this paper innovatively introduces the Gini regression method into the field of signal processing.Firstly,we use the derived Gini regression coefficient definition formula that contains rank information,and use some potential mathematical relations of the rank to convert the definition formula to obtain a new Gini regression coefficient definition formula.Secondly,according to the new definition of Gini regression coefficient,we design a Gini regression circuit based on FPGA.This circuit can not only realize parallel operation,but also detect and estimate signals quickly.Finally,the designed FPGA circuit is applied to the field of signal processing,and the feasibility and rationality of the circuit are verified by Matlab,so as to expand the application field of the method.