统计中估计量的估计精度研究

AbstractStatistics is a very old science.It is generally believed that its academic researchbegan in the Aristotelian era ofancient Greece and has a history of more than 2,300 years.It originated from the study of social and economic issues.Therefore,"mathematicalstatistics"is not a single new statistical theme.Rather,it is a comprehensive term for allthe new methods of collecting and analyzing data formed by statistics in the thirddevelopment stage.Probability theory is the theoretical basis of mathematical statisticalmethods,but it does not belong to the category of statistics,but belongs to the category ofmathematics.In statistics,an estimator is a rule to calculate the estimated value of a knownquantity based on the observation data.Estimators are sometimes called estimators,which are used to estimate unknown general parameters.One-time estimation refers tothe result of applying this function to a set of known data sets to find the function.For agiven parameter,there can be many different estimators.We choose better estimatorsfrom them by some selection criteria,but sometimes it's hard to say that one estimator isbetter than the other.This paper is divided into four parts:In the first part,the common estimators,parameter estimation and nonparametricestimation are introduced,and the point estimation and interval estimation in parameterestimation,histogram estimation,paezen window estimation,kernel density estimationand k-nearest-neighbor estimation in nonparametric estimation are described in detail.In the second part,we introduce some common inequalities,mainly prove Markovinequality,Chebyshev inequality,Hoeffding inequality,Mill inequality,Bernsteininequality,Berry Esseen inequality.The third part uses the previous knowledge points to explain and prove the propertiesof the accuracy estimation of common estimators,and summarizes the unbiasedness,consistency and effectiveness.The fourth part introduces what is deviation and uses the knowledge summarized inthe previous parts to deduce the relevant knowledge of large deviation.KEY WORDS:Parameter estimation,Nonparametric estimation,Chebyshev inequality,Bernoulli sequence,Large deviation Technology