摘要：参数估计是指从样本统计量来推断总体参数。在实际研究中，总体的参数（如总体均值、总体方差等）通常是未知的，我们需要通过抽取的样本数据进行计算和推断，对总体参数进行估计。

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思维导图

Mind mapping

参数估计

Parameter Estimation

一、定义

1. Definition

参数估计是指从样本统计量来推断总体参数。在实际研究中，总体的参数（如总体均值、总体方差等）通常是未知的，我们需要通过抽取的样本数据进行计算和推断，对总体参数进行估计。

Parameter estimation refers to inferring population parameters from sample statistics. In practical research, parameters of the population (such as the population mean, population variance, etc.) are usually unknown. We need to calculate and make inferences through sampled data to estimate the population parameters.

例如，我们想要知道某学校全体学生的平均身高（总体参数），但是不可能去测量每一个学生的身高。所以就抽取一部分学生作为样本，通过计算样本的平均身高来估计全校学生的平均身高。

For example, we want to know the average height of all students in a school (a population parameter), but it's impossible to measure the height of every student. So, we select some students as a sample and calculate the average height of the sample to estimate the average height of all students in the school.

二、参数估计的类型

2. Types of Parameter Estimation

(1) 点估计

(1) Point Estimation

定义：点估计是用样本统计量来估计总体参数，也就是用一个具体的数值来估计总体参数。例如，用样本均值估计总体均值，用样本方差估计总体方差。

Definition: Point estimation uses sample statistics to estimate population parameters, that is, a specific value is used to estimate the population parameter. For example, the sample mean is used to estimate the population mean, and the sample variance is used to estimate the population variance.

常用的点估计方法：

Common Point Estimation Methods:

矩估计法：其基本思想是用样本矩（如样本均值、样本方差等）来估计总体矩。

Method of Moments Estimation: Its basic idea is to use sample moments (such as sample mean, sample variance, etc.) to estimate population moments.

极大似然估计法：设总体有一个分布密度函数，其中包含待估计的参数。从总体中抽取一个样本，样本的联合分布密度函数与待估计参数有关。极大似然估计就是求使得这个联合分布密度函数达到最大值的参数值，作为该参数的估计值。

Maximum Likelihood Estimation Method**: Suppose the population has a distribution density function that contains parameters to be estimated. A sample is drawn from the population, and the joint distribution density function of the sample is related to the parameters to be estimated. Maximum likelihood estimation is to find the value of the parameter that maximizes this joint distribution density function and use it as the estimated value of the parameter.

(2) 区间估计

(2) Interval Estimation

定义：区间估计是在点估计的基础上，给出总体参数估计的一个区间范围，并且同时给出这个区间包含总体参数的概率（置信水平）。例如，我们估计总体均值在某个区间内，并且有95%的置信水平，这意味着如果我们重复抽样很多次，每次都构建这样的区间，大约有95%的区间会包含真实的总体均值。

Definition: Interval estimation, based on point estimation, gives an interval range for the estimation of population parameters and also gives the probability (confidence level) that this interval contains the population parameter. For example, we estimate that the population mean is within a certain interval with a 95% confidence level. This means that if we repeat sampling many times and construct such an interval each time, about 95% of the intervals will contain the true population mean.

General Steps for Constructing Interval Estimation:

确定样本统计量及其分布。例如，对于总体均值的估计，当总体方差已知时，样本均值服从正态分布；当总体方差未知时，样本均值服从自由度与样本容量有关的t分布。

Determine the sample statistics and their distributions. For example, for the estimation of the population mean, when the population variance is known, the sample mean follows a normal distribution; when the population variance is unknown, the sample mean follows a t - distribution whose degrees of freedom are related to the sample size.

根据置信水平确定分位点。如对于95%的置信水平，在正态分布下有对应的双侧分位点；在t分布下，需要根据自由度来查找对应的分位点。

Determine the critical points according to the confidence level. For example, for a 95% confidence level, there are corresponding two - sided critical points in the normal distribution; in the t - distribution, the corresponding critical points need to be found according to the degrees of freedom.

构建区间。以总体均值的区间估计为例，根据总体方差已知或未知的情况，利用样本统计量、分位点等构建相应的区间。

Construct the interval. Taking the interval estimation of the population mean as an example, construct the corresponding interval using sample statistics, critical points, etc., depending on whether the population variance is known or unknown.

三、评价参数估计量的标准

3. Criteria for Evaluating Parameter Estimators

(1) 无偏性

(1) Unbiasedness

定义：如果样本统计量的数学期望等于被估计的总体参数，那么这个统计量就是无偏估计量。例如，样本均值是总体均值的无偏估计量，样本方差是总体方差的无偏估计量。

Definition: If the mathematical expectation of a sample statistic is equal to the estimated population parameter, then this statistic is an unbiased estimator. For example, the sample mean is an unbiased estimator of the population mean, and the sample variance is an unbiased estimator of the population variance.

(2) 有效性

(2) Efficiency

定义：设两个统计量都是总体参数的无偏估计量，如果其中一个统计量的方差小于另一个统计量的方差，那么方差小的那个统计量更有效。例如，在估计总体均值时，样本均值比样本中位数作为总体均值的估计量更有效（在正态总体下）。

Definition: Suppose two statistics are both unbiased estimators of a population parameter. If the variance of one statistic is less than that of the other statistic, then the statistic with the smaller variance is more efficient. For example, when estimating the population mean, the sample mean is more efficient than the sample median as an estimator of the population mean (under a normal population).

(3) 一致性

(3) Consistency

定义：设一个估计量是基于样本容量为n的总体参数估计量，如果随着样本容量n无限增大，该估计量与总体参数的偏差大于任意给定正数的概率趋近于0，那么这个估计量就是总体参数的一致估计量。简单来说，随着样本容量的增大，估计量越来越接近总体参数。例如，样本均值是总体均值的一致估计量。

Definition: Let an estimator be an estimator of a population parameter based on a sample size of n. If as the sample size n increases infinitely, the probability that the deviation between the estimator and the population parameter is greater than any given positive number approaches 0, then this estimator is a consistent estimator of the population parameter. Simply put, as the sample size increases, the estimator gets closer and closer to the population parameter. For example, the sample mean is a consistent estimator of the population mean.

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