7.2.1 Estimation: Selection and Evaluation 估计量：选择与评价在线视频

那估计量它不完全相同
The estimators are not the same

对于我们来讲
To us

可能就会产生下一个问题
it could lead to the next problem

就是在这么多的估计量里边
In all of these estimators

哪一个可能会更好呢
which one might be better

在我们的应用里边
In our application

我们总是希望
we always hope to

找到一个好的估计量
find a good estimator

所以接下来为大家介绍
So let me introduce you

估计量选择的优良标准
good criteria for estimator selection

在上面的内容学习里边
From the above content we learned

我们可以看到
we can see that

对于同一个总体的参数
for the parameters of the same population

用不同的方法
The estimators

估计求到的估计量
found by different estimation methods

可能是不同的
may be different

当然从原则上来讲
In principle, of course

这些不同方法所计算的估计量
the estimators calculated by these different methods

都可以作为未知参数的估计量
can be used as estimators for unknown parameters

那么接下来问题来了
So here's the question

对于同一个参数
For the same parameter

究竟采用哪一个估计量更好呢
which estimator is better to use

评价估计量优劣的标准
what are the criteria for evaluating

又是什么呢
the quality of an estimator

通常情况下面
In general

确定估计量好坏的标准
the criteria for determining the quality of estimators

必须是整体性的
Must be integral

说得明确一点就是
Make it more specific

必须在大量观察的基础上的
we must base it on a large number of observations

从统计的意义上
and statistic sense

来评价估计量的好坏
To evaluate the quality of an estimator

也就是说估计的好坏
that is to say the estimated quality

取决于估计量的统计性质
depends on the statistical nature of the estimator

一般认为一个好的估计量
It is generally considered that a good estimator

应该具备如下的几个条件
should meet the following conditions

第一 无偏性
First, Unbiased

所谓无偏性指的是
So-called unbiased means that

样本某个统计量的数学期望值
the mathematical expected value of a sample statistic

等于它所估计的总体参数
is equal to the population parameter that it estimates

那么这个估计量
So this estimator

就叫做该总体参数的
is called the unbiased estimator

无偏估计量
of the population parameter

如果我们要用数学的式子
If we use a mathematical formula

来表达无偏性的定义的话
to express the definition of Unbiased

就是如果（公式如上）
That is, if (the formula above)

也就是说如果θ_hat
or if _hat

也就是我们样本的统计量
that is, the statistic of our sample

它的期望等于总体的
has an expectation equaling the population

参数的时候
parameter

我们就可以称θ_hat
we can say _hat

是θ的无偏估计量
is an unbiased estimator of θ

所以从数学性质上来讲的话
So mathematically speaking

无偏性它的描述非常地简单
unbiased is very simple to describe

就是希望样本统计量的
That is, it is hoped that the mathematical expectation

数学期望等于总体参数的
is equal to the real value

真实值
of the population parameter

无偏估计量的含义
The meaning of an unbiased estimator

指的就是θhat作为样本的函数
refers to that hat as the sample function

是一个随机变量
is a random variable

因为每一个样本
Because every sample

它是随机产生的
is randomly generated

它在θ的真值的附近在波动
It fluctuates near the true value of θ

但是不管你波动有多远
But no matter how far it is

你始终都是围绕着θ来波动的
it always fluctuates around θ

也就是你的平均值
That is, your average

恰好是等于θ的真值
is exactly the truth value of θ

所以这就是无偏估计量的
So that's the mathematics meaning of

数学的含义
an unbiased estimator

通常所谓的无偏性
Generally, the so-called Unbiased

就是要求没有系统误差
requires no system error

比如在视频中
Like in the video

为了估计旅行箱包的质量
in order to estimate the quality of the suitcases

国家质检总局
AQSIQ

抽查了118批次的产品
inspected 118 batches of products randomly

第一次抽样的合格率是64.4%
The pass rate of the first sampling was 64.4 %

如果他们愿意第二次还抽
If they wanted to do the second sampling

用同样的方法
by the same method

还抽118批次的产品
from 118 batches

进行抽查
of products

那第二次的合格率可能是68%
The pass rate on the second sampling might be 68%

他还可以继续用同样的方法
It could go on using the same method

在总产品里边
from the total products

抽取118个批次的产品
did sampling from 118 batches

来进行调查
of products

每次的结果都不尽相同
The results could be different each time

和总体的合格率之间
More or less, there must be certain errors

都或多或少存在着一定的误差
from the pass rate of the population

但是如果
But, if

我们把每次抽样的合格率
if we calculate the average

来计算平均数
of the pass rates of all samples

那么它们正好
They will exactly

会等于总体的合格率
be equal to the pass rate of the population

如果这118个批次的产品
If the 118 batches of products

是从全部的箱包生产厂家
are selected from the products

生产的产品里边抽取的
manufactured by all suitcase manufacturers,

那么每一次的合格率
Then, the error between each pass rate

和总体合格率之间的误差
and the population pass rate

就属于随机误差
belongs to random errors

因为从总体里边
Because the sampling units

抽取样本单位的时候
selected from the population

它是随即产生的
it was generated immediately

但是如果视频里边
But, if the video

说明了这118个批次的产品
explained that the 118 batches of products

是从中小型箱包生产企业的
were selected from the products of

产品里边抽取的话
small and medium suitcase manufacturers,

那么除了随机误差
so except for the random error

还存在着系统误差
there are also systematic errors

因为抽样框和真正的总体之间
Because between the sampling frame and the real population

结构不完全相同
the structure is not the same

那我要了解的
What I want to know

是所有厂家生产的箱包质量
is the quality of suitcases produced by all manufacturers

但是你抽查的时候
But in sampling

你只抽了中小型箱包生产厂家
you only select small and medium-sized manufacturers

那当然你的抽样的框
That's certain that between your sampling frame

抽样框和总体之间
and the population

就存在着结构上的差异
there is a structural differences

那这种就叫做系统性的误差
This is called a systematic error

我们所谓的无偏性
What we call Unbiased

是排除了系统性误差
is that the errors are brought

而完全是由随机因素
fully by random factors

所带来的误差
excluding the systematic errors

这就是无偏性
This is Unbiased

它的一个基本的要求
Its basic requirement is

没有系统性的误差
there is no systematic error

通常情况下来讲
In general

一个总体参数θ
for a population parameter

它的无偏估计量
its unbiased estimator

不具有唯一性
has no uniqueness

比如假定样本X{\fs10}1{\r} X{\fs10}2{\r}……X{\fs10}n{\r}
Assume there is a sample X{\fs10}1{\r} X{\fs10}2{\r}……X{\fs10}n{\r}

它是来自于总体X
Which comes from the population X

假定总体的均值
Assume the mean of the population

可以用μ来表示
can be expressed by μ

那么X_bar
Then, _bar

也就是我们通常说的
as we usually called

简单算术平均数
simple arithmetic mean

它是μ的无偏估计量
It's the unbiased estimator of μ

这个很简单
That's easy

我们可以证到
to prove

此外
Besides

在数学上也可以证到
you can prove it mathematically

（公式如上）
(The formula is as above)

当（公式如上）等于1的时候
When (the formula above) is equal to 1

也是μ的无偏估计量
it's also the unbiased estimator of μ

也就是说对于某一个样本
So for a particular sample

除了简单算术平均数
in addition to the simple arithmetic average

是μ的无偏估计量以外
which is the unbiased estimator of μ

它的加权形式的算术平均数
the arithmetic mean in its weighted form

也是μ的无偏估计量
is also the unbiased estimator of μ

那加权形式的权数
The weight of the weighted form

可以由你任意来分配
is subject to your assignment

只要它们求和等于1就可以了
As long as they add up to 1

所以对于一个总体参数来讲
So for a population parameter

对于μ来讲
for μ

它可以有很多个无偏估计量
it can have many unbiased estimators

无偏估计量是对估计量的
Unbiased is an important common

一个重要而常见的要求
requirement for estimators

实际意义是多次试验后
The real meaning is, after a lot of trials

没有系统性的偏差
there is no systematic bias

也是实际操作中
In practical operation

完全合理的要求
it’s also a fully reasonable requirement

但是我们不要一味地
But let's not just

认为估计量不满足无偏原则
consider that an estimator that does not satisfy the Unbiased principle

就是不好的估计量
is a bad estimator

所以这一点一定要弄清楚
So this has to be clear that

虽然它是一个很重要
though it is a very important

并且很常见的
and very common requirement

但是并不是说
that's not to say that

它不满足无偏的原则
an estimator that does not satisfy the principle of Unbiased

就是一个不好的估计量
should be a bad estimator

这个我们在后续的课程里边
In our later course

继续地如果会学习一些
if we continue to learn more

统计相关的专业课程的时候
statistics related professional courses

会有更多的体验
we’ll have more experiences

关于无偏性它的要求
about the requirement for Unbiased

那接下来
We’ll

我们也稍微地介绍一下
talk a little about

无偏性的弱点
the weakness of Unbiased

第一个
Firstly

一个总体参数θ
for the population parameter θ

可能不存在无偏估计量
there may be no unbiased estimator

第二个
Secondly

一个参数θ可能存在着
the population parameter θ may have

多个无偏估计量
multiple unbiased estimators

比如咱们刚才说的
As we said just now

对于μ来讲
for μ

它可能除了
in addition to

简单算术平均数以外
the simple arithmetic mean

它的加权形式的算术平均数
the arithmetic mean in its weighted form

也都是μ的无偏估计量
is also an unbiased estimator of μ

甚至中位数
Even the median

也可以是μ的无偏估计量
can also be an unbiased estimator of μ

它可以有很多个
It could have many

另外第三个
Thirdly

无偏估计值反映了
the unbiased estimate reflects that

估计量在参数真值的附近波动
the estimator fluctuates near the parameter truth value

并不表示其他更多的内容
It doesn't mean anything more

这是我们介绍的
This is the first criterion we introduced

第一个标准无偏性
Unbiased

如果对于同一个总体参数
If, for a same population parameter

它确实存在着多个无偏估计量
there are indeed multiple unbiased estimators

那么接下来
Then

我们如何来选择呢
How to choose

如何来评价它们的优劣呢
and To evaluate them

这就引出了
This lead to

接下来的一个评价标准
another evaluation criterion

有效性
Effectiveness

估计量的无偏性
The Unbiased of estimator

只保证了估计量的取值
only guarantees the values of the estimator

在参数真值周围波动
fluctuates around the parameter truth-value

但是波动的幅度有多大呢
But what is the amplitude of fluctuation

自然地我们希望估计量
Naturally we want estimators

波动的幅度越小越好
to have a smaller fluctuation amplitude as much as possible

幅度越小则估计量取值
The smaller the amplitude, the smaller the possibility for big deviation

和参数真值
of the value of the estimator

有较大偏差的可能性就越小
from the truth-value of the parameter

而衡量随机变量波动幅度的
For measuring the fluctuation amplitude of the random variable

常用的指标
there is a common indicator

我们很熟悉
that we are very familiar with

那就是方差
that is variance

所以通常情况下
In general

我们对有效性是这样来定义的
We define effectiveness this way

在无偏估计量里边
of the unbiased estimators

方差越小的估计量
the smaller the variance

它是越有效的
the more effective the estimator

如果我们要用数学符号
If we use mathematical notation

来表示的话
to express

那么我们可以借助下面的语言
we can use the following terms

假定θ1_hat
Assume θ1 _hat

和θ2_hat
And θ2_hat

都是参数θ的无偏估计量
are both unbiased estimators of the parameter θ

如果在任意的条件下
If in any case,

θ1_hat的方差
the variance of θ1_hat

都比θ2_hat的方差
compared with that of θ2_hat

要来的更小一些
is smaller

那么我们就称θ1_hat
we say θ1_hat

是比θ2_hat
is a more effective estimator

更有效的估计量
than θ2_hat

这就是关于有效性的描述
So that's a description of the effectiveness

其实它就是指的是
It actually means

方差越小越好
the smaller the variance, the better

那我们也知道
We know that

因为方差它是反映离散程度的
the variance is a measure of dispersion

那自然我们希望
Naturally we hope

θ_hat它是围着θ来波动的
as θ1_hat fluctuates around θ

那如果离着它近的话
If it is nearer to θ

那对它的代表性不就更强吗
that is more representative of it

这个我们前面
We have learned

也学习过有关的知识
the relevant knowledge

我们自己也可以应用过来
We can apply it

我们要评价一个指标
when we evaluate the representativeness

它的代表性的时候
of an indicator

你可以用什么样的指标
What indicators can you use

去对它进行分析
to analyze it

下面我们可以看一个例子
Let's look at an example

比如对于来讲
For μ

我们刚才也提到
as we mentioned just now

它可以有很多个无偏估计量
it can have many unbiased estimators

比如第一个形式可以是
For example, the first form could be

（公式如上）
(The formula is as above)

这是一个算术平均数
This is an arithmetic average

加权的形式
in weighted form

一个权重是3/4
One weight is 3/4

一个权重是1/4
One weight is 1/4

第二个无偏估计量的形式
The second form of unbiased estimators

就是简单算术平均数
It's the simple arithmetic mean

（公式如上）
(the formula is as above)

第三个无偏估计量的形式
The third form of unbiased estimators

（公式如上）
(the formula is as above)

那这些都是
These are all

它的算术平均数的形式
its arithmetic mean form

当然我们从数学上面
Of course, mathematically

可以非常简单
it is very simple

非常快地证明
and can be proven very quickly

它们都是μ的无偏估计量
They're all unbiased estimators of μ

那接下来我们就可以尝试着
Then we can try

来计算这三个不同的
to compute the variances

无偏估计量
of these three different

它们方差的大小
Unbiased estimators

当然它们都是
Of course they are all

从同样的样本里边
from the same samples

所抽取的它们的μ
their μ

以及它们的方差
and their variances

都是相同的
are the same

那我们只需要简单地
We need to just simply

把这些信息
substitute the information

代入进去进行运算
and calculate

很快我们可以得到
We'll get soon

第一个无偏估计量
the first unbiased estimator

也就是（公式如上）
That is (the formula is as above)

它的方差是（公式如上）
Its variance is (the formula is as above)

那假定X{\fs10}1{\r}和X{\fs10}2{\r}之间是独立的
Assume that X{\fs10}1{\r} and {\fs10}2{\r} are mutually independent

那这个地方我们后面不再重复
then we won't repeat that later

我们所谓的简单随机样本
The so-called simple random sample

就满足这些要求
can satisfy these requirements

后面我们都是同样的
For the following calculation, we can have the same

有这个结论
Conclusion

不再重复这一点
We won’t repeat it later

那（公式如上）因为它们来自于
Then, (the formula is as above) because they come from

同一个总体
the same population

所以它们是同方差的
So their variances are the same

因此第一个无偏估计量的方差
So the variance of the first unbiased estimator

我们可以推算出来是（字符如上）
is calculated to be (the character is as above)

那第二个无偏估计量
Then, for the second unbiased estimator

以及第三个无偏估计量
and the third unbiased estimator

我们可以用同样的方法
we can use the same method

给它推算出来
to calculate the variance

可以得到第二个无偏估计量
We can obtain the variance for the second unbiased estimator

也就是（公式如上）
It is (the formula is as above)

它的方差是等于（字符如上）
Its variance is equal to (character above)

第三个无偏估计量
For the third unbiased estimator

它的方差是（字符如上）
its variance is (character above)

那从这三个方差上面来看的话
From these three variances

很快我们可以得到一个结论
we can reach a conclusion quickly

简单算术平均数1
simple arithmetic mean

它所对应的方差是最小的
has the least variance

当然我们这里
Of course, here we

只是用具体的一个形式
only use a specific form

来验证了简单算术平均数
to verify that simple arithmetic mean

它的方差比较小
has a smaller variance

那实际上从数学上面
Actually, mathematically

我们可以利用
we can use

拉格朗日函数求极值的方法
the method of finding the extreme values of Lagrange functions

可以证明得到
to prove that

在μ的无偏估计量里边
In the unbiased estimators of μ

所有的无偏估计量里边
of all the unbiased estimators

X_bar是最有效的
X_bar is the most efficient

他说它最有效的意思就是
That it is most effective means

它的方差是最小的
it has the smallest variance

所以我们在对μ
So when we do estimation

进行估计的时候
on μ,

首选的估计量
The preferred estimator

就是X_bar简单算术平均数
is the simple arithmetic average of X_bar

是因为它符合我们评选的标准
Because it meets our criteria

它既符合无偏性的要求
It conforms to both the requirement of Unbiased

也符合有效性的要求
and the requirement of effectiveness

所以对我们来讲
So, to us

它是一个性质优良的估计量
it is an estimator of good quality

另外我们也可以证明
And we can also prove that

X_bar如果和中位数来比的话
if X_bar is compared with the median

它的方差也是更小一些
its variance is smaller

所以在所有的μ的
So, of all the unbiased estimators

无偏估计量里边
of μ

X_bar就是一个
X_bar is

方差最小的无偏估计量
the unbiased estimator with the smallest variance

通常情况下
Generally

我们把方差最小的无偏估计量
to the unbiased estimator with the smallest variance

有一个特别的名字给它
we give a special name

就叫做最小方差无偏估计量
It's called the minimum variance unbiased estimator

其实也就是最有效的一个估计量
It's actually the most efficient estimator

它的定义大家也可以了解一下
You can see the definition

假定θ0_hat是θ的一个无偏估计量
Assume that _hat is an unbiased estimator of θ

如果对于θ的任一方差存在的
If the unbiased estimator θ_hat existing

无偏估计量θ_hat
for any variances of θ

都有θ0_hat的方差
has the variance of θ0_hat

比θ_hat的方差来得
that is smaller than

更小一些的话
the variance of θ_hat

那么我们就称θ0_hat
then, we call θ0_hat

是θ的最小方差
is the minimum variance

无偏估计量
unbiased estimator of θ

一般来讲
Generally speaking

最小方差无偏估计量
minimum variance unbiased estimator

是一个最优的估计量
is an optimal estimator

所以我们在寻求估计量的时候
so when we're looking for estimators

你可以先用无偏性去考核它
you can test it first with Unbiased

如果能满足无偏性的要求
If it can satisfy the requirements of Unbiased

接下来我们可以用第二条标准
then we can use the second criterion

有效性再去考核它
effectiveness to test it

那这样如果两者能够兼顾的话
If it satisfies both,

我们就可以找到一个最小方差
we can find a minimum variance

无偏估计也就是最优估计
The unbiased estimate is the optimal estimate

这是第二个评价的优良标准
This is the second criterion of excellence for evaluation

第三个评价的优良标准一致性
The third criterion of excellence is consistency

如果估计量
If the estimator

随着样本容量n的增大
with the increase of sample size n

而越来越接近总体参数的值
is getting closer and closer to the value of population parameter

那么我们就称这样的估计量
we call such estimator

是总体参数的一致性估计量
is the consistent estimator of the population parameter

也称相合估计量
It is also called congruent estimator

估计量的一致性
consistency of estimators

是从极限意义上来讲的
is in the ultimate sense

它适用于大样本的情况
It applies to large samples

同样地
Similarly

我们给出一致性的数学表达式
we give the mathematical expression

数学表达式
for consistency

我们以样本平均数为例子
Let's take the sample mean as an example

如果它符合一致性的要求
If it meets the requirements for consistency

即存在以下关系
there are the following relationships

在这个式子里面（符号如上）
In this formula (the sign is as above)

是一个任一小的正数
If it's any small positive number

那这个式子要表达的就是
then what the formula expresses is that

当n趋近于无穷大的时候
As n approaches infinity

X_bar依概率收敛于μ
X_bar converges to μ based on probability

这就是一致性的数学的表达
That's the mathematical expression of consistency

直观上面来看
Intuitively

当n增大的时候
as n increases

样本信息增多
sample information will increase

当然希望估计量
Of course, it is hoped that the probability for the estimator

越来越靠近真值的概率
to approach the truth-value

也越来越大
will become bigger

这种想法就引出了
That’s lead to

上面的一致性的概念
the above concept of consistency

一致估计量
Consistent estimator

一般是在样本容量很大的时候
will be able to display its advantages

才能够显示它的优点
only when the sample size is very large

必须证明它依概率收敛
You have to prove that it converges with probability

这个有的时候非常地麻烦
This is sometimes very troublesome

下面是一个常用来判定
Here's a theorem that is often used to judge

一致性的定理
consistency

在这个定理里边告诉我们
It tells us in this theorem that

如果θ_hat是θ的一个估计量
if θ _hat is an estimator of θ

只要满足以下两个条件
as long as the following two conditions are met

第一个条件就是
The first condition is that

当n趋近于无穷大的时候
as n approaches infinity

θ_hat的期望值是等于θ
the expected value of θ_hat equals θ

第二个
The second is that

是当n趋近于无穷大的时候
as n approaches infinity

θ_hat的方差是趋近于0的
the variance of θ_hat approaches zero

只要能够满足这两个条件的话
As long as the two conditions are met

那么我们就可以判定
we can decide that

θ_hat就是θ的相合估计量
θ _hat is the congruent estimator of θ

或者叫做一致估计量
or called consistent estimator

最后我们稍微地来总结一下
Finally, let's just conclude a little bit

我们这一讲
in this lecture

为大家讲述的内容
the contents talked about

在这一讲里边
In this lecture

我们主要地介绍了
we mainly introduced

参数估计的一般问题
general issues of parameter estimation

包括点估计
including point estimation

包括估计量的评选标准
and Evaluation criteria for estimators

在介绍估计量的
For the Evaluation criteria

评选标准的时候
of estimators

主要有无偏性
there are mainly three criteria, Unbiased

有效性和一致性这么三个
effectiveness and consistence

一致性是对估计量的
For estimators, consistency is

一个基本要求
a basic requirement

一般来讲
Generally speaking

不具备一致性的估计量
an estimator without consistency

是不予考虑的
will not be considered

但是估计量的一致性
But the consistency of estimators

只有当样本容量相当大的时候
only if the sample size is fairly large

才能显示出优越性
can show its advantages

这个在实际工作里边
In actual work

往往难以做到
it is often hard to do

因此在操作里边
Therefore, in operation

往往使用无偏性
the criteria of Unbiased

和有效性这两个标准
and effectiveness are often used

来帮助我们考核
to help us test

一个估计量的好坏
an estimator

这一讲的内容就是这样
So much for this lecture

谢谢大家
Thank you