7.1.1 Point estimation: methods and applications 点估计：方法与应用在线视频

妞妞妈 大妞 二妞
Niuniu mom and my lovely daughters

今年我们班毕业20周年
It is the 20th anniversary of the graduation of our class in this year

大家商量着暑假挑个时间
We are discussing the time in summer vacation

到青岛聚会都带上家属
We’ll bring family members to the party in Qingdao

你们做好准备
Get ready to go

到时和我一起去
with me at that time

耶 太好啦
Oh, very good

暑假可以去海边玩喽
We can go to the beach in the summer vacation

哎呀 要出去旅游
Oh, to travel

还得重新买一两个箱子呢
We need to buy one or two more suitcases

我先来了解了解箱包
Let me take a look at the suitcase first

看来箱包的质量不容乐观呀
It seems that the quality of the suitcase is not optimistic

买的时候可要仔细了
Be careful when you buy

大家好
Hello, everybody

欢迎来到轻松学统计课堂
Welcome to the Easy Learning Statistics Class

上一章我们介绍了总体样本
In the last chapter, we introduced the population sample,

简单随机样本
simple random sample,

统计量和抽样分布等概念
concepts like statistics and sampling distribution and

介绍了统计中常用的三大分布
introduced three distributions commonly used in statistics

给出了几个重要的
and several important

抽样分布定理
sampling distribution theorems

他们是进一步
They are the basis for further

学习统计推断的基础
learning the statistical inference

通常情况下来讲
In general

统计推断包括
statistical inference includes

参数估计和假设检验两个内容
parameter estimation and hypothesis testing

今天我们要介绍的是
What we're going to talk about today is

参数估计的有关内容
parameter estimation related contents

参数估计
Parameter estimation

在我们的日常生活里边
is very common

非常的常用
in our daily life

比如咱们在炒菜的时候
For example, when cooking

拿筷子夹一点点菜尝一下
we try a little dish with your chopsticks

就可以了解整锅菜的口感如何
to know the taste of the whole pot dish

又或者我们在买食品的时候
When we buy foods

比如买松子 开心果或者葡萄
like pine nuts, pistachios or grapes

番茄等等食品的时候
tomatoes and so on,

我们经常可能会
We often might

拿起一个尝一下
pick one up and taste it

了解一下整批产品的质量如何
to know the quality of the whole batch of product

或者咱们在体检的时候
Or when we get a checkup

通过抽少量的血
a small amount of blood is drawn

来了解身体的相关指标
to understand the body's relevant indicators

这些都是参数估计
These are parameter estimation

在日常生活里边的应用
applications in daily life

在这一章里面
In this chapter,

我为大家安排了以下几个内容
I've arranged the following for you

首先来了解参数估计的
Let's start with general issues

一般问题
In parameter estimation

在参数估计的一般问题里边
In the general issues of parameter estimation

我们有估计量与估计值
we have estimators and estimates,

点估计以及估计量选择的
point estimation, estimator selection

优良标准等
good standard, etc.

三个内容要介绍给大家
There are three contents I'd like to introduce to everyone

我们一个一个来看
Let's look at them one by one

首先我们来看估计量与估计值
Firstly, let's look at estimators and estimates

在估计量与估计值
For the content of

这个内容里边
estimators and estimates,

首先我们来介绍
first we introduce

参数估计的概念给大家
the concept of parameter estimation

所谓参数估计
So called parameter estimation

指的就是利用样本的信息
means using the sample information

去估计总体的参数
to estimate the parameters of a population

或参数的函数
or a function of a parameter

比如咱们想了解
For example, we want to know

南昌七月份的平均的降水量
the average rainfall in Nanchang in July

是什么样的一个状况
What is the situation

又或者在视频中大妞的妈妈
or the mother of the girl in the video

想要了解箱包的合格率
wants to know the pass rate of suitcases

是如何等等
and so on

这些都可以是总体的参数
These all can be parameters of the population

那接下来我们提到的一个概念
Then one of the concepts we mentioned next

是估计量
is the estimator

估计量是用来估计总体参数的
Estimators are used to estimate population parameters

统计量的名称
A name of the statistic

这个概念我想
This concept

在上一章抽样
was introduced in the last chapter: sampling

以及抽样分布这一个内容里边
and sampling distribution

已经有给大家介绍到
already

这里我们不再详细地打开
We won't open it in detail here

通常情况下
In general

我们用θ
we use θ

表示总体的参数
to represent the population parameters

用（字符如上）表示估计量
use the above character to represent estimators

通常我们可能关注
The population parameters

that we might focus on include

总体的均值μ
population mean μ

总体的乘数P
population multiplier P

或者是总体的方差σ平方
or the population variance σ squared

这些都是我们可能关心的
These are all population parameters

总体的参数
that we might care about

那为了了解总体的这些参数
To understand these parameters of the population,

我们就需要根据样本的统计量
we need to estimate them

来对它们进行估计
based on the sample statistics

所以我们需要有一个估计量
So we need to have estimators

作为准备
as preparation

那有了估计量还不够
That's not enough to have estimators

我们还需要估计量对应的值
We also need the corresponding values of these estimators

通常我们把这个概念
Usually, we call this concept

称之为估计值
as estimates

它是根据一个具体的样本
They are the values of estimators calculated

计算出来的估计量的数值
from a specific sample

比如咱们刚才提到的
For example, we just mentioned

大妞的妈妈
the girl’s mother

她想了解箱包的合格率
She wants to know the pass rate of suitcases

那么根据国家质检总局
According to the samples of 118 batches

2015年抽查的118批次的样本
selected by AQSIQ in 2015

得到的箱包的合格率为64.4%
The pass rate of bags % suitcases was 64.4%

这个就是一个估计值
This is an estimate

它是根据某一次具体抽样
It was the specific value

所产生的样本
of the pass rate calculated based on

计算出来的合格率的
the samples generated

具体的数值了
in a specific sampling

那把估计量和估计值
Now, after introducing

介绍给大家以后
estimators and estimates,

接下来我们就来介绍
we are going to

第二个内容
the second content

点估计
point estimation

那么点估计
Then, point estimation,

它是什么样的一种方法呢
What kind of method is it

实际上面点估计
In fact,

它是用样本估计值
point estimation is a method

直接作为总体的未知参数
that uses sample estimates directly as the true values

真实值的一种方法
of the unknown parameters of the population

这种方法最初是
This method was initially proposed

由卡尔·皮尔逊
by Karl Pearson

在1894年的时候提出来的
in 1894

它目前在我们的生活中
Presently, in our life

仍然非常地常见
it is still very common

比如咱们刚才举到的例子
For example, in the example cited just now

在视频里边
In the video

国家质检总局2015年抽查的
the pass rate of travel suitcases sampled in 2015

旅行箱包的合格率为64.4%
by State Quality Inspection Administration was 64.4%

那这个就是利用样本的信息
This used the sample information

来作为总体合格率的
as a representative value

一个代表值
of the population pass rate

又或者我们在关注
When we focus

空气净化器的
on air purifier

这一产品的时候
Products,

在广告里边
in advertisements

我们经常可能听到
we can frequently hear

有这样的描述
such description

某某品牌的空气净化器
The air purifier of a certain brand

宣称它的PM2.5的去除率为99%
is claimed to have a PM2.5 removal rate of 99%

它的甲醛净化率为99.5%
and a formaldehyde purification rate of 99.5%

那这样的数字
Such figures

其实也是点估计的应用
are also the applications of point estimation

又或者我们有同学想要就业
Or when our students want to be employed

想要找工作
want to find jobs

那我们通常说的一句话
What we usually say is

我们想去北、上、广、深
we want to go to Beijing, Shanghai, Guangzhou and Shenzhen

为什么去这些地方呢
Why do we want to go to these places

那根据某招聘网站的调查数据显示
On a job site

调查数据显示
the investigation data shows,

在2016年春季求职期
during the spring 2016 job search period,

各城市职工的平均工资的
the investigation result of average wages of the urban employees

结果显示
shows

上海以平均工资8825元
the average salary in Shanghai was 8,825 yuan

位列全国的首位
ranking the first in the country

紧随其后的就是北京8717元
It was followed by 8,717 yuan in Beijing

深圳8141元 广州7178元
8141 yuan in Shenzhen and 7178 yuan in Guangzhou

那北、上、广、深这几个城市
Beijing, Shanghai, Guangzhou and Shenzhen

在平均工资的排行榜
In terms of the ranking of average salary,

上面来看的话
they really

确实也是排在前几位的
rank at the top places

当然南昌地区
How about Nangchang

我也找了一下6008（元） 平均工资
The average salary is 6008 yuan as I searched

排在全国的第17位
ranking 17th in the country

那这些数字对我们来讲
For us, these figures are

其实都是点估计
all point estimations in fact

我们都可以通过这些信息
We can, through such information

去估计总体的情况
to estimate the population situation

所以点估计的方法
So the point estimation method,

在我们的日常生活里边
in our daily life,

是非常地常见的
is very common

那接下来我们来介绍一下
Next, we‘ll introduce about

点估计的有关的内容
point estimation related contents

我们再来看一个例子
Let’s see another example

比如某地区新生婴儿的体重
Let’s say the weight of newborn babies in a particular area

我们假定是用变量X来表示它
Assume we represent it with variable X

它可以服从正态分布
It can follow a normal distribution

那么这个正态分布
This normal distribution

一般有两个重要的参数
generally have two important parameters

一个是μ 一个是平方
One is μ and the other is σ squared

当然现在我们暂时不知道
Of course, we don't know right now

μ和σ平方
μ and σ squared

那接下来我们就可以通过
Then, we can

随机抽查一些婴儿的体重
select the weight of some babies at random

来了解
to know

或者来估计μ和σ平方的值
or estimate the values of μ and σ squared

假如现在
Suppose now

我随机抽查了100个婴儿
I select 100 babies by random

得到了100个体重数据
and get 100 pieces of weight data

我把它们记录下来
I record them

比如8.5公斤 7公斤 6公斤
such as 8.5 kg, 7 kg, 6 kg

6.5公斤 5公斤 5.2公斤等等
6.5 kg, 5 kg, 5.2 kg, etc.

我把100个体重数据
I record all

全部给它记录下来
of these 100 pieces of weight data

那我有了这100个数据
So I have these 100 pieces of data

接下来
Then

我要如何去估计μ和σ呢
how do I estimate μ and σ

那一种常用的方法
Use that common method

就是点估计的方法
That is point estimation

为了估计μ 也就是
In order to estimate μ

总体的平均数
the population mean,

我们可以构造一个
we can construct a

样本的函数（字符如上）
sample function (character above)

当然这个（字符如上）
Of course this (character above)

它是X{\fs10}1{\r} X{\fs10}2{\r}到X{\fs10}n{\r}的一个函数
is a function of X{\fs10}1{\r} X{\fs10}2{\r} up to X{\fs10}n{\r}

X{\fs10}1{\r}到X{\fs10}n{\r}代表的就是
X{\fs10}1{\r} up to X{\fs10}n{\r} represents

一个简单随机样本了
a simple random sample

这个我想在前面的抽样
This was introduced to you

这一章里边已经介绍给大家
in the chapter of sampling

我们后面提到样本的时候
The samples when mentioned later

一般就是指简单随机样本了
generally, refer to simple random samples

每当有了样本
When we have a sample,

我们就把我们的实际的观测值
substitute our actual observations

代入到这个函数θ_hat里边
into this function θ_hat[]

那么接下来
Then,

你通过这个函数的形式
through this function

就可以计算出一个
we may calculate

θ_hat的值
the value of θ_hat[]

那这个值我们就可以
We may take this value

把它当做μ的估计值
as the estimate of μ

通常情况下
Generally,

我们就可以把θ_hat
we may call θ_hat[]

称为参数μ的点估计量
as the point estimation estimator of parameter μ

把样本值代入
Substituting the sample values

（字符如上）函数里边
into (character as above) function

得到的值就称之为是
The value obtained is called

μ的一个点估计值了
a point estimation estimate of μ

所以我们如果已经采集到100个
Therefore, if we have collected 100 pieces of

新生婴儿的体重的数据的话
data of new born baby weights,

那么我们接下来要做的工作
what we will do is to

就是构造一个函数
construct a function

那这个函数
Then this function

可以有什么样的形式呢
What form can it take?

大家肯定要问
You are sure to ask

那通常情况下
Generally

我们可以用什么样的估计量
what kind of estimator can we use

去估计μ呢
to estimate μ

一般我们可以用样本的均值
In general, we can use the mean of the sample

也可以用样本的中位数
We can also use the median of the sample

甚至还可以用别的统计量
You can even use other statistics

也就意味着
That means

对于某一个总体参数来讲
for a certain population parameter,

我们的统计量
our statistics

可以有很多个不同的形式
can take many different forms

那接下来我们来了解一下
So let's take a look at

常用来构造估计量的方法
methods commonly used to construct estimators

简单跟大家提一下
Just give you an idea

这个在概率论和数理统计里边
In probability theory and mathematical statistics,

通常也会详细地介绍
they are generally introduced in detail

比如通常有矩估计法
For example, there are usually such methods as moment estimation,

极大似然估计法
maximum likelihood estimation,

最小二乘法 贝叶斯方法等等
least square and Bayesian method

这样的一些方法
These methods

都可以帮助我们来构造估计量
can help us construct the estimators

当然矩估计法和极大似然估计法
Of course, the methods of moment estimation and maximum likelihood estimation

是最常用的
are two mostly used

两种构造估计量的方法
methods for constructing estimators

那如何利用这些方法
How to use these methods

来构造估计量
to construct estimators

我们在这个里就不再打开
will not be further introduced here

感兴趣的同学
Those who are interested in that

还是可以通过
may through learning

概率论与数理统计的学习
probability theory and mathematical statistics

来了解它详细地构造
to see the detailed process of

估计量的过程
construction of estimators

那接下来
Then

大家可能就会有一个疑问
You might have a question

老师
Teacher,

有这么多种方法构造估计量
there are so many ways to construct estimators

那这些估计量它们会相同吗
will these estimators going to be the same

确实
Really

用不同的方法所构造的估计量
Estimators constructed by different methods

可能不完全相同
may not be the same

比如在正态总体的假定下
For example, under the assumption of normal population

矩估计法和极大似然估计法下面
When the moment estimation method and maximum likelihood estimation method are used

我们推算的μ和σ平方的
the estimators of μ and σ squared calculated

估计量均可以用下面的
can use the following

式子来表达
formula to express

（公式如上）
(The formula is as above)

样本的平均数了
The sample average

或者叫样本的均值
or the sample mean

那方差的估计量σ平方（字符如上）
The estimator of variance σ squared (character as above)

它是等于
is equal to

（公式如上）
(The formula is as above)

也就是样本的方差
or the sample variance

那用矩估计法和极大似然估计法
The estimators obtained by

所得到的估计量
using moment estimation method and maximum likelihood estimation method

它们在形式上来看是相同的
They are the same in form

但是如果
But if

我们换成贝叶斯估计法
we use Bayesian estimation

去推算的话
for calculation

那么它所得到的估计量
The obtained estimators, when compared

就和矩估计法以及极大似然估计法
with those obtained by moment estimation method and maximum likelihood estimation method

不完全相同
will not exactly the same

所以用不同的方法
Therefore, the estimators obtained

得到的估计量
by different methods

它是不完全相同的
are not exactly the same