Econometrics：Random Sampling 笔记
Econometrics
0. Random Sampling
几组对比
对比 1：population vs sample
对比 2:random sampling or random assignment
random sampling: 随机抽样，从总体中选取参与者进入研究的方法
random assignment: 将已入选的参与者分配到不同的实验组别
对比 3:parameter or sample statistics
population quantity, parameter: fixed feature of the population, doesn’t reference sample size, e.g. \(E(Y_{i})\)
sample statistics: differing from sample to sample, e.g. sample mean
对比 4:average or mean
population average, mean, mathematical expectation, expectation
\(E(Y_{i})=\int tf_{Y}(t)dt\) sample statistics, estimator: a function of sample data used to estimate parameters
\(\bar{Y}=\frac{1}{N}\sum _{i=1}^{N}Y\)
对比 5:LLN vs unbiasedness
they are two properties of the sample mean (sample statistics)
LLN (大数定律):
\(N \rightarrow \infty,\bar{Y}\rightarrow^{P}E(Y_{i})\)
unbiasedness of the sample mean (given the sample size):
\(E(\bar{Y})=E(Y_{i})\)
对比 6:variability
population variance
\(V(Y_{i})=E[(Y_{i}-E[Y_{i}^{2}])]\equiv \sigma^{2}_{Y}\) sample variance
\(S(Y_{i}^{2})=\frac{1}{n}\sum^{n}_{i=1}(Y_{i}-\bar{Y})^{2}\)
\(S(Y_{i}^{2})=\frac{1}{n-1}\sum^{n}_{i=1}(Y_{i}-\bar{Y})^{2}\) (无偏)
对比 7:standard deviations vs standard errors
standard deviation: the square root of the population variance
- 标准差 (SD)：衡量的是数据本身的变异程度（无论是总体数据还是样本数据）。它回答的是“个体 typically 偏离平均值多远”
- the sample counterpart, $S(Y_{i})$, which is the squared root of $S(Y_{i})^{2}$
- variance or SD is a descriptive fact about the distribution of $Y_{i}$
- also known as descriptive variance
standard error: the property of the sample statistics, the standard deviation of the sample statistics
- 标准误 (SE)：衡量的是样本统计量（如样本均值 $\bar{Y}$）的变异程度，即抽样分布的离散程度。它回答的是“样本均值（作为总体均值的估计） typically 偏离总体真值多远”，它衡量的是估计的精确度
样本均值的标准误 (SE of the mean) 可以通过样本标准差来估计：$SE=\frac{s}{\sqrt{ n }}$。这意味着，样本量 n 越大，样本均值的波动就越小，我们对总体均值的估计就越精确。
| Standard Deviation | Standard Error | |
|---|---|---|
| Measures variability | Within a sample | Between samples |
| Defines width of a | Distribution of individual values | Sampling distribution |
| Assesses distances between | Data values and sample mean | Sample statistics and population parameter (i.e., accuracy) |
| As sample size increases, there is | No tendency to change | A tendency to decrease. |
SD: 求和均值的距离 SE：求样本均值之间的变异程度
Sampling variance
the population variance of a sample statistics, such as the sample mean
\[V[\bar{Y}]=E[(\bar{Y}-E[\bar{Y}])^{2}]\]- Sampling variance (vs. descriptive variance): The quantity measures the variability of a sample statistic in repeated samples, as opposed to the dispersion of raw data.
- Sampling variance is related to the descriptive variance, but it is also determined by sample size (when n approaches infinity, the sample variance disappears)
Standard Error
The standard error: the standard deviation of a statistic like the sample average; the square root of sampling variance
\[SE(\bar{Y})=\frac{\sigma}{\sqrt{ n }}\]- The standard error summarizes the variability in an estimate due to random sampling
- The estimated standard error: quantifying the standard error
The t-Statistic and Central Limit Theorem
- the t-statistic
- 中心极限定理
- If the null hypothesis is true, then, as long as the sample is large enough, the sampling distribution of the t-statistic (regardless of whether Yi itself is normally distributed) is very close to the standard normal distribution.
- Or, the large sample distribution of a t-statistic is independent of the distribution of the underlying data used to calculate it 1 2