chapter_7_statistical_estimate_and_sampling_distribution.ipynb

Kernel: Python 3

Chapter 7 Statistical Estimation and Sampling Distribution

7.1 Point Estimates

Parameters: used to discrabe a probability distribution, e.g. mean, variance, etc.

Statistics: a function of random sample of a distribution, i.e. randomly choose some samples from a distribution.

Estimate: guess the parameter by the statistics.

7.2 Properities of Point Estimate

Estimate bias: $bias(\hat{\theta}) = E(\hat{\theta})-\theta$, if $bias = 0$, we call this estimate unbias estimate.

Estimate variance: $Var(\hat{\theta})$ is the same with the definition of variance.

Mean squared error (MSE): $MSE(\hat{\theta})=E(\hat{\theta}-\theta)^2=Var(\hat{\theta}) + bias^2(\hat{\theta})$

7.3 Sampling Distribution

$X\sim B(n, p)\Longrightarrow \hat{p}\sim N(p, \frac{p(1-p)}{n})$

$X\sim N(\mu, \sigma)\Longrightarrow T = \frac{\sqrt{n}(\bar{X}-\mu)}{S}\sim t_{n-1}$

7.4 Constructing Parameter Estimate

Moment estimate method

For parameters $\hat{\theta}=[\theta_1, \theta_2, \dots, \theta_n]$, we create

$$\bar{x}=E(X)$$$$\bar{x}^2=E(X^2)$$$$\dots$$$$\bar{x}^n=E(X^n)$$

Solve these functions we can get the $\hat{\theta}$.

Maximum likehood estimate method

$MLE(\hat{\theta})=max_{\theta}L(\theta)=max_{\theta}f(x_1;\theta)\times f(x_2;\theta)\times \dots f(x_n;\theta)$

where $L(\theta)$ is the likehood function.