4.5 Introduction to Hypothesis Testing

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Definition: Null and alternative hypothesis

Definition

Let $X$ : Random variable with pdf : $f(x;\theta ),\theta \in \Omega$

Suppose due to theory/preliminary experiment

• $w_0,w_1$ disjoint subsets of $\Omega$, (i.e., $w_0\cup w_1=\Omega$)
• $\theta \in {\omega }_0$ or $\theta \in {\omega }_1$

Then

• We label these hypotheses as $$\begin{array}{rl}{H}_0& :\theta \in {\omega }_0\\ H_1& :\theta \in {\omega }_1\end{array}$$
• We refer the hypothesis $H_0$ as the null hypothesis
• We refer the hypothesis $H_1$ as the alternative hypothesis

Definition: Test, critical region, decision rule

Definition

Let

• $X$ : Random variable wth pdf $f(x;\theta ),\theta \in \Omega$
• $X_1,\dots ,X_n$ : Random sample of $X$
• $\mathcal{D}=\mathrm{space}\{(X_1,\dots ,X_n)\}$ : Sample space

Suppose

• $H_0$ : Null hypothesis
• $H_1$ : Alternative hypothesis

Then

• A test of $H_0$ versus $H_1$ is based on a subset $C\subset \mathcal{D}$
• We call $C$ the critical region
• Decision rule (test) of $C$ is

$$\begin{array}{rl}\text{Reject }{H}_0(\text{Accept }{H}_1)& \text{if }(X_1,\dots ,X_n)\in C\\ \text{Retain }{H}_0(\text{Reject }{H}_1)& \text{if }(X_1,\dots ,X_n)\in C\end{array}$$

Definition: Test error types

Definition

Suppose

• $H_0$ : Null hypothesis
• $H_1$ : Alternative hypothesis

Then

• We say a Type I error occurs if $H_0$ is rejected when $H_0$ is true
• We say a Type II error occurs if $H_0$ is accepted when $H_1$ is true

Definition 4.5.1: Size of a critical region (or significane level)

Definition

if $ \alpha=\max_{\theta\in \omega_{0}}P_{\theta}[(X_{1},\dots,X_{n})\in C] $$ Then

• We say critical region $C$ is of size $\alpha$
• We often say $\alpha$ is the significance level of the test associated with $C$
• The size/significance level $\alpha$ is the probability of making a type I error

Definition: Power of a test

Definition

Let $\theta \in {\omega }_1$[^1]

Then

• $ 1-P_{\theta}[\text{Type II Error}] = \textcolor{yellow}{P_{\theta}[(X_{1},\dots,X_{n})\in C]} $$
• We say $P_{\theta }[(X_1,\dots ,X_n)\in C]$ the power of the test at $\theta$

Definition: Power function

Definition

We define power function of a critical region to be $ \gamma_{C}(\theta)=P_{\theta}[(X_{1},\dots,X_{n})\in C];\quad\theta\in \omega_{1} $$

Suppose $C_1,C_2$ : Critical regions, both of size $\alpha$

Then $C_1$ is better than $C_2$ if ${\gamma }_{C_1}(\theta )\ge {\gamma }_{C_2}(\theta ),\forall \theta \in {\omega }_1$

Remark: Types of hypothesis tests

Definition

A statistical hypothesis is a question a distribution from one or more random variables

A simple statistical hypothesis specifies completely about a distribution. e.g., $H_0:\theta =75$ and $H_1:\theta =77$

A composite statistical hypothesis specifies in-completely about a distribution. e.g., $H_0:\theta \le 75$ and $H_1:\theta >75$

Remark: Equivalent terms

The following terms are equivalent:

• Significance level of test ($\alpha$)
• Size of critical region ($\alpha$)
• Power of a test if $H_0$ is true
• Probability of type I error

Example

Let $X$ : Random variable with $X\sim N(\theta ,100)$

Let $n=25$, thus the sample space is

$$\mathcal{D}=\{(x_1,x_2,\dots ,x_{25}):-\infty <x_i<\infty ,i=1,2,\dots 25\}$$

Let the critical region $C\in \mathcal{D}$ be $C=\{(x_1,x_2,\dots ,x_{25}):x_1+x_2+\cdots +x_{25}>(25)(75)\}$

Notice that ${\sum }_{i=1}^{25}{x}_i>(25)(75)⟺\bar{x}>75$. Therefore, $C=\{(x_1,x_2,\dots ,x_{25}):\bar{x}>75\}$

• We reject $H_0:\theta \le 75$ and accept $H_1:\theta >75$ if and only if the sample mean is $\bar{x}>75$
• If $\bar{x}<75$, we accept $H_0:\theta \le 75$

Because $X\sim (\theta ,100)$, then $\bar{X}\sim (\theta ,4)$ and $\frac{\bar{X}-\theta }{2}\sim N(0,1)$. We can then obtain

$$\begin{array}{rl}{\gamma }_C(\theta )& =\mathrm{Pr}(\bar{X}>75)\\ & =\mathrm{Pr}\left(\frac{\bar{X}-\theta }{2}>\frac{75-\theta }{2}\right)\\ & =1-\mathrm{Pr}\left(\frac{\bar{X}-\theta }{2}\le \frac{75-\theta }{2}\right)\\ & =1-\Phi \left(\frac{75-\theta }{2}\right)\text{(Because }\frac{\bar{X}-\theta }{2}\sim N(\theta ,1))\end{array}$$

where $\Phi (x)$ is a distribution function from Normal distribution