Interval Estimation

A $100\ast p$% likelihood Interval is $\{\theta :R(\theta )\ge p\}$

• where $R(\theta )$ is the Relative Likelihood Function

For instance, if we want to the 50% likelihood interval, we would find $\{\theta :R(\theta )\ge 0.5\}$. When we solve $\theta$, since this in an inequality, we will get a range $[a,b]$, which gives us our interval.

title: Plausibility
We use the [[Relative Likelihood Function]] to quanitfy plausibility.
- $R(\theta) \geq 0.5 \rightarrow$  very plausible
- $R(\theta) \geq 0.1 \rightarrow$ plausible
- $R(\theta) \lt 0.1 \rightarrow$ implausible
- $R(\theta) \lt 0.01 \rightarrow$  very implausible

Example

Example: Suppose a coin is tossed $200$ times with $\theta =P(H)$. Suppose that we observe $y=80$ successes. Find the 10% ($p=0.1$) likelihood interval for $\theta$.

We first find $\hat{\theta }$, which we know is $\frac{y}{n}$ since this is data follows a Binomial Distribution. $\hat{\theta }=\frac{80}{200}=0.4$ Find all $\theta$ s.t. $R(\theta )\ge 0.1$ $R(\theta )=\frac{L(\theta )}{L(\hat{\theta })}\ge 0.1$ $\frac{\left(\genfrac{}{}{0ex}{}{200}{80}\right){\theta }^{80}(1-\theta {)}^{120}}{\left(\genfrac{}{}{0ex}{}{200}{80}\right)0.4^{80}0.6^{120}}\ge 0.1$

By the magic of computers, we find the interval $\theta \in [0.33,0.47]$, which is at a minimum 10% as likely as $\hat{\theta }=0.4$.

Using this interval, we can then conclude for example that $\theta =0.36$ is plausible, since $R(0.36)\ge 0.1$.

Related

• Confidence Interval