Hypothesis tests and Confidence intervals

it may be impossible to discover the true value of $$\theta$$, hence some estimate of parameter $$\theta$$ is required and is denoted $$\tilde{\theta}$$

confidence intervals provide a way to quantify the uncertainty in $$\hat{\theta}$$

there are some issues in discussing random values and their realizations using the normal convention of X and x

estimates of $$\hat{\theta}$$ tend to $${\theta}$$ when the MLE is used.

Confidence intervals are used to summarize the uncertainty in $$\tilde{\theta}$$.

There is a common theory underlying hypothesis tests and confidence intervals.

Hypotheses
a hypothesis is a statement about the particular value that some value or paramater $$\theta$$ might take from parameter space $$\Omega $$

$$H:\theta = 0$$

scientific questions are typically formed in terms of hypotheses or statements about the world, and a hypothesis is a statement some particular value that a parameter might take.

so some $$\theta$$ in $$\Omega$$

$$\theta \in \omega$$

null and alternative hypothesis
The null hypothesis is denoted with the zero subscript like so $$H_{0}:\theta = 0$$

$$H_0:\theta \in \omega$$ then $$H_1:\theta \in \Omega - \omega$$

(is is also accurate to put) $$\notin \omega$$

The alternative hypothesis is often denoted with a 1 subscript

$$H_1:\theta \notin \omega$$

rejection region
ermm.. why? $$R \in \mathbb{R}^n$$

R = \{ x:t(x) > c \}

steps in testing
@todo


 * define a null hypothesis
 * collect data $$x_1, n_2, x_n$$
 * define a rejection region
 * usually compared to a test statistic
 * critical value c
 * critical value c

Composite Hypotheses
where there are multiple parameters

Fixed level testing
approach to hypothesis testing

Likelihood Ratio
The likelihood ratio provides a measure with which to evaluate the relative strength of evidence for two simple hypotheses.

If $$X \sim f(x|\theta) $$ and $$L(\theta)=L(\theta|x_1,x_2,...,x_n)$$

$$ LR = \frac{ L(\theta_1)}{L(\theta_0) }$$

$$ log( LR ) = log ( \frac{ L(\theta_1)}{L(\theta_0) } ) = l(\theta_1)-l(\theta_0)$$

Neyman-Pearson Lemma
is that when performing a hypothesis test between two simple hypotheses

$$H_0:\theta=\theta_0$$ and $$H_1:\theta=\theta_1$$

the likelihood ratio test is the most powerful test of size $$\alpha$$ for any value of $$\alpha$$

generalized likelihood ratio test
Is the likelihood ratio test applied to composite hypotheses

as composite hypotheses do not determine exact values of $$\theta$$

Free parameters
the number of unspecified parameters under a given hypothesis is also called the number of free parameters

Other approximate statistics
the likelihood ratio test can always be used to construct a confidence region

various approximations to the likelihood ratio test, which are equivalent asymptotically as the sample size grows large

alternatives are sometimes easier to apply.

Wald tests
"The Wald tests are derived by expanding the log-likelihood at its null value $$\theta_0$$, i.e. $$l(\theta_0)$$, around its value at the MLE theta hat using Taylor series" M347

type 1
this approximates 2 log (LR)

$$W_1 = (\theta_0 - \hat{\theta})^2 E \{-l''(\theta)\}|_\hat{\theta}$$

type 2
$$W_2 = (\theta_0 - \hat{\theta})^2 E \{-l''(\theta)\}|_{\theta_0}$$

Score Tests
The $$l'(\theta)$$ is known as the score function.

by the defintion of the MLE, the score function takes the value of 0 at the MLE $$\hat{\theta}$$

$$S = \frac{\{l'(\theta_0)\}^2}{E\{-l''(\theta)\}|_{\theta_0}}$$

if the null hypothesis is true, then S approximates 2 log (LR) in large sames, and its asymptotic null distribution is X^2(1)

Confidence Intervals
The first derivative of the log-likelihood is known as the score function.

A confidence interval represents a plausible range of values for an estimate

also known as an interval estimator if the variance is large, then the interval will be wide