Lecture 3

Notation

Likelihood

Parameter

Nuisance parameter

Probability being a function of the parameter stochastic variable

observed value

Density function

MLE Maximum likelihood estimate

Standard error

Score function, ie first derivative of Fisher Information, minus the second derivative of

Definition of the likelihood Pawitan (2001) page 22:

Assuming a statistical model parameterized by a fixed and unknown , the likelihood is the probability of the observed data considered as a function of .

Notation (cont.)Suppose we have collected n observations: …

The ordered values from smallest to largest are then given by: …

Hence, = maximum value and = minimum value.

Possible to combine different sources of information in a common likelihood:

Example 2.7 (page 28)Two independent samples from Sample 1: The maximum of 5 observations, , is reported.Sample 2: The average of 3 observations,

The likelihood for Sample 2 easiest to construct.We have So, )

Possible to combine different sources of information in a common likelihood:

Example 2.7 (cont.)

The likelihood for Sample 1 is a little bit more tricky (see Example 2.4 for more details).Let be the cumulative distribution function for a standard normal distribution, and the probability density function.

We have The probability density function is the derivative of this function:So,

Possible to combine different sources of information in a common likelihood:

Example 2.7 (cont.)

We have )

The two log-likelihoods can now simply be added:

The MLE, is computed by maximizing .But we also get the uncertainty in this estimated parameter.(See Figure 2.4 in Pawitan 2001).