Mathematical Statistics - 3

Question 1

Define the observed likelihood function and the log likelihood

Answer 1

l(theta) = log L(theta)

Question 2

Define the maximum likelihood estimate

Answer 2

The maximum likelihood estimate (theta)^hat is the value of theta which maximises L, assuming it exists and is unique

Question 3

Define the likelihood ratio

Answer 3

Question 4

Define the generalised likelihood ratio

Answer 4

Question 5

Define

i) Bias
ii) Mean Square Error
iii) Standard Error

Answer 5

Question 6

Answer 6

Question 7

When is a sequence of estimators T₁,……, consistent for theta

Answer 7

If T_n converges to theta in probability

Question 8

Define the normal percentile

Answer 8

Question 9

Define a confidence interval

Answer 9

Question 10

Define a pivot

Answer 10

A pivot is a random variable whose distribution doesn’t depend on the statistical parameters

Question 11

Define the students t(n) distribution

Answer 11

Question 12

Define the p-value of a test statistic, when should we reject the null hypothesis?

Answer 12

When p-values are small

Question 13

Define a power function and the conclusions we draw from it

Answer 13