Final

Question 1

What are the general steps for finding a MOM estimator?

Answer 1

1. Equate the sample moments to the population moments
2. Solve the sytems of equations for pop. in terms of sample

Question 2

What is an MLE?

Answer 2

The paramater value at which the likelihood function L(Ø|x) is maximized

Question 3

What is the formula for a continuous likelihood function?

Answer 3

π_{i =1} f(x_i| ø₁ … ø_k)

Question 4

What is the formula for a discrete likelihood function?

Answer 4

π P( X = x_i | ø)

Question 5

What are the steps for finding an MLE?

Answer 5

1. ** Optional: take the log of the function
2. Take the first derivative & set equal to 0
3. Take the second derivative to check that the max/min is global

Question 6

What is convergence in distribution?

Answer 6

{X_n}^∞ converges to a r.v., X_∞ , if and only if

lim F_xn(t) = F_x∞(t)

Question 7

What is convergence in probability?

Answer 7

{X_n}^∞ converges in prob. to X_∞ if and only if

lim P( |X_n - X_∞| > ɛ) = 0

** Convergence in probability implies convergence in distribution

Question 8

What is the likelihood for a Bern (p) distribution?

Answer 8

L(p|x) = π p^x_i (1 - p)^1-x_i

Question 9

What is the general form for Bayesian estimators?

Answer 9

(Likelihood * prior) / ∫ (Likelihood * prior) dθ

Question 10

What is the general form posterior mean in Bayesian estimation?

Answer 10

E(θ|X) = ∫ θ f**_θ (θ|X) dθ

where, f_θ (θ|X) is the posterior density

Question 11

What is the general formula for bias?

Answer 11

Bias(W) = E(W) - θ

where, W = estimator

Question 12

What is the general formula for MSE?

Answer 12

MSE(W) = E [(W - θ)²]

= Var(W) + [E(W) - θ]²

= Variance + Bias²

Question 13

What is a hypothesis test?

Answer 13

A function mapping the values of the t-stat (T_n = T(x₁ … x_n)) into {0,1} where:

0 = accept the null

1 = reject the null

Question 14

What is the general form of the Likelihood Ratio Test (LRT)?

Answer 14

λ(X) = sup_{θ<span>0 </span>}L(θ|X) ÷ sup_θ L(θ|X)

or

restricted (to H₀ parameters) ÷ unrestricted

Question 15

What are the two types of errors we wish to avoid?

Answer 15

Type I error: Reject the null when it is true

Type II error: Accept the null when it is false

Question 16

What is a power function?

Answer 16

A function of θ that provides the Type I error probability [ß(θ)] for a null hypothesis

Question 17

What is coverage probability for an interval estimator?

Answer 17

The probability that the random interval includes θ

Question 18

What is the confidence coefficient of an interval estimator?

Answer 18

Minimum coverage probability where θ is still “covered” by the interval

Question 19

Do the probability statements associated with interval estimators refer to the CI or the parameter of interest?

Answer 19

The CI

Question 20

What is the most common way of finding an interval estimator?

Answer 20

Invert the t-stat & solve for the parameter of interest

Question 21

What is a pivotal quantity?

Answer 21

A random variable [Q(X, θ)] that has the same distribution over all values of θ