Confidence intervals and Hypothesis testing

Question 1

What is an estimator serving as - what kind of estimate

Answer 1

Once data is observed, the estimators give us a point estimate of the
parameter of interest

Question 2

What does the observed value of the estimator not tell us about

Answer 2

The uncertainty regarding the result

Question 3

Name a method for constructing confidence intervals

Answer 3

Pivot method

Question 4

What are the properties of a pivot

Answer 4

A pivot is a function of the random sample {Xi }i and the unknown
parameter, θ.
The distribution of the pivot does not depend on the parameter of
interest.

Question 5

What is the goal of the pivot method

Answer 5

Use the sampling distribution of a pivot to determine the

bounds of the interval - may have to transform to change the distribution to one we can calculate

Question 6

How to use the pivot method

Answer 6

Using true statement P(quantile 1< statistic < quantile 2 ) = 1- alpha
Manipulate the equation to get the parameter of interest in the middle of the interval

Question 7

What is the problem with pivots

Answer 7

They are not always available

Question 8

Give another result to calculate confidence intervals

Answer 8

Let theta hat be the MLE estimator for θ. Then √E [I (θ)] x (θ hat − θ)
converges in
probability to a standard Gaussian

Question 9

What do we replace in the result that converges to a standard normal distirbution in probability

Answer 9

Replace Theta in the fisher information variance as theta is unknown as cannot calculate this value on an unknown value. Replace with theta hat

Question 10

What are two consequences of the result in theorem that converges in probability to a standard normal distribution

Answer 10

The MLE is consistent for the parameter of interest.
We can use this theorem to construct an approximate confidence interval
for any parameter of interest, provided that the MLE is available and that
n is large enough.

Question 11

What is a consistent estimator

Answer 11

An estimator is consistent if, as the sample size increases, the estimates (produced by the estimator) “converge” to the true value of the parameter

Question 12

How to use theorem that converges in probability to a standard normal distribution step by step

Answer 12

Find pdf/pmf
Find likelihood
Find log likelihood and its derivatives
If necessary find the MLE to construct the confidence interval around

Next find the observed fisher information
Now find the expectation of the fisher information with respect to the sample (so this is a function In X)
Now find expectation of the estimator (sub in theta hat value in stead of theta and then sub in the MLE)
Use pivot method by stating a true probability from the convergence to the standard normal model

Question 13

What can the interpretation of confidence interval be

Answer 13

The random confidence interval has approximately 95% chance of containing the true value of the parameter of interest

Question 14

What question does hypothesis tests seek to answer

Answer 14

Is the relationship observed in the sample clear enough to be called statistically
significant, or could it have been due to chance?

Question 15

What are the four steps to hypothesis testing

Answer 15

Determine null and alternative hypothesis
Collect data and summarise with test statistic
Determine how unlikely the test statistic would be if the null hypothesis were true
Make a decision

Question 16

Define the null hypothesis

Answer 16

Null hypothesis is an unsurprising baseline(denoted H0). This usually says that
nothing is happening, i.e. observed relationship is due to chance

Question 17

Define the alternative hypothesis

Answer 17

The alternative hypothesis (denoted HA) is the research hypothesis:
observed relationship is a symptom of an incorrect null hypothesis

Question 18

If we fail to reject H0 what does this not mean

Answer 18

Does not mean H0 is necessarily true

Question 19

Define the test statistic and how we use it

Answer 19

The observed data is summarised in a relevant estimator or statistic called test statistic.
We use this to determine whether we reject the null hypothesis based on
the observed value of the test statistic.

Question 20

What is involved in creation of a decision process

Answer 20

We partition the sample space into two disjoint regions. A Fail to reject region and the subset of all points that will make us reject is called the rejection region.
We pivot the test statistic to define the numeric dividing line between these subsets

Question 21

How do we make a decision in a hypothesis test

Answer 21

We check if the observed value of the test statistic is in the rejection
region.
1. If yes, then we reject the null hypothesis and accept the alternative.
2. If no, we fail to reject the null hypothesis

Question 22

What is a simple vs simple hypothesis

Answer 22

H0 : θ = θ0 vs

HA : θ = θ1

Question 23

What is a simple vs one tailed composite hypothesis

Answer 23

H0 : θ = θ0 vs

HA : θ > θ0

Question 24

What is a simple vs two tailed composite hypothesis

Answer 24

H0 : θ = θ0 vs

HA : θ not equal to θ0

Question 25

What is a compositive vs composite hypothesis

Answer 25

H0 : θ < θ0 vs

HA : θ ≥ θ0

Question 26

What is a type 1 error and the probability of is happening

Answer 26

A type I error is made if H0 is rejected when H0 is true.

Has probability α

Question 27

What is a type 2 error and the probability of it happening

Answer 27

A type II error is made if we fail to reject H0 when HA is true
Has probability β

Question 28

Define the significance level and power of a test

Answer 28

The value α is called significance level of the test.

The value 1 − β is called power of the test

Question 29

What is the goal for the relationship between power and significance level

Answer 29

The goal is to find an optimal test that maximises 1 − β for a given α.

Question 30

What result provides a way to determine the rejection region for simple
hypotheses

Answer 30

Neyman Pearson Lemma

Question 31

What method do we employ to find the rejection region for simple hypothesis

Answer 31

Neyman Pearson Lemma - using fixed alpha and pivot we can determine K

Question 32

What is the Neyman Pearson Lemma value examined

Answer 32

Examines a ratio between the likelihood under null hypothesis and alternative hypothesis

Question 33

How do we used the Neyman Pearson Lemma method

Answer 33

1. find condition of what value the data under h0 has to be above/ below to be rejected
2. Then put alpha= P(Rejecting H0|H0 is true)
   Or alpha= P(Condition in K|H0)
3. Pivot this probability

Question 34

Define the p value

Answer 34

The p-value of a test is the probability of observing a test statistic more
extreme than the one observed if the null hypothesis were true

Question 35

How do we use the p value to make a decision. How does it relate to the Neyman pearson lemma

Answer 35

We reject H0 if the p-value is smaller than a given threshold, or fail to
reject otherwise.
If we choose the threshold to be α, then this method leads to the same
conclusion as that of Neyman-Pearson lemma

Question 36

If H0 or HA are composite meaning they don’t specify the distribution of an RV when does test have significance level alpha?

Answer 36

If its size is less than or equal to α (these

two definitions coincide when we have a simple null hypothesis)

Question 37

What will the error probabilities be a function of for composite hypothesis

Answer 37

the hypothesised values of the parameter of interest

Question 38

Why do we use the max of alpha value when h0 or HA are composite

Answer 38

Worst case scenario we are begin cautious.

Question 39

What is UMP

Answer 39

A Uniformly Most Powerful (UMP) test is a test that maximises the
power for a given level of significance.
Note that, generally, UMP tests are available only for simple hypotheses,
with the Neyman-Pearson lemma

Question 40

What test do we use instead for hypothesis that are composite to define the rejection regions

Answer 40

Likelihood ratio test rejection region. Will have likelihood functions calculated in theta while theta varies int he set H0This test is not UMP, but it works generally well

Question 41

How can the rejection region be found for composite hypothesis after the likelihood ratio test

Answer 41

Once the form of the likelihood ratio test is found, κ can be calculated by
fixing the value of α and using an appropriate pivot.

Question 42

What is the purpose of the wald test

Answer 42

Extends possibilities to testing. We cna use knowledge of asymptotics to create a pivot and an approximate rejection region. The sampling distribution of the test statistic may not be known or we may not find a suitable pivot so we can use the result showing the square root of the expected value of the observed fisher information converges in probability to normal distirbution