Confidence intervals and Hypothesis testing Flashcards
What is an estimator serving as - what kind of estimate
Once data is observed, the estimators give us a point estimate of the
parameter of interest
What does the observed value of the estimator not tell us about
The uncertainty regarding the result
Name a method for constructing confidence intervals
Pivot method
What are the properties of a pivot
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.
What is the goal of the pivot method
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
How to use the pivot method
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
What is the problem with pivots
They are not always available
Give another result to calculate confidence intervals
Let theta hat be the MLE estimator for θ. Then √E [I (θ)] x (θ hat − θ)
converges in
probability to a standard Gaussian
What do we replace in the result that converges to a standard normal distirbution in probability
Replace Theta in the fisher information variance as theta is unknown as cannot calculate this value on an unknown value. Replace with theta hat
What are two consequences of the result in theorem that converges in probability to a standard normal distribution
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.
What is a consistent estimator
An estimator is consistent if, as the sample size increases, the estimates (produced by the estimator) “converge” to the true value of the parameter
How to use theorem that converges in probability to a standard normal distribution step by step
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
What can the interpretation of confidence interval be
The random confidence interval has approximately 95% chance of containing the true value of the parameter of interest
What question does hypothesis tests seek to answer
Is the relationship observed in the sample clear enough to be called statistically
significant, or could it have been due to chance?
What are the four steps to hypothesis testing
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
Define the null hypothesis
Null hypothesis is an unsurprising baseline(denoted H0). This usually says that
nothing is happening, i.e. observed relationship is due to chance
Define the alternative hypothesis
The alternative hypothesis (denoted HA) is the research hypothesis:
observed relationship is a symptom of an incorrect null hypothesis
If we fail to reject H0 what does this not mean
Does not mean H0 is necessarily true
Define the test statistic and how we use it
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.
What is involved in creation of a decision process
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
How do we make a decision in a hypothesis test
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
What is a simple vs simple hypothesis
H0 : θ = θ0 vs
HA : θ = θ1
What is a simple vs one tailed composite hypothesis
H0 : θ = θ0 vs
HA : θ > θ0
What is a simple vs two tailed composite hypothesis
H0 : θ = θ0 vs
HA : θ not equal to θ0
What is a compositive vs composite hypothesis
H0 : θ < θ0 vs
HA : θ ≥ θ0
What is a type 1 error and the probability of is happening
A type I error is made if H0 is rejected when H0 is true.
Has probability α
What is a type 2 error and the probability of it happening
A type II error is made if we fail to reject H0 when HA is true
Has probability β
Define the significance level and power of a test
The value α is called significance level of the test.
The value 1 − β is called power of the test
What is the goal for the relationship between power and significance level
The goal is to find an optimal test that maximises 1 − β for a given α.
What result provides a way to determine the rejection region for simple
hypotheses
Neyman Pearson Lemma
What method do we employ to find the rejection region for simple hypothesis
Neyman Pearson Lemma - using fixed alpha and pivot we can determine K
What is the Neyman Pearson Lemma value examined
Examines a ratio between the likelihood under null hypothesis and alternative hypothesis
How do we used the Neyman Pearson Lemma method
- find condition of what value the data under h0 has to be above/ below to be rejected
- Then put alpha= P(Rejecting H0|H0 is true)
Or alpha= P(Condition in K|H0) - Pivot this probability
Define the p value
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
How do we use the p value to make a decision. How does it relate to the Neyman pearson lemma
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
If H0 or HA are composite meaning they don’t specify the distribution of an RV when does test have significance level alpha?
If its size is less than or equal to α (these
two definitions coincide when we have a simple null hypothesis)
What will the error probabilities be a function of for composite hypothesis
the hypothesised values of the parameter of interest
Why do we use the max of alpha value when h0 or HA are composite
Worst case scenario we are begin cautious.
What is UMP
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
What test do we use instead for hypothesis that are composite to define the rejection regions
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
How can the rejection region be found for composite hypothesis after the likelihood ratio test
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.
What is the purpose of the wald test
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
