M ATH S TAT T RIVIAL P URSUIT (S ORT OF )F OR R EVIEW ( MATH 30)
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Transcript of M ATH S TAT T RIVIAL P URSUIT (S ORT OF )F OR R EVIEW ( MATH 30)
COLORS AND CATEGORIES
Blue – Basics of Estimation Pink – Properties of Estimators and Methods
for Estimation Yellow – Hypothesis Testing Brown – Bayesian Methods Green – Regression Orange – Nonparametric Procedures and
Categorical Data Analysis
BLUE 1
Suppose you have an estimator theta-hat, and you want to know its bias. How is bias computed?
BLUE 3
What is a common unbiased point estimator for a population mean and what is its standard error?
BLUE 4
What is a common unbiased point estimate of a difference in two population proportions, and what is its standard error?
BLUE 5
A very important result related to samples from a normal distribution is that: The sample mean is ____________ distributed. The sample variance, appropriately scaled, is
____________ distributed. The sample mean and sample variance are
____________________.
(Fill-in all three blanks for credit).
BLUE 7
How would you use the asymptotic normal distribution of many unbiased point estimators to create a confidence interval for their respective parameters?
(You can just give the formula).
Hint: Think of a specific case and generalize.
PINK 1
If relative efficiency is computed between two estimators, it means that both estimators were _______________, and if the numerical value of the relative efficiency is 2, then it means that the _____________ (first or second) estimator is better.
PINK 2
What is the definition of consistency for an estimator?
Bonus: What concept of convergence is this equivalent to?
PINK 3
For an unbiased estimator, what is the “fast” way of showing consistency?
Bonus: Do you remember what convergence result this was derived from?
PINK 4
If you have a RS of n observations from a distribution with unknown parameter theta, and T is sufficient for theta, what does that mean?
PINK 5
What is the result you can use to show sufficiency without resorting to computing conditional pdfs?
PINK 6
What does the Rao-Blackwell Theorem say?
Bonus: What’s the fast way of finding the quantity RB refers to in the end?
PINK 10
If an estimator is NOT admissible (i.e. inadmissible), what does that mean?
Give an example of an inadmissible estimator.
YELLOW 2
Describe the relationships between the two types of error in a hypothesis test, as well as their connection to power.
YELLOW 3
If you have a test statistic, you can use either a rejection region approach or a p-value approach to determine if the null hypothesis should be rejected. What is the difference in the 2 approaches? (Describe).
YELLOW 4
For the common large sample asymptotically normal z-tests, what is the rejection region for a 2-tailed test?
Bonus: If the significance level is .05 for this test, what is the range of test statistics where you would NOT reject the null hypothesis (numerical values).
YELLOW 6
What is the difference between the pooled and unpooled t-tests for 2 independent samples when considering tests for means?
YELLOW 7
In order to determine which 2-sample t-test for small sample sizes is appropriate, you might have to run a test to check for equality of _______________, and in order to control your overall significance level, you might have to use a ____________ _____________.
YELLOW 8
What does the Neyman-Pearson Lemma say?
(Get the gist of it, what does it let you find, and how?)
YELLOW 10
How do you construct a likelihood ratio test?
What is the asymptotic distribution related to LRTs?
BROWN 1
What is the major difference between Frequentist and Bayesian approaches to statistics in terms of how the parameter theta is treated?
BROWN 2
What is the difference between a proper and improper prior?
What is the difference between an informative and uninformative prior?
BROWN 5
How would you find the Bayes estimate of: theta theta(1-theta)
if you had the posterior density of theta?
BROWN 6
A Bayes estimator is ALWAYS a function of a _______________ statistic because of the _______________ ________________.
BROWN 8
Is it possible for Bayesian and Frequentists intervals to agree? If yes, how might this happen?
BROWN 9
Bayesian hypothesis testing is performed using ______ ________, which are Bayesian analogues of ________ test procedures, and which can allow you to find evidence in favor of your ___________ hypothesis.
GREEN 1 Relationships between two variables, X and Y
can be deterministic or ________________. Regression is used when the relationship is _______________. This means that ….
GREEN 3
If your regression model was:
Then how many parameters do you need to estimate?
3322110)( xxxYE
GREEN 4
In least squares solutions for regression, what quantity is minimized to find the solution?
(You can just give the simple LR quantity).
GREEN 5
The least squares estimates are all ____________, and their variances are functions of _____________, which in turn can be estimated by _______, which is equal to (1/(n-2))SSE.
GREEN 6
What is the full set of conditions on the error terms in order to get normal sampling distributions for the parameter estimates if sigma is known?
GREEN 7
Why do we end up using a t distribution for inference about slope parameters in regression instead of a normal distribution?
GREEN 8
What is the main difference between a confidence interval for a mean response and a prediction interval for an individual response in regression?
GREEN 9
How are CIs for mean responses and prediction intervals for individual responses affected as the chosen x moves further from the mean of the x’s?
ORANGE 5
How does a Kolmogorov-Smirnov one-sample test work? Is the null hypothesis in the procedure simple or composite?
ORANGE 7
When performing categorical data analysis, the main distribution you need to understand for the theoretical setup of problems is the ______________ distribution, but the test statistics turn out to have a different distribution, which is the ________________ distribution.
ORANGE 10
For 2x2 tables, inference is also possible with: _________ exact test for small sample sizes _________ ratios, which relies on an asymptotic
______ distribution for it’s natural log.
Takehome Final Exam is due Friday, May 13th at 5 p.m. SHARP.
Office Hours (see front cover of exam): Monday 9-12 during my other course’s exam Tuesday 10-12 Wednesday 1-3 Thursday 1-3 Any other time by appt. – just send me an email!
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