STAT 512 - Test 3 Review Sheet

I. Continuation of Confidence Interval Section
  C.  Large-Sample Confidence Intervals
     1. General Formula for CIs when unbiased estimator has approximately normal sampling distn
     2. Meaning of confidence level
     3. Examples of Large-sample CIs
       a. CI for mu
       b. CI for p
       c. CI for mu1 - mu2
       d. CI for p1 - p2
     4. Interpreting CIs in the context of the variable(s) in the problem
  D.  Sample Size Determination Formulas
  E.  Small-sample CIs for mu and for mu1 - mu2
     1. Role of the t-distribution
     2. One-sample and Two-sample situations with small sample sizes
     3. Robustness of t-based CI procedures
  F.  CIs for variances
     1. Chi-square-based CI for one variance
     2. F-based CI for ratio of two variances
II. Brief Introduction to Hypothesis Testing
  A.  Null Hypothesis and Alternative (research) Hypothesis
  B.  Type I and Type II errors in Hypothesis Testing 
     1. Rejection Region
     2. Significance Level alpha
  C.  Relationship between Hypothesis Testing and CIs
     1. Connection Between Two-sided Tests and Two-Sided CIs
     2. Connection Between One-sided Tests and Lower (or Upper) Confidence Bounds
III. Properties of Point Estimators
  A.  Unbiasedness and Small Variance
  B.  Relative efficiency
     1. Why is it used?
     2. Calculating relative efficiency of one estimator compared to another
  C.  Cramer-Rao Lower Bound
     1. Calculating CRLB
     2. Definition of Efficient estimator
  D.  Sufficiency
     1. Definition of sufficient statistic
     2. What sufficiency intuitively means
     3. Likelihood Function
     4. Factorization theorem and how it is used
     5. One-to-one function of a sufficient statistic
  E.  Minimum Variance Unbiased Estimation
     1. Rao-Blackwell Theorem
     2. Minimal sufficient statistic
     3. Complete sufficient statistic
     4. Lehman-Scheffe Theorem and its usefulness
     5. One-parameter Exponential Family
     6. Finding a complete sufficient statistic when pdf in exponential family
     7. Finding MVUE for a parameter, based on the complete sufficient statistic
     8. Finding MVUE for a function of a parameter, based on the complete sufficient statistic
IV.  Method of Moments
  A.  Population Moments and Sample Moments
  B.  Setting up equation(s) to estimate parameter(s)
  C.  Solving equation(s) for parameter(s)
  D.  A typical weakness of MMEs
V. Method of Maximum Likelihood
  A.  Writing out the likelihood function based on a sample of data Y1,...,Yn
  B.  Maximization of the (log) likelihood by taking its derivative with respect to target parameter
  C.  Setting derivative to zero and solving for target parameter
  D.  Checking second-derivative condition to ensure maximum was found
  E.  Finding MLEs of two (or several) parameters simultaneously
  F.  Relationship of MLEs to sufficient statistics
  G.  Invariance Property of MLEs and how it is useful