STAT 512 - Test 3 Review Sheet

I. Continuation of Confidence Interval Section
  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
VI. Asymptotic (Large-sample) Properties of Point Estimators
  A.  Consistency
     1. Formal Definition of Consistency
     2. Theorem relating consistency to unbiasedness and asymptotic variance
  B.  Convergence in Probability 
     1. Using the helpful 4-part theorem about convergence in probability
  C.  Examples of Consistent Estimators
     1. Law of Large Numbers (about the sample mean)
     2. Consistency of the sample variance
  D.  Slutsky's Theorem
     1. How does it validate many of our large-sample inferences?
     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.  Large-Sample properties of MLEs
     1. Large-sample distribution of any MLE (assuming regularity conditions)
     2. Formula for the large-sample variance of an MLE
     3. "Consistent, asymptotically normal, asymptotically efficient" property
     4. Large-sample CI formula based on any MLE
  F.  Delta Method
     1. Large-sample distribution of a FUNCTION of an MLE
     2. Formula for the large-sample variance of a FUNCTION of an MLE
     3. Large-sample CI formula for a FUNCTION of our parameter, based on an MLE