1. Hypothesis Testing
   1. randomized test vs. nonrandomized test
   2. simple test
      1. Prop. 12.1
         1. Lagrange multiplier
      2. def. of LRT
      3. Neyman-Pearson lemma
      4. Prop. 12.3
      5. Corr. 12.4
   3. optimization problem
      1. generalized Neyman-Pearson lemma
   4. Optimal tests
      1. simple test
         1. LRT
      2. composite test
         1. UMP test
            1. method: Thm 12.9
            2. condition: Monotone likelihood ratio
            3. identifiability
            4. L(x) is monotone
            5. results
            6. UMP test
            7. power non-decreasing
            8. minimize
         2. UMPU test
            1. definitions
            2. unbiased test
            3. SOB test
            4. UMPU test
            5. UMP SOB test
            6. nuisance parameter
            7. Neyman structure
            8. bounded complete
            9. lemmas
            10. 1. UMP test is unbiased
            11. 2. unbiased + continuous power = SOB
            12. 3. continuous power + level alpha + UMP SOB = UMPU
            13. cases
            14. one parameter
            15. idea: use UMP SOB to find UMPU
            16. four problems in exponential family
            17. UMP
            18. problem I
            19. problem II
            20. UMPU
            21. problem III
            22. problem IV
            23. high dimensional
            24. idea: UMP conditional test => UMP SOB test => UMPU test
            25. UMP conditional test
            26. use conditioning
            27. need Neyman structure
            28. reduce to univariate case
            29. UMP SOB test
            30. UMPU test
            31. lemmas & properties
            32. unbiased + continuous power = SOB
            33. SOB + bounded complete = Neyman structure
            34. exponential family
            35. dist. of T
            36. dist. of U given T
            37. power function is continuous
            38. four problems in exponential family
            39. reparameterization
            40. directly compute conditional distribution
            41. use TSH Thm 5.1.1
            42. construct V
            43. use Basu's thm
            44. problem I
            45. V=h(u,t) is increasing
            46. problem IV
            47. V=h(u,t) is linear in u and increasing
2. Permutation Test
   1. problem settings
      1. null: Xi, Yj are from the same distribution
      2. alternative: joint distribution is h
   2. test
      1. power
         1. H0: phi(z) has uniform distribution
         2. H1: depends on h
      2. optimal test
         1. use NP-lemma
         2. h(z) > c(z)
            1. is UMP test
         3. choose c(z) s.t. alpha*(m+n)! permutations are rejected
3. Rank Test
   1. problem settings
      1. H0: F=G
      2. H1: F <s G
      3. focus on the rank of Y
   2. distributions of rank S
      1. H0: uniform
      2. H1: Hoeffding's formula
         1. two forms
   3. test
      1. general form: \sum h(Sj) > c
         1. Wilcoxon: h(j) = j
         2. normal-score test: h(j) = E(W(j))
      2. locally most powerful test
         1. g = f(x - delta)
         2. NP-lemma
         3. Wilcoxon case
         4. normal-score case
4. General Linear Models
   1. problem settings
   2. transformations
      1. transform parameter yita
      2. transform coordinate system
         1. find the O
   3. find the complete ss
   4. find the UMVUE for a'\xi
      1. the properties of P
   5. distribution of \xi
   6. distribution of \beta
   7. UMVUE for \sigma^2
   8. Gauss-Markov theorem
5. Point estimation
   1. Sufficient Statistics
      1. sufficient statistics
         1. def.
         2. checking
      2. minimal sufficient statistics
         1. def.
         2. checking
         3. exponential family
      3. complete
         1. def.
         2. sufficient + complete => minimal
         3. full rank exponential
      4. ancillary
         1. def.
         2. Basu's Theorem
      5. Rao-Blackwell Theorem
   2. Unbiased Estimation
      1. U-estimable
      2. methods to find unbiased estimator
      3. def of UMVU
      4. theorem: T is complete and sufficient, g is U-estimable, then ...
      5. MSE
      6. distributions
         1. normal
         2. gamma
         3. chi-squared
      7. variance bound
         1. Cov^2 <= VarX * VarY
         2. Hammersley-Chapman-Robbins inequality
         3. Cramer-Rao lower bound
         4. Fisher information
            1. definition
            2. reparametrization
            3. exponential family: lower bound achieved by T
            4. invariance in location family
            5. independent X, Y, (X, Y)
         5. High dimensional case
      8. Steps to calculate UMVU
   3. Bayesian Estimation
      1. def. of Bayes estimator
      2. Thm 7.1: Bayes estimator minimize E[L(\theta, \delta) | X = x] a.e. x
      3. commonly used formulas to get Bayes estimator
   4. Empirical Bayes
      1. problem settings
      2. James-Stein lemma
   5. Minimax Estimation
      1. def. of minimax estimator
      2. def. of Bayes risk
      3. def. of least favorable prior
      4. one theorem and two corollaries on minimax
      5. least favorable sequence of prior
      6. Theorem to find minimax estimator
   6. equivariant Estimation
      1. def. of equivariant estimator
      2. def. of invariant loss
      3. def. of invariant function
      4. location family
      5. maximal invariant
      6. minimum risk equivariant estimator
      7. Steps for equivariant estimator
      8. Pitman estimator
6. Decision Theory
7. Basics
   1. Generating Functions
      1. MGF: def, uniqueness
      2. moment; cumulant
         1. condition
      3. EXY = EX * EY
         1. independent sum
            1. moment
            2. cumulant
            3. mean
            4. variance
      4. exponential family
   2. Exponential Family
      1. definition
      2. full rank
         1. def.
         2. T is complete
      3. fisher information
      4. continuous power
   3. order statistics
      1. joint distribution
      2. U(i) - U(j)
   4. Distributions
      1. Normal
         1. Multivariate Normal
            1. moment generating function
            2. equivalent definition
            3. generate multi-normal vectors
            4. density
            5. conditional distribution
         2. bivariate normal
            1. density
         3. univariate normal
            1. MGF
            2. characteristic function
            3. (Xbar, S^2) are complete and sufficient
      2. Beta
         1. density
         2. expectation
      3. Gamma
         1. density
         2. MGF
         3. moment
         4. additivity
         5. Chi-squared
            1. MGF
            2. distribution of V
         6. exponential
      4. Poisson
         1. additivity
   5. Taylor Series
      1. e^x
      2. sinx, cosx
      3. 1/(1 - x)