1. Basic Large Sample Theory
   1. convergence
      1. a.s.
      2. in probability
      3. in r-th mean
      4. in distribution
      5. uniformly integrable
      6. convergence implications
         1. a.s. -- p
         2. p -- r
            1. Vitali's thm
         3. p -- d
   2. inequality
      1. c_r inequality
      2. Holder inequality
      3. Cauchy-Schwarz inequality
      4. Liapunov inequality
      5. Minkowski's inequality
      6. Basic inequality
         1. even, increasing on [0, inf), positive
      7. Markov's inequality
      8. Chebychev's inequality
      9. Jensen's inequality
   3. CLT
      1. WLLN
      2. SLLN
      3. classical CLT
      4. Liapunov CLT
      5. Berry-Essen Theorem
         1. evaluate the speed of convergence
      6. Lindeberg-Feller CLT
      7. Cramer-Wold Device
      8. Mann-Wald(Continuous Mapping) Theorem
      9. Slutsky's Theorem
      10. The Delta Method
      11. bounded in probability
      12. Big Oh Pee and Little Oh Pee
      13. Polya-Cantelli lemma
   4. Skorokhod's Theorem
      1. def. of F-1
      2. properties of F-1
         1. continuous F
         2. general F
      3. inverse transformation
      4. skorokhod's theorem
      5. Helly Bray theorem
         1. g is bounded and continuous
   5. lemmas
      1. Fatou's lemma
      2. MCT
      3. DCT
   6. Empirical Measures and Empirical Processes
      1. uniform case
      2. general case
      3. Glivenko-Cantelli theorem
         1. Gn and I are close
      4. Theorem 5.2
         1. for a special construction, Un and U are close
      5. Donsker's theorem
         1. g is ||.||inf - continuous
         2. g(Un) ->d g(U)
   7. Quantiles and Quantile Processes
      1. quantiles
         1. uniform case
         2. general case
      2. Gn-1 and I are close
      3. relationship between Vn and Un, V and U
      4. for a special construction, Vn and V are close
      5. Theorem 7.2
2. Lower Bounds for Estimation
   1. Cramer-Rao Bounds
      1. conditions
         1. M1: open set
         2. M2
            1. A:derivative of density
            2. B: support
         3. M3: information
         4. M4:differentiated under integral
         5. M5: twice differentiated under integral
      2. efficient influence function
   2. Geometry
      1. adaptive estimator
      2. efficient score for v
      3. efficient influence for v
      4. information for v
      5. information bound for v
      6. projections
         1. score
         2. influence
   3. Regular Estimates and Superefficiency
      1. def. of locally regular
      2. Hodge's estimator
   4. Hajek-Le Cam convolution and LAM
      1. regular + LAN(DQM) = convolution
      2. lower bounds
         1. regular estimator + bowl-shaped loss
         2. general estimator + LAN(DQM) + bowl-shaped loss
      3. asymptotically linear estimator is best regular
3. Efficient Likelihood Estimation and Tests
   1. Maximum likelihood
      1. K-L divergence
         1. motivates MLE
      2. definitions
         1. score function
         2. influence function
      3. regularity conditions
         1. A0: identifiability
         2. A1: support
         3. A2: density
         4. A3
            1. open neighborhood
            2. (i)
            3. (ii)
         5. A4
            1. (i)
            2. (ii)
            3. (iii)
      4. Theorem 1.2
         1. existence and consistency
         2. asymptotically linear
         3. likelihood ratio statistics
         4. Wald statistics
         5. score statistics
         6. LAN
         7. corollary
      5. one-step estimator
         1. Theorem 1.3
         2. start with moment estimator or quantile
   2. Three tests
      1. test statistics
         1. H0 is fully given
         2. H0 is partially given
      2. distribution
         1. H0 is fully given
            1. under H0
            2. under fixed alternative
            3. 1/n (.) ->0
            4. under local alternative
            5. non-central chi-square
         2. H0 is partially given
            1. under H0
            2. under fixed alternative
   3. strong consistency of MLE
      1. Wald's theorem
         1. compact
         2. upper semi-continuous
         3. integrable envelope
         4. measurability
         5. identifiability
   4. EM algorithm
4. LAN
   1. contiguity
      1. definition
      2. check contiguity
         1. Le Cam's first lemma
         2. example: e^N(\mu, \sigma^2)
   2. Le Cam's third lemma
      1. assumption
      2. conclusion
      3. abstract version
      4. normal version
   3. DQM
      1. def.
      2. DQM is a property of P_theta
      3. checking
         1. lemma3.1
            1. continuously differentiable for every x
            2. I_theta is well defined & continuous in theta
      4. common examples
         1. Poisson
         2. Cramer regular
         3. location family
         4. uniform is not DQM
   4. LAN
      1. def.
      2. LAN is a property of P_n,theta
      3. Theorem 3.1
         1. in i.i.d. case, DQM implies LAN
         2. open set assumption
      4. limiting distribution under shrinking alternative
   5. asymptotic normality of local likelihood
      1. matching
         1. MLE usually matches with MLE
      2. asymptotic normality of MLE
   6. Hajek-Lecam Lower Bound
      1. locally regular
      2. LAM
5. Nonparametrics
   1. KDE
   2. MSE
   3. Taylor expansion
   4. MISE
   5. AMISE
6. Dependent Data
   1. stationary sequence
      1. def. of stationary
      2. m-dependent
         1. m-dependent CLT
      3. ergodic
         1. Birkhoff's theorem LLN
      4. mixing
         1. alpha mixing
         2. mixing CLT
   2. Martingale Difference Sequence
      1. def. of Martingale and MDS
      2. CLT for MDS
7. Semiparametrics