Statistics, based loosely on DeGroot & Schervish
----------------------------------------
Week 3: Estimation (cont'd)

6.5  Maximum Likelihood Estimators:

     - Simple EGs: Bi(n, th); Po(th); Ex(th); Ge(th).

     - Moderate EGs: Be(alp, bet); Ga(alp, bet); NB(alp, beta).

     - Important EG: No(mu, sig).

     - Cautionary EG: Uniform on [0,\th]

     - Discouraging EG: Un[\th-1, \th+1]

     - Alarming EG: 1/2 No(0,1) + 1/2 No(mu, sig^2), mu & sig unknown.

6.6  Properties of MLE's:

     - Invariance to change of vbl's
     - Optimization: e.g. est'g \alph for Gamma dist'n, or Cauchy median
     - Consistency:  asymptotically, MLE \hat\th_n(X) -> \th
       (proof for L2 distributions; brief chat about convergence notions)

=================================================================================
Week 4: Estimation (cont'd)

6.7  Sufficient Statistics:
     - "Statistic" is function T = r(X_1,...,X_n) of observations, such
       as sample mean, maximum, 17.

     - If joint distn of X given T=t doesn't depend on \th T is "suff for \th"
     - Factorization Criterion:  f(x|\th) = u(x) v[r(x),\th] -> T=r(x) suff
     - Jointly sufficient (vector-valued T)
     - Examples of INsufficiency; examples where x-bar, s^2 are not sufficient;

6.9  Improving an Estimator: Rao-Blackwell
     - Warning re: robustness (eg: median, trimmed mean for No location)
     - Apply Rao-Blackwell to median

=================================================================================