STA250: Statistics, based loosely on DeGroot & Schervish
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Week 4: MLE Estimation           [ This week's notes need expansion ]

7.5  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] (work out in detail)
     - Discouraging EG: Un[\th-1, \th+1]
     - Alarming EG:     1/2 No(0,1) + 1/2 No(mu, sig^2), mu & sig unknown.

7.6  Properties of MLE's:

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

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For this material see latex notes "suff.pdf" 

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

     - T is "suff for \th" if joint distn of [ X | T ] doesn't depend on \th
       Illustrate with Bernoullis & T=sum

     - Factorization Criterion: f(x|\th) = h(x) v[r(x),\th] <-> T=r(x) sufficient

7.8  - Jointly sufficient (vector-valued T)
     - Examples of INsufficiency; examples where x-bar, s^2 are not sufficient;

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

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