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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