s2 - Stat 5102 Lecture Slides Deck 2 Charles J. Geyer...

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Unformatted text preview: Stat 5102 Lecture Slides Deck 2 Charles J. Geyer School of Statistics University of Minnesota 1 Statistical Inference Statistics is probability done backwards. In probability theory we give you one probability model, also called a probability distribution. Your job is to say something about expectations, probabilities, quantiles, etc. for that distri- bution. In short, given a probability model, describe data from that model. In theoretical statistics, we give you a statistical model , which is a family of probability distributions, and we give you some data assumed to have one of the distributions in the model. Your job is to say something about which distribution that is. In short, given a statistical model and data, infer which distribution in the model is the one for the data. 2 Statistical Models A statistical model is a family of probability distributions. A parametric statistical model is a family of probability distribu- tions specified by a finite set of parameters. Examples: Ber( p ), N ( , 2 ), and the like. A nonparametric statistical model is a family of probability dis- tributions too big to be specified by a finite set of parameters. Examples: all probability distributions on R , all continuous sym- metric probability distributions on R , all probability distributions on R having second moments, and the like. 3 Statistical Models and Submodels If M is a statistical model, it is a family of probability distribu- tions. A submodel of a statistical model M is a family of probability distributions that is a subset of M . If M is parametric, then we often specify it by giving the PMF (if the distributions are discrete) or PDF (if the distributions are continuous) { f : } where is the parameter space of the model. 4 Statistical Models and Submodels (cont.) We can have models and submodels for nonparametric families too. All probability distributions on R is a statistical model. All continuous and symmetric probability distributions on R is a submodel of that. All univariate normal distributions is a submodel of that. The first two are nonparametric. The last is parametric. 5 Statistical Models and Submodels (cont.) Submodels of parametric families are often specified by fixing the values of some parameters. All univariate normal distributions is a statistical model. All univariate normal distributions with known variance is a sub- model of that. Its only unknown parameter is the mean. Its parameter space is R . All univariate normal distributions with known mean is a differ- ent submodel. Its only unknown parameter is the variance. Its parameter space is (0 , ). 6 Statistical Models and Submodels (cont.) Thus N ( , 2 ) does not, by itself, specify a statistical model. You must say what the parameter space is. Alternatively, you must say which parameters are considered known and which unknown....
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s2 - Stat 5102 Lecture Slides Deck 2 Charles J. Geyer...

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