Lect14_2700_s09

# Lect14_2700_s09 - ENGRD 2700 Basic Engineering Probability...

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Statistical Inference Point Estimation Desirable Properties Standard Error Title Page JJ II J I Page 1 of 33 Go Back Full Screen Close Quit ENGRD 2700 Basic Engineering Probability and Statistics Lecture 14: Point Estimation: Concepts David S. Matteson School of Operations Research and Information Engineering Rhodes Hall, Cornell University Ithaca NY 14853 USA [email protected] March 11, 2009

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Statistical Inference Point Estimation Desirable Properties Standard Error Title Page JJ II J I Page 2 of 33 Go Back Full Screen Close Quit 1. Statistical Inference Given data x 1 ,...,x n , we want to Test a hypothesis, for example: Is data random sample from normal distribution? The model for the data can be represented as a linear model. Design speciﬁcations for the dam will not be ex- ceeded in the next 10,000 years. Estimate parameters in an assumed statistical model. Assuming data is normal (based on histograms, QQ-plots), what values of μ,σ should we use? Various methods are possible for estimation of pa- rameters. How do we compare diﬀerent methods? Which method is best? What does best mean?
Statistical Inference Point Estimation Desirable Properties Standard Error Title Page JJ II J I Page 3 of 33 Go Back Full Screen Close Quit 2. Point Estimation What is a statistical model? A Probability Model is a triple S = sample space A =[ events; subsets of S ] P = rule for assigning probabilities to events. A Statistical Model is a family S = sample space A =[ events ] { P θ Θ } = parametric family of probabilties. Note θ could be multidimensional.

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Statistical Inference Point Estimation Desirable Properties Standard Error Title Page JJ II J I Page 4 of 33 Go Back Full Screen Close Quit Example: Suppose we have a normally distributed exper- iment with S = R , P μ,σ where P μ,σ ( -∞ ,x ] = Φ( x ; μ,σ 2 ) . Then what is θ ? What is Θ ? We could deﬁne a random variable X N ( μ,σ 2 ) by X ( s ) = s, s S = R , which would make X a normally distributed rv.
Statistical Inference Point Estimation Desirable Properties Standard Error Title Page JJ II J I Page 5 of 33 Go Back Full Screen Close Quit Example: Suppose we have an experiment where we want the outcome exponentially distributed: S =[0 , ) A = subsets of positive numbers { P λ ,λ > 0 } = probabilitiy assignments. Here P λ ([0 ,x ]) = 1 - e - λx , x > 0 , and θ = , Θ = We can deﬁne an exponential random variable by X ( s ) = s, s S = [0 , ) .

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Statistical Inference Point Estimation Desirable Properties Standard Error Title Page JJ II J I Page 6 of 33 Go Back Full Screen Close Quit Example: Build a statistical model for the experiment: Randomly sample a normal variable n times . S
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## This note was uploaded on 04/05/2009 for the course ENGRD 2700 taught by Professor Staff during the Spring '05 term at Cornell.

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Lect14_2700_s09 - ENGRD 2700 Basic Engineering Probability...

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