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# 20 Estimation

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### 20 Estimation

1. 1. Stat310 Estimation Hadley Wickham Sunday, 11 April 2010
2. 2. Backup! http://mozy.com/ Sunday, 11 April 2010
3. 3. Misc. • Test grading in progress. Should have back by Thursday. • Will also calculate overall ytd grades. • Two party planning commissioners needed to help plan end of class party - please email me if interested. Sunday, 11 April 2010
4. 4. 1. Recap 2. Method of moments 3. Maximum likelihood 4. Feedback Sunday, 11 April 2010
5. 5. Recap Sunday, 11 April 2010
6. 6. Inference Up to now: Given a sequence of random variables with distribution F, what can we say about the mean of a sample? What we really want: Given the mean of a sample, what can we say about the underlying sequence of random variables? Sunday, 11 April 2010
7. 7. Marijuana Up to now: If I told you that smoking pot was Binomially distributed with probability p, you could tell me what I would expect to see if I sampled n people at random. What we want: If I sample n people at random and ﬁnd out m of them smoke pot, what does that tell me about p? Sunday, 11 April 2010
8. 8. Your turn Let’s I selected 10 people at random from the Rice campus and asked them if they’d smoked pot in the last month. Three said yes. Let Yi be 1 if person i smoked pot and 0 otherwise. What’s a good guess for distribution of Yi? Sunday, 11 April 2010
9. 9. 3.5e−08 3.0e−08 2.5e−08 2.0e−08 prob 1.5e−08 1.0e−08 5.0e−09 0.0e+00 0.0 0.2 0.4 0.6 0.8 1.0 p Sunday, 11 April 2010
10. 10. Aim Given data, we want to ﬁgure out what the true parameters of the distribution are. We also want to know how much error is associated with our estimate. Sunday, 11 April 2010
11. 11. Deﬁnitions Parameter space: set of all allowed parameter values Estimator: process/function which takes data and gives best guess for parameter Point estimate: estimate for a single value Sunday, 11 April 2010
12. 12. Hat notation Usually write the estimate of a parameter with a little hat over it. Subscript identiﬁes type of estimator used. ˆ µM M ˆ µ ˆ2 σ 2 σ ˆ ˆ µM L Sunday, 11 April 2010
13. 13. Lower/upper case X1, X2, X3, ... = random variables that deﬁne an experiment. IID. x1, x2, x3, ... = results of single experiment So we collect x1, x2, x3, ... and want to say something about the distribution of X Sunday, 11 April 2010
14. 14. Plug-in principle A good ﬁrst guess at the true mean, variance, or any other parameter of a distribution is simply to estimate it from the data. Two more ways: method of moments and maximum likelihood estimator Sunday, 11 April 2010
15. 15. Method of moments Sunday, 11 April 2010
16. 16. Method of moments We know how to calculate sample moments (e.g. mean and variance of data) We know what the moments of the distribution are in terms of the parameters. Why not just match them up? Sunday, 11 April 2010
17. 17. Gamma distribution X ~ Gamma(α, β) E(X) = αβ Var(X) = αβ2 Sunday, 11 April 2010
18. 18. Steps • Write down formulas for mean and variance. • Rewrite to use sample estimates • Solve for the parameters • (If there are more parameters, you’ll need to use more moments, but that won’t come up since we haven’t learned any distributions with more than two parameters) Sunday, 11 April 2010
19. 19. Your turn What are the method of moments estimators for the mean and variance of the normal distribution? What about the Poisson distribution? Is there anything different about the Poisson? Sunday, 11 April 2010
20. 20. But... Let X ~ Unif(0, θ) What is a method of moments estimator for θ? For one experiment we got the values 3, 5, 6, 18. What is the method of moments estimate for θ? Is it a good estimator? Sunday, 11 April 2010
21. 21. Pros/cons Pro: Simple! Con: Doesn’t always work. Con: Often not best estimator Con: Don’t know anything about the error in the estimate Sunday, 11 April 2010
22. 22. Maximum likelihood Sunday, 11 April 2010
23. 23. Example X ~ Binomial(10, p) We repeat the random experiment 10 times and get the following value: 4515324224 What is p? Sunday, 11 April 2010
24. 24. 3.5e−08 3.0e−08 2.5e−08 2.0e−08 prob 1.5e−08 1.0e−08 5.0e−09 0.0e+00 0.0 0.2 0.4 0.6 0.8 1.0 p Sunday, 11 April 2010
25. 25. Maximum likelihood Write down the joint pdf (pdf of iid random variables is?) We have the data, so we can work out how likely each possible value of p is. Then pick the p that is most likely. Can do this numerically (like that graphics) or algebraically (with calculus) Sunday, 11 April 2010
26. 26. likelihood = joint pdf Sunday, 11 April 2010
27. 27. Steps Write out likelihood (=joint pdf) Write out log-likelihood (Discard constants) Find maximum: Differentiate and set to 0 (Check second derivative is positive) (Check end points) Sunday, 11 April 2010
28. 28. Binomial 1. Joint pdf - be careful about x’s 2. Log-likelihood - why? 3. Differentiate and set to zero Sunday, 11 April 2010
29. 29. Feedback Sunday, 11 April 2010