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Nonparametric Methods I

Nonparametric Methods I

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Nonparametric Methods I

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  1. Nonparametric Methods I Henry Horng-Shing Lu Institute of Statistics National Chiao Tung University hslu@stat.nctu.edu.tw http://tigpbp.iis.sinica.edu.tw/courses.htm

  2. Parametric vs. Nonparametric • MLE: probability distribution and likelihood • Bayes: conditional, prior and posterior distributions • Distribution free? • http://en.wikipedia.org/wiki/Non-parametric_statistics

  3. Motivation (1) In many applications, direct access to a measurement and is not possible. However, an estimation of the measurement is needed. Most of the time, the large scale repetition of an experiment is not economically feasible. What can one do?

  4. Motivation (2) • Q1: What estimator for the problem of interest can be used? • Q2: Having chosen an estimator, how accurate is it? What is the bias and variance of an estimator? • Q3: How to make inference? What is the confidence interval? What is the p-value for a hypothesis testing?

  5. References • B. Efron (1979) Computers and the theory of statistics: thinking the unthinkable, SIAM Review, 21, 460-480. • B. Efron and R. J. Tibshirani (1993) An Introduction to the Bootstrap. Chapman & Hall. • J. I. De la Rosa and G. A. Fleury (2006) Bootstrap methods for a measurement estimation problem. IEEE Transactions onInstrumentation and Measurement, 55, 3, 820–827. • http://en.wikipedia.org/wiki/Resampling_%28statistics%29#Jackknifehttp://en.wikipedia.org/wiki/Bootstrapping_%28statistics%29

  6. Resampling Techniques • Data resampling • PART 1: Jackknife • Resampling without replacement • PART 2: Bootstrap • Resampling with replacement

  7. PART 1: Jackknife • Naming • Illustration • Math Expression • Examples • R codes • C codes

  8. Why the funny name of Jackknife? Jackknife: a pocket knife http://en.wikipedia.org/wiki/Jackknife Mosteller and Tukey (1977, p. 133) described a predecessor resampling method, the jackknife, in the following way: “The name ‘jackknife’ is intended to suggest the broad usefulness of a technique as a substitute for specialized tools that may not be available, just as the Boy Scout’s trustworthy tool serves so variedly…” http://mrw.interscience.wiley.com/emrw/9780470013199/esbs/article/bsa321/current/abstract

  9. Illustration of Jackknife Population, Estimate by sampling inference resampling N times statistics

  10. Math Expression

  11. An Example of Jackknife (1) HW HW

  12. An Example of Jackknife (2)

  13. Summary of the Jackknife Method

  14. How do quartiles lead to an estimate?

  15. Jackknife by R 1. Open “R”

  16. 2. Install add-on packages

  17. 3.Select a mirror site, like Taiwan (Taipeh)

  18. 4.Select the package of “bootstrap”

  19. 5. type: library(bootstrap)

  20. If you want to see the manual, you can type “?jackniffe”.

  21. R-package

  22. Select the menu to open the editor in R

  23. You can edit your program in this box and then store this program.

  24. You can save your program……

  25. main.jackknife.function

  26. (1) Use mouse to select the R commands you want to run. (2) Press “F5” to run

  27. output

  28. Jackknife by C define functions

  29. An example for jackknife

  30. PART 2: Bootstrap • Naming • Illustration • Math Expression • Examples • R codes • Three approaches • Package(bootstrap) • Package(boot) • Write your own R codes • C codes

  31. The Bootstrap • Bootstrap technique was proposed by Bradley Efron (1979, 1981, 1982) in literature. • Bootstrapping is an application of intensive computing to traditional inferential methods.

  32. Why the funny name of bootstrap? • Bootstrap: http://www.concurringopinions.com/archives/Bootstrap_1.jpg • In the book of ‘Singular Travels, Campaigns and Adventures of Baron Munchausen’ by R. E. Raspe (1786), the main character, finding himself in a deep hole, extracts himself using only the straps of his boots. • http://tigger.uic.edu/~slsclove/stathumr.htm

  33. Illustration of Bootstrap Population, estimate by sampling inference resampling B times statistics

  34. Math Expression

  35. Population,

  36. Population step1 sampling

  37. step2 resampling B times

  38. Step 3: statistics

  39. Summary of the Bootstrap Method

  40. Bootstrap by R • Approach 1 • Use package “bootstrap” • Approach 2 • Use package “boot” • Approach 3 • Write your own R codes

  41. Approach 1 http://finzi.psych.upenn.edu/R/library/bootstrap/DESCRIPTION

  42. 1. Install the add-on package