6.041 / 6.431 25. Classical Statistical Inference - III

LECTURE 25 Outline • Reference: Section 9.4 • Review of simple binary hypothesis tests – examples • Testing composite hypotheses – is my coin fair? – is my die fair? – goodness of fit tests Simple binary hypothesis testing – null hypothesis H0: X ! pX(x; H0) [or fX(x; H0)] – alternative hypothesis H1: X ! pX(x; H1) [or fX(x; H1)] – Choose a rejection region R; reject H0 iff data " R • Likelihood ratio test: reject H0 if pX(x; H1) pX(x; H0) > ! or fX(x; H1) fX(x; H0) > ! – fix false rejection probability "; (e.g., " =0.05) – choose ! so that P(reject H0; H0)= " Example (test for normal mean) • n data points, i.i.d. H0: Xi ! N(0, 1) H1: Xi ! N(1, 1) • Likelihood ratio test; rejection region: (1/#2#)n exp{− !i(Xi − 1)2/2}(1/#2#)n exp{− !i X2 i /2} > ! – algebra: reject H0 if: " i Xi > !% • Find !% such that P # n" i=1 Xi > !%; H0 $ = " – use normal tables Example (test for normal variance) • n data points, i.i.d. H0: Xi ! N(0, 1) H1: Xi ! N(0, 4) • Likelihood ratio test; rejection region: (1/2#2#)n exp{− !i X2 i /(2 · 4)}(1/#2#)n exp{− !i X2 i /2} > ! – algebra: reject H0 if " i X2 i > !% • Find !% such that P # n" i=1 X2 i > !%; H0 $ = " – the distribution of !i X2 i is known (derived distribution problem) – “chi-square” distribution; tables are available Composite hypotheses Is my die fair? • Got S = 472 heads in n = 1000 tosses; • Hypothesis H0: is the coin fair? P(X = i)= pi =1/6, i =1,..., 6 – H0: p =1/2 versus H1: p =1&/2 Observed occurrences of i: Ni• • Pick a “statistic” (e.g., S) • Choose form of rejection region; Pick shape of rejection region chi-square test: • (e.g., |S − n/2| > !) reject H0 if T = " (Ni − npi)2 > ! i npi • Choose significance level (e.g., " =0.05) Choose ! so that: Pick critical value ! so that: • • P(reject H0; H0)=0.05 P(reject H0; H0)= " Using the CLT: P(T> !; H0)=0.05 P(|S − 500| ' 31; H0) ( 0.95; ! = 31 Need the distribution of T :• (CLT + derived distribution problem) • In our example: |S − 500| = 28 < ! H0 not rejected (at the 5% level) – for large n, T has approximately a chi-square distribution – available in tables Do I have the correct pdf? What else is there? • Partition the range into bins • Systematic methods for coming up with – npi: expected incidence of bin i shape of rejection regions(from the pdf)– Ni: observed incidence of bin i Methods to estimate an unknown PDF • (e.g., form a histogram and “smooth” it – Use chi-square test (as in die problem) out) • Kolmogorov-Smirnov test: form empirical CDF, FˆX, from data Efficient and recursive signal processing • Methods to select between less or more • complex models – (e.g., identify relevant “explanatory variables” in regression models) • Methods tailored to high-dimensional unknown parameter vectors and huge number of data points (data mining) etc. etc....•(http://www.itl.nist.gov/div898/handbook/) • Dn = maxx |FX(x) − ˆFX(x)| • P(#nDn ) 1.36) ( 0.05 MIT OpenCourseWare http://ocw.mit.edu 6.041 /6.431 Probabilistic Systems Analysis and Applied Probability Fall 2010 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.