Statistics for Applications Fall Problem Set 7

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Nigel Miller
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1 Statistics for Applicatios Fall 016. Proble Set 7 Due Friday, Oct. 8 at 1 oo Proble 1 QQ-plots Recall that the Laplace distributio with paraeter λ > 0 is the cotiuous probaλ bility easure with desity λ x fλ (x) = e, x IR ad the Cauchy distributio is the 1 1 cotiuous probability easure with desity g(x) = π 1 + x, x IR. Cosider five saples of i.i.d. rado variables with the followig distributios: The stadard Gaussia distributio; The uifor distributio o [ 3, 3]; The Cauchy distributio; The expoetial distributio with paraeter 1; The Laplace distributio with paraeter. For each of these saples, we have draw the oral QQ-plots below. Idetify which plot correspods to which saple. QQ-Plot 1 QQ-Plot 1

2 QQ-Plot 3 QQ-Plot 4 QQ-Plot 5

3 Proble Koloogorov-Sirov test for two saples Cosider two idepedet saples X 1,..., X ad Y 1,..., Y of idepedet real valued cotiuous rado variables, ad assue that the X i s are iid with soe cdf F ad that the Y i s are iid with soe cdf G. Note that the two saples ay have differet sizes (if = ). We wat to test whether F = G. We cosider the followig hypotheses: H 0 : F = G ad H 1 : F = G. For siplicity, we will assue that i additio to be cotiuous, F ad G are icreasig. 1. Propose a exaple of experiet i which testig whether two saples are geerated by the sae distributio would be of iterest.. For i = 1,...,, deote by U i = F (X i ) ad for j = 1,...,, let V j = G(Y j ). What are the distributios of the U i s ad the V j s? 3. Let F be the epirical cdf of the saple { X 1,..., X} ad G be the epirical cdf of {Y 1,..., Y }. a) Let T, = sup F (t) G (t). Prove that T, ca be writte as the axit IR u value of a fiite collectio of ubers. b) If H 0 is true, show that T, = sup 0 x Ui x j=1 1 Vj x. c) If H 0 is true, what is the joit distributio of the + rado variables U 1,..., U, V 1,..., V? d) Coclude that the test statistic T, is pivotal, i.e., if H 0 is true, the distributio of T, does ot deped o the ukow distributio of the saples. e) Let α (0, 1) ad let q α be the (1 α)-quatile of the distributio of T, uder H 0. I practice, eve if this quatile ay be available o tables for soe values of ad, you ay ot be able to fid it olie for your values of ad. Describe a algorith that you could ru o a software (e.g., R) i order to get a approxiate value of q α, for a give α. f) Defie a test with o asyptotic level α for the hypotheses H 0 v.s. H 1. Proble 3 Test of idepedece for saples with cotiuous cdf Cosider i.i.d. pairs of real rado variables (X 1, Y 1 ),..., (X, Y ) with soe cotiuous distributio. We would like to test whether X 1 ad Y 1 are idepedet. Defie the followig hypotheses: H 0 : X 1 Y 1 ad H 1 : X 1 ad Y 1 are ot idepedet. 3

4 For i = 1,...,, defie R i as the rak of X i i the saple X 1,..., X : E.g., if X i is the sallest uber of this saple, the R i = 1; If X i is the largest, the R i =. I a siilar fashio, defie Q i as the rak of Y i i the saple Y 1,..., Y. 1. Propose a exaple of experiet i which testig idepedece of two saples would be of iterest.. Without a rigorous proof, explai why R 1,..., R are ot idepedet rado variables. 3. Prove that the distributio of (R 1,..., R ) does ot deped o the (ukow) distributio of the X i s. Siilarly, the distributio of (Q 1,..., Q ) does ot deped o that of the Y i s. 4. Prove that if H 0 is true, the the two vectors of raks (R 1,..., R ) ad (Q 1,..., Q ) are idepedet. 5. Coclude that if H 0 is true, the the joit distributio of the rado variables R 1,..., R, Q 1,..., Q is kow ad does ot deped o the distributio of the origial saple. 6. Cosider the followig test statistic: R )(Q i Q (R i ) T =. R ) (R Q ) i (Q i T is the epirical correlatio betwee the R i s ad the Q i s. If H 0 is true, the T should be very close to zero. We are first goig to show that T has a very sipler expressio. a) Prove that R = Q = + 1, (Ri R ) = (Qi ( 1) Q ) =. 1 ( + 1) ( + 1)( + 1) Hit: Recall that i = ad i =. 6 b) Coclude that 1 3( + 1) T = R i Q i. ( 1) 1 7. Usig all the previous questios, prove that if H 0 is true, the T has the sae distributio as 1 3( + 1) S R i ' Q ' = i, ( 1) 1 4

5 where (R 1 ',..., R ' ) ad (Q ' 1,..., Q ' ) are the respective rak vectors of two idepedet saples of i.i.d. uifor rado variables i [0, 1]. 8. Let α (0, 1). Deote by q α the (1 α)-quatile of S. Describe a algorith that you could ru o the software R i order to get a approxiate value of q α, for a give value of. 9. Defie a test for H 0 v.s. H 1 that has o asyptotic level α. 5

6 MIT OpeCourseWare / Statistics for Applicatios Fall 016 For iforatio about citig these aterials or our Ters of Use, visit:

Contents Two Sample t Tests Two Sample t Tests

Contents Two Sample t Tests Two Sample t Tests Cotets 3.5.3 Two Saple t Tests................................... 3.5.3 Two Saple t Tests Setup: Two Saples We ow focus o a sceario where we have two idepedet saples fro possibly differet populatios. Our