Problem set 4 - Berkeleyyuekai/201b/pset4.pdf · PROBLEM SET 4 Due at the beginning of Section on...
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PROBLEM SET 4 Due at the beginning of Section on November 19, 2015. 1. The MSE of soft-thresholding. 1. Let z ∼N (0,I n ). Recall the Gaussian large-deviation inequality: P(|z 1 | >λ) ≤ 2 exp ( - λ 2 2 ) . By a union bound, show that P ( max i∈[n] |z i | > √ 4 log n ) ≤ 2 n . 2. Let x = μ + z. Consider the soft-thresholding estimator: ˆ μ ST,i := x i - λ x i ≥ λ 0 x i ∈ [-λ, λ] x i + λ x i ≤-λ. Show that as long as λ> max i∈[n] |z i |, (ˆ μ ST,i - μ i ) 2 ≤ 4λ 2 . 3. Let x ∼N (μ, I n ). Show that by soft-thresholding x at (4 log n) 1 2 P ( k ˆ μ ST - μk 2 2 > 16s log n ) ≤ 2 n , where s is the sparsity (number of non-zero components) of μ. 2. Do Problem set 3, Question 4. 3. Let x i i.i.d. ∼ N (μ, 1). We wish to test H 0 : |μ|≤ 1 versus H 1 : |μ| > 1. Consider the test φ(x)= 1 [c,∞) ( | ¯ x| ) . (a) What is its power function? (b) Find c so that the test is α-level. 4. There is a theory that people can postpone their death if an im- portant event is coming up. To test the theory, Phillips and King (1988) collected data on the deaths of Jews around Passover. Of 1919 deaths, 922 died the week before Passover, and 997 died the week after. Consider the number of deaths before Passover as a bin(1919,p) random variable. Report a p-value for testing H 0 : p = 1 2 . 1
Transcript of Problem set 4 - Berkeleyyuekai/201b/pset4.pdf · PROBLEM SET 4 Due at the beginning of Section on...
PROBLEM SET 4
Due at the beginning of Section on November 19, 2015.
1. The MSE of soft-thresholding.
1. Let z ∼ N (0, In). Recall the Gaussian large-deviation inequality:
P(|z1| > λ) ≤ 2 exp(−λ2
2
).
By a union bound, show that
P(maxi∈[n] |zi| >
√4 log n
)≤ 2
n .
2. Let x = µ+ z. Consider the soft-thresholding estimator:
µ̂ST,i :=
xi − λ xi ≥ λ0 xi ∈ [−λ, λ]
xi + λ xi ≤ −λ.
Show that as long as λ > maxi∈[n] |zi|, (µ̂ST,i − µi)2 ≤ 4λ2.
3. Let x ∼ N (µ, In). Show that by soft-thresholding x at (4 log n)12
P(‖µ̂ST − µ‖22 > 16s log n
)≤ 2
n ,
where s is the sparsity (number of non-zero components) of µ.
2. Do Problem set 3, Question 4.
3. Let xii.i.d.∼ N (µ, 1). We wish to test H0 : |µ| ≤ 1 versus H1 : |µ| > 1.
Consider the testφ(x) = 1[c,∞)
(|x̄|).
(a) What is its power function?(b) Find c so that the test is α-level.
4. There is a theory that people can postpone their death if an im-portant event is coming up. To test the theory, Phillips and King (1988)collected data on the deaths of Jews around Passover. Of 1919 deaths, 922died the week before Passover, and 997 died the week after. Consider thenumber of deaths before Passover as a bin(1919, p) random variable. Reporta p-value for testing H0 : p = 1
2 .
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2 STAT 201B
5. (Optional). Let p0(x) and p1(x) be two densities on X . Consider
testing H0 : xii.i.d.∼ p0 versus H1 : xi
i.i.d.∼ p1. Under H1, show that the powerof the αlevel Neyman-Pearson test
ϕ(x) = 1(t,∞)
(∏i∈[n]
p1(x)p0(x)
)approaches 1 as n→∞. You may assume the log-likelihood ratio is asymp-totically normal.
References.
Phillips, D. and King, E. (1988). Death takes a holiday: mortality surrounding majorsocial occasions. The Lancet 332 728–732.
