E1 244: Detection and Estimation Theory (2017) Homework 1
1. Consider the following hypotheses for Y ∈R: H1:Y =X2 H0:Y =X3,
where X ∼ N(0, σ2). Write down an expression for the likelihood ratio between H0 and H1, for an observation y.
2. SupposeY is a random variable that, under hypothesisH0, has probability density function (pdf),
p0(y) = (2
3(y+ 1), 0≤y≤1
0, otherwise
and, under hypothesis H1, has pdf
p1(y) =
(1, 0≤y≤1 0, otherwise.
(a) Find the Bayes rule and minimum Bayes risk for testingH0 versusH1, with uniform cost and equal priors.
(b) Plot the two pdfs on a common plot and identify the threshold τ in the Bayes rule assuming uniform cost and equal priors. Discuss the effect of π0 on the thresholdτ (Hint: you can use the posterior probabilities πi(y) to illustrate).
(c) Find the minimax rule and minimax risk under uniform costs.
(d) Find the Neyman-Pearson rule and the corresponding detection probability for false- alarm probabilityα∈(0,1).
3. Consider the hypothesis pair
H0: Y =N−s versus
H1: Y =N+s
where s >0 is a fixed real number andN is a continuous random variable with density pN(n) = 1
π(1 +n2), n∈R
(a) Find the likelihood ratio betweenH0 andH1, for an observationy.
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(b) Find the Bayes rule and the minimum Bayes risk, under the costsC00 =C11 = 0, C01= 2C10= 1, and the priorπ0=14.
(c) Find the minimax decision rule and the corresponding risk under uniform cost.
(d) Find the threshold and detection probability for an α-level Neyman Pearson test, α∈(0,1).
4. (a) Show the basic min-max inequality: mina∈Amaxb∈Bf(a, b)≥maxb∈Bmina∈Af(a, b).
(b) Letf andgbe two affine (i.e., linear plus a constant), real-valued functions defined on the interval [u, v]∈R. Show that the functionh, defined byh(x) = min(f(x), g(x)) on [u, v], is concave; in other words, show that ∀a, b∈[u, v] andλ∈[0,1], we must have
h(λa+ (1−λ)b)≥λh(a) + (1−λ)h(b).
5. For the binary channel as discussed in class, find (a) the minimax risk, (b) a randomized decision rule δ(y) which achieves the minimax risk, and (c) the least favorable priorπL, for each of the following cases of the channel:
a) λ0= 0.4, λ1= 0.2, b) λ0= 0.6, λ1= 0.45.
Assume costs to be uniform.
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