math LateX案例

Problem 1. [Estimating a Hilbert’s density] (12 pts, 4 ⇥ 3)

因编辑器不支持公式 所以乱码 pdf预览查看

Let {Xi}n be an i.i.d. sample from an unknown distribution P with (Lebesgue) density f supported on

some compact X ⇢ Rd with Lebesgue measure 1. For simplicity, we assume furthermore that f is bounded,

i.e., supx f(x) F for some known F. .

R, X g2 dx < 1}. Suppose we know an orthonormal basis {fj}1j=1 of F (under the inner-product hg, hi =

X g(x) · h(x) dx); then the idea is to estimate the expansion coefficients ↵j of f, where f = P1j=1 ↵jfj,



In particular, one idea is to minimize (an approximation to)

2 D E

fbN — f    = kfk  + kfbN k2 — 2    fbN ,f   .



In particular, this is equivalent to minimizing kfbN k2 — 2hfbN ,fi,  where kfbN k2  =  PN   ↵2,  and where



hfbN ,fi R   fbN (x)f(x) dx  =  EfbN (X)  can be estimated from a sample as 1 Pn   fbN (Xi).  In other

N n

EkfN — fkp ———! 0, provided N = o(n),N ! 1.X ↵b2 — 2 X fb (X ). (1)

(1). Show that equation (1) is minimized by setting ↵bj = 1 Pn   fj(Xi), for j = 1,. .., N.



Show that fNso defined is consistent, i.e., for any 1 p 2, we have




Supposethe tail coefficients of f  satisfy 1j=N+1    2
.

Ø(N), for some known Ø(N)

2





—N——!—1!  0.



Derive an asymptotic L2 confidence ball of the form B6 = {g 2 F : kfbN — gk r6}, where r6 depends

on n, N, Ø(N), confidence level 1 — 6 (for any fixed 0 < 6 < 1), and z-critical values of N (0, 1).

Problem 2. [When is classification well-defined?] (8 pts, 2 ⇥ 4)

Let  X  be  a  random  variable  (⌦, ⌃)  7!  (R, B(R))  with  range  R.   In  classification,  a label y  2  {0, 1} is associated to every value x of X in R.  Usually, we posit the existence of a labeling function h(x),  h : R ! {0, 1}. If such h is not measurable (R, B(R)) 7! ({0, 1}, 2{0,1}), classification is not well-defined as a statistical problem, e.g., we cannot talk about estimating h under given Risk measures (e.g. P(h 6= h)).

Usingthe fact that B(R) 6= 2R, argue that, without additional assumptions, a labeling function h might indeed not be measurable (R, B(R)) 7! ({0, 1}, 2{0,1}).
SupposeY  : (⌦, ⌃) 7! ({0, 1}, 2{0,1}) is jointly distributed with X (on (R2, B(R2))). Argue that, there always exists a labeling function h of the form {E[Y |x] ≤ 1/2} (so-called Bayes classifier), where h is measurable (R, (R)) ( 0, 1 , 2{0,1}).
Hint: The rights answers should be a few lines following from basic definitions and results.

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