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Definitions |
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How is F distributed?
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(Q1/k1)
_____
(Q2/k2)
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Rigorously define E[y | g]
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It's the unique element of L2 such that
E[y | g] is g-measurable
E[E[y | g]z] = E[yz] for all g-measurable z memberof L2
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Define the inner product and norm
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Inner product of x and y is
||xy|| = (x'y)
<x,y> := E[xy]
Norm:
||xy|| = sqrt((x'y))
<x,y> := sqrt(E[xy])
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What is the ECDF?
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The empirical distribution of the sample is the discrete distribution that puts equal probability on each sample point. The cdf for the empirical distribution is called the empirical cumulative distribution function, or ecdf.
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What is the first moment assumption and some conclusions from it?
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E[u | X] = 0 for all X.
From this E[u] = 0
E[um | xnk] = 0
E[umxnk] = 0
Cov[um,xnk] = 0
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What makes a sequence {Ft} a filtration? What makes a sequence {Ft} adapted to the RV sequence {mt}?
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If Ft is always a subset of Ft+1.
If mt is always Ft measurable.
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What is the triangle inequality?
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||x+y|| < ||x|| + ||y||
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What is a test?
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Formally, a test is a binary function f mapping the observed data x into {0, 1}.
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span(X) := ?
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sum of all akXk such that (a1, . . . , aK) members of RK
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What is Weierstrass theorem?
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Given continuous function f, there exists a polynomial function g such that g is arbitrarily close to f.
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OPT Mark III?
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y = Py + My
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What is the LLN
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If xn is an IID sequence with finite second moment, then xbarN -p-> E[xn] = Int sF(ds) as N --> Inf
E[(sigmahat-sigma)^2]
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What is a martingale difference sequence?
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E[Mt+1|Ft] = 0 for all t
and {Ft} and {Mt} are adapted.
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What are some other expressions for u?
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My and Mu. Useful.
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Chebychev's inequality
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P{|y| >= delta} <= E[y^2] / delta^2
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What is ||| x - y ||| when x and y are RVs?
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sqrt(E[(x-y)^2])
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What is the plugin estimator?
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The average of the realizations (variance) across a sample.
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Global stability in markov series is equivalent to what?
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Ro < 1. Then Cov(x_t , x_t+j) ~ 0 when j is large. Asymptotically independent.
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What comes out of a CDF F being symmetric?
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F(-s) = 1-F(s)
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If y is g-measureable then
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It can be written as a deterministic function of the contents of g.
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What are the three main assumption necessary for OLS?
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y = Xbeta + u
E[u | X] = 0
E[uu' | X] = sigma^2 I
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What is necessary for vector: xn -p-> x ?
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xnk -p-> xk for all k
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What is the Kth moment?
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E[x^k] = Int s^k F(ds)
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What is global stability?
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Regardless of x_0 you end up at the CapPi_inf stationary distribution.
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rank(A) := ?
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dim(rng(A))
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What is the quantile function F-1(q)?
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:= the unique s such that F(s) = q
0 <= q <= 1
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What is L_2?
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L_2 := { all RVs x with E[x^2] < inf }
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What is a transition density?
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p(.|s) := conditional density on xt+1 when xt = s.
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What is identical distribution and independence for distributions?
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Fn = Fm for all n and m
F(s1, ..., sN) = F(s1) * ... * F(sn)
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What is the power function?
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The power function associated with the test f is the probability that the test rejects when the data is generated by Mo.
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OPT for random variables and some resulting properties
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Given linear subspace S of L2 and y in L2, there is a unique y memberof S, such that
|||y-^y||| <= |||y-z||| for all z member of S
Py memberof S
y - Py orthogonal to S
Py = y iff y memberof S
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Requirements for OPT Mark II?
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Py member of S
y-Py orthogonal to S
||y||^2 = ||Py||^2 + ||y-Py||^2
||Py|| =< ||y||
Py = y if y member of S
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Define covariance and variance of random vectors x (and y) conditional on Z.
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Cov: E[xy' | Z] - E[x | Z] E[y | Z]'
Var: E[xx' | Z] - E[x | Z] E[x | Z]'
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What is the principle of maximum likelihood?
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Choose the parameters that would make the data you saw most likely.
L(sigma)
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What are the requirements for S to be a linear subspace of L2?
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If a and b are both real numbers, and x and y are members of S then ax + by is a member of S.
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What is the continuous mapping theorem?
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If g is a continuous function and xn converges in probability to x then g(xn) converges in probability to x.
If g is a continuous function and xn converges in distribution to x then g(xn) converges in distribution to x.
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(AB)-1 = B-1A-1?
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Yes, if A and B are both invertible.
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What's L2(g)? What a property has it?
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All of the g-measurable RVs in L2.The set is a linear subspace of L2.
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Expand: Var(alpha x + beta y)
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alpha^2 Var(x) + beta^2 Var(y) + 2 * alpha * beta Cov(x,y)
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What is asymptotic normality and what is the asymptotic variance?
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sqrt(N) * (sigmahat - sigma) -d-> Normal(0, v(sigma))
v(sigma) is the asymptotic variance of the estimator
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What is the conditional density?
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p(sk+1 etc | s1 ... sk) = p(sk+q etc) / p(s1 ... sk)
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What is slutsky's theorem?
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If Yn converges in probability to a matrix C and xn converges in distribution to a random matrix x, then
Ynxn -d-> Cx and Yn + xn -d-> C + x
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Describe the CLT
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If the second moment of an xn is finite and draws are IID, then sqrt(n) (xbarN - mu) -d-> Normal(0, Var(xn))
Where mu = E[xn]
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What is an intuitive idea of conditional expectation in F2?
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The E[y | g] is the closest g measurable RV to y.
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What is a parametric versus nonparametric class?
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More generally, a parametric class of densities is a set of densities pq indexed by a vector of parameters. This is a large set of densities that cannot be expressed as a parametric class. In such
cases, we say that the class of densities is nonparametric
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How is t-test distributed?
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Z(k/Q)^1/2
z = normal
Q = chisq
k = dof of chisq
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What is the difference between the joint and marginal density.
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Joint: F(s1,..., sn) = |P(x1<s1, ..., xn < sn)
Marginal: F(s1) = |P (x1 < s1)
Joint may not exist, is a large integral. It's not just the same.
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What is required for the LLN to hold in time series?
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IID, or at least that Cov(x_t , x_t+j) ~ 0 when j is large, then it hold asymptotically.
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What is the TSS, SSR, ESS, R^2?
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TSS: ||y||^2
SSR: ||My||^2
ESS: ||Py||^2
R2: ||Py||^2/||y||^2
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Why does the trace of P = K?
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Trace(P) = Trace(X(X'X)-1X') = Trace((X'X)-1X'X) = Trace(I) = K
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What is OPT mark I?
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There exists a solution to ^y := argmin ||y-z||
^y member of S
y - ^y orthogonal to S
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What is the mean square error?
And of an estimator?
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E[(xn - x)^2]
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What is required for Markov stability?
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g() is continuous, lambda < 1 and L is a positive constant
littlePhi is density with finite mean and positive probability at all points
||g(s)|| <= lambda ||s|| + L for all s memberof Rk
then there exists a unique globally stable stationary density pi_inf
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What is the stationary distribution?
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CapPi_inf
If x_o = CapPi_inf then {x_t} is identically distributed.
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Fact. If A idempotent, then what do we know about the trace?
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rank(A) = trace(A)
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What is the formula for f to be a density of x?
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Integral B f(s)ds = P{x memberof B} for all B subset R^k.
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What are some conditions for a density f?
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Integrates to 1 and is nonnegative at all points. One to one correspondence with an F.
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What are some conditions for a probability mass function and the formula?
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all 0 <= pj <= 1 and the sum of pj = 1
F(s) = Sum ( 1(sj < s) * pj)
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What is the kolmogorov distribution good for?
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Producing asymptotic confidence sets of F.
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