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Last active April 2, 2019 13:58
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vec space def and ex

  • set which allows linear combination of vectors to form another vector in the set

basis vecotrs def

  • vectors which span vec space

linear combination

linear span

  • all possible linear combination

linear transformation

  • L(ax+by)=aL(x)+bL(y)

centering operation as idempotent operation

  • I-(1/n)ii'



partitioned matrices

  • make sure dimension consistency

vec operator

  • transform a matrix into vector

In linear algebra, a symmetric (n x n) real matrix M is said to be positive definite if the scalar z'Mz is positive for every non-zero column vector z n real numbers.

symmetric with positive eigenvalues is positive definite

Hessian matrix :second derivative symmetric matrix

symmetric matrix has real eigenvalues and real orthogonal eigenvectors


  • spectral composition symmetric A=C$\Lambda$C'

  • diagonalize symmetric A, CAC' = $\Lambda$

  • rank, tr, det is the same as A

  • easy to multiply

for non-singular square matrix

  • $A^{-1}$ exist

  • det(A) != 0

  • columns independent

  • full rank


probability space

  • sample space S

    • set of elements
  • power set ($\sigma$ field)

    • set of all subset of S
  • probability measure

    • P: power set -> [0,1]


  • f(x,y) = f(x)·f(y)

no correlation

  • cov(x,y) = 0 or corr(x,y) = 0

  • indep -> var = 0

moment generating func

  • $E(e^{tx})$

  • E(x) = M'(0), E(x^2) = M"(0)

normal distribution



good estimator

  • unbias

  • efficiency

    • var(theta1) < var(theta2)

    • exist cramer-rao low bound

  • sufficiency: statistics can convey same information as parameter

  • asymptotic consistency: convergence in probability

  • asymptotic normality

method of estimation

  • LSE

  • GMM: combine 2nd and 4th moment

  • MLE: choose theta to max L(theta, x) = joint pdf of f(x;theta)


parametric statistical steps

  1. set model x~f(x;theta)
  2. sample
  3. estimate
  4. test
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