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STATS 730: Lecture 3

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Tangent at point a is the function. f(a) (x-a) f'(a) ... ith smallest observation must be in here. x. x h. 9/23/09. 730 lecture 3. 13. Order Statistics(cont) ... – PowerPoint PPT presentation

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Title: STATS 730: Lecture 3


1
STATS 730 Lecture 3
More Introductory Stuff
2
Todays lecture
  • Taylor Series
  • Order Statistics
  • Examples

3
Taylor Series
4
Taylor approximation
f(a) (x-a) f(a)
f(x)
  • Tangent at point a is the function
  • f(a) (x-a) f(a) approximates curve f(x) if
    x is
  • close to a

5
Taylor series(2)
  • Alternative form

6
Taylor series (3)
  • Generalization

7
Multinomial distribution
8
Multinomial (cont)
  • Out of n trials
  • Y1 are A
  • Y2 are B
  • Y3 are C
  • Probability is

9
Order statistics
  • Sample X1,Xn
  • Arrange in ascending order
  • X(1) , X(2) ,,X(n)
  • X(1) smallest, X(n) largest
  • What is density of X(k)?

10
Order statistics(cont)
  • Divide observations X1,Xn into 3 groups
  • A those x,
  • happens with prob F(x)
  • Bthose with xltXi xh,
  • happens with prob F(xh)-F(x)
  • C those ³xh,
  • happens with prob1-F(xh)
  • Let Y1, Y2,Y3 be the counts for A,B,C.

11
Order Statistics (cont)
  • Will calculate density fi(x) of X(i) as
  • Observe that xltX(i) xh iff
  • Y1lti, and
  • Y1Y2³i.

12
Order Statistics (cont)
  • ith smallest observation must be in here

x
xh
13
Order Statistics(cont)
  • To get P(xltX(i) xh), we have to add up all the
    multinomial terms corresponding to these ys
  • To get the required density, we must then divide
    by h, and let h0
  • Terms with y2gt1 converge to 0
  • Leaves only terms with y1i-1, y21,y3n-i.

14
Order Statistics(final!!)
  • Result is

15
Example
  • Order statistics from Uniform distribution
  • F(x)x, f(x)1

Beta(i,n-i1) distribution
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