# STATS 730: Lecture 3 - PowerPoint PPT Presentation

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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