STATS 730: Lecture 3

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STATS 730 Lecture 3
More Introductory Stuff
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Todays lecture

• Taylor Series
• Order Statistics
• Examples

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Taylor Series
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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

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Taylor series(2)

• Alternative form

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Taylor series (3)

• Generalization

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Multinomial distribution
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Multinomial (cont)

• Out of n trials
• Y1 are A
• Y2 are B
• Y3 are C
• Probability is

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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)?

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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.

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Order Statistics (cont)

• Will calculate density fi(x) of X(i) as

• Observe that xltX(i) xh iff
• Y1lti, and
• Y1Y2³i.

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Order Statistics (cont)

• ith smallest observation must be in here

x
xh
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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.

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Order Statistics(final!!)

• Result is

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Example

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

Beta(i,n-i1) distribution