Statistical inference form observational data

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Statistical inference form observational data

• Parameter estimation
• Method of moments
• Use the data you have to calculate first and
  second moment
• To fit a certain distribution, use relation to
  moments formulae
• Method of maximum likelihood
• (too difficult)
• Interval estimation confidence interval

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Method of moments

• Suppose you have 10 data about x
• 0.3, 4, 5, 1, 1.3, 6.5, 0.85, 2.5, 4.56, 3.14
• After calculation, mean 2.915, var 4.2981

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Method of moments

• Suppose we want to fit with uniform,
• Now
• Solving,
• b 6.5142, a -0.684

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Method of moments

• Suppose we want to fit with normal,
• Now E(X) 2.915 µ
• Var(X) 4.2981 s2
• N (2.915, 2.07) is suitable
• Try Lognormal yourself

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Confidence interval of µ

• To calculate confidence interval, you need to
  know
• 1) One sided / two sided?
• 2) (true) variance known / unknown?
• Normal student-t

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Confidence interval of µ- one sided

• Suppose you have 25 samples, sample mean 9,
  sample s.d. 2. Assume sample s.d. true s.d.
• (why confidence interval?)
• P (True mean) lt 10?

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Confidence interval of µ- one sided
true mean is smaller than a certain value with
probability 0.98?
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Confidence interval of µ- two sided
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Confidence interval of µ
Compare k0.02
k0.025 k1-a
k1- a/2 a0.02
a0.05 k depends on 1)
Confidence level a you want 2) One sided / two
sided
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Confidence interval of µ- student-t

• When the true variance is unknown, we use t and
  sample variance
• Suppose you have 25 samples, sample mean 9,
  sample s.d. 2
• You keep everything the same but just check on
  another table!
• To check t, you need 1) confidence level, 2)
  d.o.f.

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Confidence interval of µ- student-t

• true mean is smaller than a certain value with
  probability 0.98?
• T depends on
• 1) Confidence level a you want
• 2) One sided / two sided
• 3) Degree of freedom

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Confidence interval of µ
Compare k0.02
k0.025
T0.02, 24 k1-a
k1- a/2
T1- a, 24 a0.02 a0.05
a0.02 compare 1.96 and 2.2066
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Variance of variance?

• As you only have limited data points, your sample
  variance will also subject to variation JUST AS
  variation of sample mean
• Example DO data n 30, s2 4.2

• To check chi-square, you need
• Probability level a
• d.o.f.
• We usually construct one-sided confidence
  interval of variance (why?)

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Goodness of fit test of distribution

• Probability Paper (old)
• Chi-square test (?2) (common)
• Kolmogorov-Smirnov test (K-S) (difficult to use)
• Chi-square test
• ei
• No. of parameters in the model