Insurance 260 Estimation ReviewPierre Lemaire - PowerPoint PPT Presentation
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Insurance 260 Estimation ReviewPierre Lemaire
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Maximum Likelihood Estimation. Bayesian Estimation. Matching. Method of Moments ... As n , the probability that the likelihood equation has a solution goes to 1 ... – PowerPoint PPT presentation
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Title: Insurance 260 Estimation ReviewPierre Lemaire
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Insurance 260 Review NotesFrequency Estimation
- Methodologies
- Maximum Likelihood
- Poisson Example
- Negative Binomial Example
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Methodologies
- Matching
- Optimization-Based Estimators
- Maximum Likelihood Estimation
- Bayesian Estimation
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Matching
- Method of Moments
- Percentile Matching Method
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Example
- Negative Binomial
- p(k)
- ?1 r?
- ?2 r? (1 ?)
- ?
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Optimization-Based Estimators
- Minimum distance estimators
- Define order statistic x(0) ? x(1) ? ? x(n)
- then Gn(x) j 1, , n
- Cramér-von Mises estimators
- G (x?) F (x?) and wj1
- Anderson-Darling estimation
- G (x?) F (x?) and wjF (x?) 1- F (x?)-1
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Maximum Likelihood Estimation
- The likelihood function
- Log-likelihood function
- 0 for all i
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MLE for Truncated Data
- Suppose that we have truncated data such that
- where n1 n2 n
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Example Poisson Distribution Fit
- Details
- p(k)
- L(?)
- l(?)
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Poisson Distribution Fit
- 1 Parameter - ?
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ExampleNegative Binomial Distribution
- Details
- p(k)
- l(?)
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Neg. Binomial Fit
- 2 Parameter r, ß
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Information Matrix
- Fishers Information Matrix
- Theorem 1 Variance of MLE
- Assuming that f(x?) satisfies some conditions,
then - As n ? ?, the probability that the likelihood
equation has a solution goes to 1 - As n ? ?, the distribution of the maximum
likelihood estimator converges to a normal
distribution with mean ? and variance such that
I(?) Var(?) ? 1
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Example The Lognormal Distribution
- Let ln x N(?, ?2) ?
- and
- and
- and
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Example The Lognormal Distribution
- Therefore
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