Lecture 1- Parameter Estimation

# Lecture 1- Parameter Estimation - Parameter estimation...

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Parameter estimation ActSCi 432/832 1 / 31

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Distributions Summary X is Binomial with parameters n (positive integer) and p (0 , 1) p . f . f ( x ) = ± n x ² p x (1 - p ) n - x for x = 0 , 1 , 2 , 3 , ...... n E ( X ) = np , Var ( X ) = np (1 - p ) X is Bernoulli with parameters p (0 , 1) X is Binomial with n = 1 X is Poisson with parameter λ > 0 p . f . f ( x ) = λ x e - λ x ! for x = 0 , 1 , 2 , 3 , ...... E ( X ) = λ, Var ( X ) = λ 2 / 31
Distributions Summary X is Gamma with parameters α > 0 and β > 0 p . f . f ( x ) = x α - 1 e - x Γ( α ) β α for x > 0 E ( X k ) = β k Γ( α + k ) Γ( α ) , if k > - α Note that Γ( α + 1) = α Γ( α ) for α > 0 and Γ( α + 1) = α ! for α = 0 , 1 , 2 , .... X is Exponential with parameters β > 0 X is Gamma with α = 1 X is Pareto with parameters α > 0 and β > 0 p . f . f ( x ) = αβ α ( x + β ) α +1 for x > 0 E ( X ) = β α - 1 for α > 1 , Var ( X ) = αβ 2 ( α - 1) 2 ( α - 2) for α > 2 3 / 31

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Introduction Uncertainties in ﬁtting actuarial loss data 1 Model uncertainty 2 Parameter uncertainty given the model 3 Sampling error This chapter is about the second item above. Example: When you have somehow concluded that Exponential distribution is suitable for the given data, how do you obtain the parameter β ? Several popular methods available with diﬀerent implications 4 / 31
Method of moments estimation (MME) Consider an iid sample { x 1 , ..., x n } of size n from the common cdf F ( x | Θ) Θ = ( θ 1 , ..., θ p ) is the vector of unknown parameters. Eg, for the normal distribution, Θ = ( μ, σ ) Write μ 0 k (Θ) := E ( X k | Θ) A method of moments estimate of Θ is any solution of the p eqns below: μ 0 k (Θ) = ˆ μ 0 k = 1 n n X 1 x k i , k = 1 , ..., p . (1) Example (Gamma) Determine the method of moments estimate of Θ = ( α, β ) from a sample of size n generated from a Gamma distribution. 5 / 31

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Percentile matching estimation (PME) The q -th percentile π q of F ( x | Θ) is deﬁned from F ( π q | Θ) = q , 0 < q < 1 (2) Note that π q is a function of Θ, so π q (Θ) would be more accurate expression A percentil matching esimtate of Θ is any solution of the p eqns below: π q k (Θ) = ˆ π q k , k = 1 , ..., p , (3) where ˆ π q k are sample percentiles Perhaps more readable version is obtained by applying F to both sides: q k = F π q k | Θ) (4) Idea is to match the theoretical percentiles to the empirical (sample) ones, using p diﬀerent points 6 / 31
Sample quantiles ˆ π q k may not be uniquely deﬁned due to its discreteness. Thus we use a midpoint approach. The smoothed empirical estimate of a

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## This note was uploaded on 02/08/2012 for the course ACTSC 432 taught by Professor Davidlandriault during the Fall '09 term at Waterloo.

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Lecture 1- Parameter Estimation - Parameter estimation...

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