Notes-Part2 - Review Notes for Loss Models 1 - ACTSC...

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Unformatted text preview: Review Notes for Loss Models 1 - ACTSC 431/831, FALL 2008 Part 2 Severity Models 1. Severity models are distributions that are used to model the amount of a claim. 2. A parametric distribution is a set of distribution functions. Any member in the set is determined by one or more parameters. 3. A parametric distribution is called a scale distribution if cX is still a member of the set of the distributions whenever X is a member and c > 0 is a constant. Parameter is called a scale parameter if the parameter for cX is c , all other parameters (if any) are unchanged. 4. The distribution function F ( y ) of a random variable Y is called an n-point mixture distribution if F ( y ) = 1 F X 1 ( y ) + + n F X n ( y ) for all y, where j 0, F X j ( y ) is the distribution function of a random variable X j for j = 1 ,...,n , and 1 + + n = 1. (a) The survival function of the n-point mixture distribution is S ( y ) = 1 S X 1 ( y ) + + n S X n ( y ) for all y. (b) If F X j ( y ) has pdf f X j ( y ) for j = 1 ,...,n , then the pdf of the n-point mixture distribution is f ( y ) = 1 f X 1 ( y ) + + n f X n ( y ) for all y. (c) The k-th moment of the n-point mixture distribution is E ( Y k ) = 1 E ( X k 1 ) + + n E ( X k n ) . 5. The tail probability or tail of a random variable X or its distribution F is the probability that X exceeds a positive value t , or Pr { X > t } = 1- F ( t ) = S ( t ) , t > . (a) Comparison between the tails of two random variables: Suppose that X and Y have distribution functions F X ( x ) and F Y ( x ), respectively. Assume that lim x Pr { X > x } Pr { Y > x } = lim x S X ( x ) S Y ( x ) = c. If c = 0, we say that X ( F X ( x )) has a lighter tail than Y ( F Y ( x )); if 0 < c < , we say that X ( F X ( x )) and Y ( F Y ( x )) have proportional tails ; if c = , we say that X ( F X ( x )) has a heavier tail than Y ( F Y ( x )). 1 (b) If X has a heavier tail than Y , then exists a constant x > 0 such that for all x > x , Pr { X > x } > Pr { Y > x } . (c) By the LH opital rule, lim x Pr { X > x } Pr { Y > x } = lim x S X ( x ) S Y ( x ) = lim x f X ( x ) f Y ( x ) ....
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Notes-Part2 - Review Notes for Loss Models 1 - ACTSC...

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