| Mean excess function |
| Written by theoretic |
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Let Y be a non-negative random variable with finite mean, distribution F and xi = inf{x : F(x) > 0} and xr = sup{x : F(x) < 1}. Then its mean excess or mean residual life function is given by ep(u) = E(Y — и | Y > и), и G (xi,xr). For our purposes, we mostly consider distributions on [0, oo) which have support unbounded to the right. The quantity ep(u) is often referred to as the mean excess over the threshold value u. In an insurance context, ep(u) can be interpreted as the expected claim size in the unlimited layer, over priority u. Here ep(u) is also called the mean excess loss function. In a reliability or medical context, ep(u) is referred to as the mean residual life function. In a financial risk management context, switching from the right tail to the left tail, ep(u) is referred to as the expected shortfall. The mean excess function of the distribution function F can be written in the form 1 Г°т^ , г ej?(«i)= ( t (y) ay , eq0,ir). (3.2.10) Fu) u This formula is often useful for calculations or for deriving theoretical properties of the mean excess function. Another interesting relationship between eF and the tail F is given by — eF(0) ( fx 1 1 t (x)= ——— exp — ——— ay , x > 0 . (3.2.11) eF(x) o eF(y) Here we assumed in addition that F is continuous and F(x) > 0 for all x > 0. Under these additional assumptions, F and eF determine each other in a unique way. Therefore the tail F of a non-negative distribution F and its mean excess function eF are in a sense equivalent notions. The properties of F can be translated into the language of the mean excess function eF and vice versa. Derive (3.2.10) and 3.2.11 yourself. Use the relation FY = L F(Y > y) dy which holds for any positive random variable Y. Example 3.2.7 (Mean excess function of the exponential distribution) Consider Y with exponential Exp(A) distribution for some A > 0. It is an easy exercise to verify that eF(u)= A-1, и > 0 . (3.2.12) |