I do think it is even clearer if you amend two lines.
If you replace:
>How many of the N observations will be < x?
With: What are the EXPECTED number of claims < x ?
And:
>If we have k losses that are all < x, then k = N * Fn(x)
With: The empirical distribution function defined on page 61 states
that Fn(x)=(number of Xi<x)/N. Therefore the number of claims less
than Xi is estimated by Fn(x)*N.
>From: "Damon Lay" <dnhlay@earthlink.net>
>To: donald_morrison@es.adp.com (Donald
Morrison),studygroup4b@lists.casact.org
>Subject: Re: <no subject>
>Date: Wed, 21 Apr 1999 21:59:22 -0700
>
>Response to Donald Morrison.
>This item gave me fits until a friend and I sat down to work it
out. I
>apologize in advance if I go into too much detail, but you may find
it
>instructive to have the derivation of why a Chi-square is used. Here
>goes...
>Begin with the prob distrib F(x), the distribution of losses.
>It is important to note that F(x) is our underlying continuous
distribution
>that is unknown.
>Choose any loss amount x.
>For any other loss amount L, Prob (L < x) = F(x)
>Define Success as loss L < x
>Therefore we have a Bernoulli random variable: Prob (Success) = F(x)
>Prob (Failure) = 1 - F(x)
>Pick N events (i.e. you observe N losses) this implies N Bernoulli
trials
>This is a Binomial distribution
>Prob(k successes) = Combinatorial(N,k) * (F(x))**k * (1-F(x))**(N-k)
>How many of the N observations will be < x?
>(Number of losses) * (Prob that L < x) = N * F(x)
>Now if F(x) is unknown, what can we use?
>We can use the empirical distribution Fn(x) (formed from our
observations).
>N * Fn(x) is the number of observations < x.
>Remember that a loss < x was defined to be a "Success".
>If we have k losses that are all < x, then k = N * Fn(x)
>k = N * Fn(x) is distributed as a Binomial rand var.
>Now continue...
>N * Fn(x) is Binomial (N trials, Prob Success = F(x))
>Fn(x) is (1/N)Binomial (N trials, Prob = F(x))
>E(Fn(x)) = (1/N) * N * F(x) = F(x)
>Var(Fn(x)) = (1/N**2) * N * F(x) * (1-F(x)) = (1/N) * F(x) * (1-F(x))
>( Fn(x) - E(Fn(x)) ) / ( Var(Fn(x)) )**(1/2) is Normal(0,1) by C.L.T.
>( Fn(x) - F(x) ) / ( (1/N)(F(x))(1-F(x)) )^(1/2) is Normal(0,1) by
C.L.T.
>Square a Normal and you have a Chi-square. Square that thing above
and you
>have the foundation of equation (4.3) and an explanation of why the
weight
>function (4.4) was chosen. See page 135.
>I hope this is helpful.
>
>- Damon
>----------
>From: donald_morrison@es.adp.com (Donald Morrison)
>To: studygroup4b@lists.casact.org
>Date: Apr 21, 1999, 9:03
>
>
> can someone explain to me this thing it mentioned on page 135
hogg
>
>
> klugman Fn(X) is a binomial nFn(X)
>
>
> var(nFn(X)) = n**2 var(Fn(X)) = n * p * (1-p)
>
>
> var(Fn(X)) = p * (1-p) / n there have several questions that
use
>
>
> thisa concept
>
>
>
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