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I'm faced with a situation where there are few observations of a life insurance product in some cells, where the observations are deaths. The problem is that I need to calculate standard deviations for the aggregate face amounts for all cells for use in other purposes. One of my colleagues has calculated that we need 200 death claims per cell for the aggregate face to be normally distributed.

How should I deal with the cells that are much lower than 200? Is it all right to use bootstrapping in this circumstance? The aggregate standard deviation needs to be reasonable, otherwise I'll have to exercise "actuarial judgement" aka judgmental adjustments, which we're trying to move away from.

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  • $\begingroup$ You might need to provide more information about the analytic approach that you intend to use after the bootstrap. Sure you could bootstrap cells lower than 200 until they hit 200, but the number of bootstrapped values you would need before you hit 200 would depend on the particular run of the bootstrap you stumbled into. How is that any better than simply inflating the number of deaths and the number of cases by ($200/N_{obs}$)? $\endgroup$ Commented Sep 17, 2014 at 18:41
  • $\begingroup$ If I just inflated to number of cases, the added cases wouldn't be independent of the original ones. $\endgroup$
    – JenSCDC
    Commented Sep 17, 2014 at 18:43
  • $\begingroup$ As for what I've going to use the results for, it's very simple- I need to calculate a percentage of the mortality rates that's high enough so that there's a good change actual mortality won't exceed expected by more than a certain margin. $\endgroup$
    – JenSCDC
    Commented Sep 17, 2014 at 18:47
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    $\begingroup$ Whether or not the approximating distribution for aggregate face amounts is normally distributed remains to be established as relevant for any analyses you propose to conduct. I suspect it may be useless, or if its' not, that another type of modeling approach would do much better. Unlike the name suggests, bootstrapping does not magically fix everything. I like Tukey's suggestion of calling it the shotgun, "...blows the head off any problem you feed it, provided you can put the pieces back together." $\endgroup$
    – AdamO
    Commented Sep 17, 2014 at 18:47
  • $\begingroup$ The added cases aren't independent in a bootstrap either because they are just drawn at random from the observed distribution. That is, unless you plan to carry out your bootstrap of this value and then go on and bootstrap the entirely subsequent analysis... which leads us back to "provide more information about the analytic approach". $\endgroup$ Commented Sep 17, 2014 at 18:47

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I am not certain where your assumption of 200 deaths per cell comes from, I have certainly never encountered such reasoning before. I think using a log linear model (Poisson regression) for your life table will give you adequate estimates and confidence intervals for face amounts. You can simply do a quick simulation to verify this, as it would be no less complicated than actually obtaining bootstrapped estimates... which should agree very closely anyway.

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  • $\begingroup$ I think he used simulation to come up with the 200 figure. In this case Poisson regression would be overkill, as all I need to do is calculated a margin that's a percentage of our expected mortality. $\endgroup$
    – JenSCDC
    Commented Sep 17, 2014 at 18:56
  • $\begingroup$ In what sense would it be overkill? Also, the simulation is to assess sufficient normality of a (something... mean? prediction?) when you're modeling count data.. which are inherently nonnormal. Yes in big $n$, means are normal by the CLT, but in small samples exact distributions work much better. It's like using a normal approximation for a small Bernoulli trial (say n=10) CI for $p$ when you can just use the 0.025 and 0.975 quantiles from a binomial($10, p$) RV. $\endgroup$
    – AdamO
    Commented Sep 17, 2014 at 19:33
  • $\begingroup$ a) The normality is of the aggregate face amount of the policies which have died. b) it doesn't have to be overly accurate. One big reason is that our tables of expected mortality (which my analysis is based on) has been over fitted by managers so many times over the years that this is GIGO. $\endgroup$
    – JenSCDC
    Commented Sep 17, 2014 at 19:51
  • $\begingroup$ You have not addressed the question why normality is of interest. Models do not just exist on their own, the intended application has far reaching implications about the kinds of assumptions for their successful use. Except in very contrived scenarios is normality actually of any interest. $\endgroup$
    – AdamO
    Commented Sep 17, 2014 at 21:18
  • $\begingroup$ Good point. I'll get to it when I have some more time. $\endgroup$
    – JenSCDC
    Commented Sep 17, 2014 at 21:39

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