Say you had a sample of ages from a population, but the ages are in buckets...Such as <1, 1-4, 5-14, 15-24, ..., 55-64, 65+...And you want to get an estimate for the average of the age distribution and get a confidence interval for the mean.

I know if you assume a parametric distribution you could just estimate by maximum likelihood, using likelihoods over the given interval. Simple. But what if you have no idea what age distribution to use? Is there some kind of non-parametric approach that could be used?

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    $\begingroup$ The problem with any approach nowadays pertains to estimating the upper tail. It used to be that age distribution data terminated at 65+ or perhaps went out to 85+, but in many areas there is a sizable subpopulation in that upper group--and it influences the estimated mean. Thus, an approach that exploits additional information, even if it's only in the form of a very flexible parametric family, is going to be your best bet. $\endgroup$ – whuber Jan 30 '20 at 15:02
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    $\begingroup$ @whuber, thank you. Do you have any suggestion for the kind of "very flexible parametric family" that may be appropriate? $\endgroup$ – Do not reinstate Monica Jan 30 '20 at 16:19
  • $\begingroup$ Seeing that your age groups are fairly broad and that many age distributions are multimodal, the most promising approach might be to find a more detailed age distribution for a related population and emulate that one. If you get close in the qualitative shape, the mean will likely be most sensitive to how you model the upper tail, so that's where you might contemplate (say) a one or two parameter of distributions with varied tail behavior. $\endgroup$ – whuber Jan 30 '20 at 16:52

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