Suppose you have one variable, $x$, with 8 data points, with a sample mean of 60%, and a sample standard deviation of 7%, and let’s also assume you know the sample comes from a lognormal distribution (or from a distribution with a heavier tail than lognormal).

A sample with mean = 60%, SD = 7%, could be produced by an underlying distribution with, say, mean = 55%, SD = 10%; or it could be produced by a true mean of 53% and SD = 9%; or… Any number of other combination of mean and SD could produce the data we see.

Now, I looked at true means running from 30% to 100% and SD’s running from 1% to 50% (all in 1% increments) – in other words, 3,550 combinations. For each combination, I simulated 10,000 groups of 8 data points from a lognormal distribution, calculated the mean and SD of those 8 points, and figured out how many of the 10,000 were “close” to my sample mean of 60% and SD of 7%. I defined “close” as being within 0.5% either way – in other words, if the simulated mean was between 59.5% and 60.5%, and the SD was between 6.5% and 7.5%, I assumed it “close.”

Let’s suppose with a true mean of 53% and true SD of 9%, 10 out of the 10,000 simulations were “close” to my sample. After doing that for each of the 3,550 combinations, I had a distribution of the likelihood that each combination is really the true underlying combination.

Next, given the likelihood of each combination, we can ask what is the probability that $x$ is, say, 70% or less. Again, we run through each of the 3,550 combinations, and get the answer. For example, if the true mean is 40%, with a 15% SD, that probability is, say, 0.903; if the true mean is 65% with a 12% SD, the probability is, say, 0.539. If those were the only 2 combinations, and they were equally likely, then we’d say the probability of $x$ being 60% or less is, say, (0.903+0.539)÷2 = 0.721.

So my question is do you think that these calculations make sense? If no, please correct me, if yes, can you please provide me with any other way that is more neat than this one. Thanks.

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    $\begingroup$ At first glance this seems like a very roundabout and probably inefficient way of performing parameter estimation; can you give any intuition as to why you might prefer a simulation-based approach over e.g. maximum likelihood estimation? $\endgroup$ – bnaul Jul 12 '12 at 18:43
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    $\begingroup$ @bnaul I agree in the sense that it sounds like a conceptually sound but computationally incredibly inefficient way to compute a confidence set for the parameters. $\endgroup$ – whuber Jul 12 '12 at 18:47
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    $\begingroup$ This looks to me like an approximation to a Bayesian analysis, with discrete uniform priors on the mean and standard deviation, and the likelihood calculated via simulation. $P(X<x)$ is essentially the marginal distribution of $X$ after the parameters have been integrated out. I agree, inefficient. $\endgroup$ – jbowman Jul 12 '12 at 19:16
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    $\begingroup$ Is this related to your previous question? If so, how? (You state here that you have 8 data, and there you had 5.) In both cases you seem to only be working with proportions. Are you sure your data are lognormal? Eg, can they go from -Inf to +Inf, or only (0,1)? I wonder if you should look into the Beta distribution, which can only include values in (0,1), but can take on many shapes, including skewed. $\endgroup$ – gung - Reinstate Monica Jul 12 '12 at 19:46
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    $\begingroup$ This looks very much like Approximate Bayesian computation... $\endgroup$ – Elvis Jul 12 '12 at 19:53

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