# Estimating a distribution based on three percentiles and a mean

I have a set of percentiles the 10-th, 50-th and 90-th. Furthermore I have the mean value. I am trying to reconstruct the underlying distribution. The question is similar to

Estimating a distribution based on three percentiles

except I have a mean value as well.

My present approach was to fit a lognormal distribution based on the 50th percentile in which the variance was an unknown. Subsequently I determined the variance_1 based on the 10th percentile keeping the mean as found from the 50th. I did the same for the 90th percentile giving me variance_2. Then I searched for the variance somewhere between variance_1 and variance_2 which gave me the mean I was given in the first place.

10-th percentile: 0.004
50-th percentile: 0.007
90-th percentile: 0.02
arithmetic mean : 0.009

I came up with a log-normal (\mu = log(0.007), \sigma = 0.70825) distribution

What do you think of the approach, and how would you tackle this problem?

• I would just like to remind you of the cautionary notes included in my answer to the other question, because they apply here, too. How I would tackle your problem would depend strongly on why you are fitting a distribution to these values and on the size of the dataset they are estimated from. Regardless, neither ad hoc methods (such as yours) nor "standard" methods such as a least-squares fit are truly appropriate due to the correlations among these statistics and the different levels of error that occur in measuring them.
– whuber
Commented Jun 26, 2014 at 3:48
• Thanks @whuber. I realise that this is not a lot of data to make strong conclusions about the underlying data distribution. Commented Jun 26, 2014 at 4:10
• It could be a huge amount of data, willem, as summarized with four statistics. So the amount of data isn't the issue. The keys to focus on in explaining your problem are (1) what is the purpose in fitting a distribution to these statistics and (2) what is the basis for assuming a lognormal distribution?
– whuber
Commented Jun 26, 2014 at 14:00
• Are these statistics (the quantiles and mean) based on the usual samples estimates of quantiles and mean, or were they inferred from full data and estimation of lognormal parameters? What sort of sample sizes are they based on? Commented Jun 26, 2014 at 22:36
• If we had lognormal data, and those quantiles were sample quantiles, and I only had the three quantiles, I'd use $k_n \log(x_{0.9}/x_{0.1})$ to estimate $\sigma$ (where $k_n$ is a scaling constant that will change with $n$ but in large samples will approach $1/2.56$), and I'd estimate $\mu$ by $\frac{1-a_n}{2} \log(x_{0.1})+ a_n \log(x_{0.5})+ \frac{1-a_n}{2} \log(x_{0.9})$, where the best choice of $a_n$ varies with $n$, but $a_n=0.5$ is very close to the optimum for both large and small $n$ (so I'd probably stick with that). Adding the mean (assuming it's a sample mean) complicates it Commented Jun 28, 2014 at 10:49