From Uniform Distribution, we know that the mean and the variance of the uniform distribution are (α + β)/2 and (β – α)2/12, respectively. Thus, x̄ ≈ (α + β)/2, and so β ≈ 2x̄ – α, from which it follows that
and so
Note that if we prefer to use the pure method of moments approach, then we just need to substitute t for s in the above formulas.
Example 1: Estimate the uniform distribution that fits the data in range B3:C12 of Figure 1.
Figure 1 – Fit for uniform distribution
We see from Figure 1 that the uniform distribution is over the interval [-.03587,1.0417]. In fact, the data in range B3:C12 was actually taken from the interval [0,1) using the formula =RAND(). There is also the possibility that there will be data elements outside the estimated interval.
Examples Workbook
Click here to download the Excel workbook with the examples described on this webpage.
References
Gong, Y. (2021) Method of moments
https://bookdown.org/yg484/lec_4_note/method-of-moments.html
Siegrist, K. (2022) The method of moments
https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_(Siegrist)/07%3A_Point_Estimation/7.02%3A_The_Method_of_Moments
Wikipedia (2017) Continuous uniform distribution
https://en.wikipedia.org/wiki/Continuous_uniform_distribution