Basic Concepts
Since the pdf for the uniform distribution on [α, β] is
the likelihood estimate for a random sample {x1, …, xn} is
provided that all the sample elements are in the interval [α, β] and 0 if not. Suppose that the random sample is in increasing order x1 ≤ …≤ xn. Thus, to maximize L, we need to ensure that α ≤ x1 ≤ xn ≤ β. But the larger β is, the smaller L is, and the smaller α is, the smaller L is. Thus, to maximize L, we need α = x1 and β = xn.
Property
Note that α = x1 and β = xn are biased estimates. If we fix α, we can find an unbiased estimator for β of the uniform distribution in the interval [α, β], as follows.
Property 1: z = α + (n + 1)(xn – α)/n is an unbiased estimator for β, assuming we know α. Similarly, if we fix β, we can find an unbiased estimator for α of the uniform distribution in the interval [α, β], as z = β + (n + 1)(x1 – β)/n.
When α and β are both unknown, we will use x1 + (n + 1)(xn – x1)/n as a quasi-unbiased estimator for β and xn + (n + 1)(x1 – xn)/n as a quasi-unbiased estimator for α. We say “quasi” since neither α nor β is known.
We can also iterate this process as follows. Convergence is quite rapid.
Example
We now repeat Example 1 of Method of Moments: Uniform Distribution using the MLE approach as shown in Figure 1.
Figure 1 – Fitting a uniform distribution using MLE
The fit using the MLE approach for the uniform distribution is the interval [.004308,99923] as shown in range F7:F8. Range G7:G8 shows a quasi-unbiased version, and J7:J8 shows the iterative version.
Note that cell G7 contains the formula =F8+(F4+1)*(F5-F8)/F4 and cell G8 contains the formula =F7+(F4+1)*(F6-F7)/F4.
Links
Examples Workbook
Click here to download the Excel workbook with the examples described on this webpage.
References
Forbes, C., Evans, M., Hastings, N., Peacock, B. (2011) Statistical distribution. Wiley
https://www.academia.edu/49056503/Statistical_distributions
Siegrist, K. (2022) Maximum-Likelihood
https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_(Siegrist)/07%3A_Point_Estimation/7.03%3A_Maximum_Likelihood
Millard, S. P. (2023) Estimate parameters of a uniform distribution
https://search.r-project.org/CRAN/refmans/EnvStats/html/eunif.html