For known location, we can find the scale parameter of a normal distribution by calculating the sum of squared differences to the location, then dividing by n-1 and taking the square root. This is the maximum-likelihood and most-efficient scheme for parameter estimation for the normal distribution.
However, for the double-exponential distribution (Laplace) using the sum of absolute values is the MLE scheme. So the power p is not 2 but 1! And for the uniform distribution the maximum difference to the center gives the most-efficient parameter estimation scheme. Also that can be described as sum of power, by let the power p go to infinite (Chebyshev norm).
So for these 3 distributions with $p=1,2,\ldots,\infty$ we know both the MLE and the PDF well.
Question: If we generalize the estimation scheme by using any positive power (like p=4, 6, or whatever), what is the according distribution PDF(p,x)?
This would lead to quite a general PDF and can form a generalized normal distribution (like the Student-t extends the normal for more long-tailed distributions).