# Light tailed symmetric distribution

Is there a family of distributions that resemble the normal distribution (symmetric, spanning all real numbers, and approximately bell-shaped) but have lighter tails than normal distribution?

I'm looking for a suitable prior for psychological trait, but I want to penalize outliers more strictly than is usual for the normal distribution.

The school organises written entrance examinations. The test has many parallel versions, so it will be scored with an IRT model using anchor items. The Bayesian parameter estimation is used for many reasons.

My employer has a requirement that the score a student receives on the test must be an integer between 0 and 100. Therefore, I linearly transform the parameter estimate $$\theta$$, round, and winsorize the values that lie outside the interval $$[0,100]$$.

A histogram of the resulting score may look like this (not real data - simulated ideal case):

What I don't like about this solution is that the vast majority of the values (about 95%) are concentrated in the interval $$[25,75]$$ - so half of the range is practically unused.

If I adjust the standard deviation of the data higher, then the solution changes like this:

Again I am not satisfied, this time because of the cumulation of values at both extremes.

I was thinking (and this is why I asked the question) that it might help if the estimate of the $$\theta$$ parameter came from a prior that has lighter tails. The data would then not contain so many outliers that leave the desired interval.

Ideally, I would imagine something like this:

• This is such a weak, vague condition that you would be better served by posing the problem you actually face. What statistical procedure are you contemplating and how does it "penalize outliers"?
– whuber
Commented Apr 16, 2023 at 15:38
• @whuber: Thanks for the reply, I have added the technical details. Commented Apr 17, 2023 at 15:13

$$P(x) \propto \exp(-|(x-\mu)/s|^\theta))$$
for $$\theta>2$$.
You could also use a scaled Beta distribution, which would have no probability outside of the range, e.g. $$\textrm{Beta}(3,3)$$ scaled by a factor of 100.