The Wrong Model
# The Wrong Model
Statistical estimation requires choosing a model. The principled choice is the correct model — the distribution that actually generated the data. If the data come from a t-distribution, fit a t-distribution. Using the wrong model (say, a normal distribution when the data are actually heavy-tailed) introduces bias and should reduce precision. This is the textbook argument for model selection: the right model gives the right answer.
The authors of arXiv:2603.26862 (March 2026) prove that when the true distribution is a t-distribution with m degrees of freedom and the sample size is n, fitting the wrong model — the normal distribution — produces more precise estimates than fitting the correct three-parameter t-distribution, provided m ≥ 1.458√n. The threshold is exact. Below it, the correct model wins. Above it, the wrong model wins.
The mechanism is estimation efficiency versus model complexity. The t-distribution has three parameters (location, scale, and degrees of freedom). The normal distribution has two (location and scale). Estimating the extra parameter — the degrees of freedom that control the tail behavior — costs precision. When the true degrees of freedom are large enough that the tails are only slightly heavier than Gaussian, the information gained by correctly modeling those tails does not compensate for the precision lost in estimating the additional parameter. The wrong model is simpler, and simplicity wins when the difference between right and wrong is small.
The 1.458√n scaling is the precise point where the information content of the tails (which grows slowly with n) equals the estimation cost of the third parameter (which grows as √n). Below this threshold, the tails contain enough signal to justify the extra parameter. Above it, the tails are too Gaussian-like to pay for their own estimation.
The structural observation: using the right model is not always better than using the wrong one. The correct model extracts signal from the tails but pays for it with an extra parameter. When the signal is weak — when the data are almost but not quite normal — the payment exceeds the extraction, and the wrong model delivers better performance precisely because it ignores a real feature of the data.
