# Is there a binomial regression model that captures data with fat tails?

Specifically, are there any binomial regression models that use a kernel with heavier tails and higher kurtosis than the standard kernels (logistic/probit/cloglog)?

As a function of the linear predictor $\textbf{x}'\mathbf{\hat{\beta}}$, the logistic distribution

• Underestimates the probability of my data being in the tails of the distribution
• Underestimates the kurtosis, or clustering of data, in the middle of the distribution:

This can be seen from a diagnostic plot of my fit:

• The red line is the logistic CDF, representing a perfect fit
• The black line represents the fitted probabilities from my dataset (calculated by binning observations into 0.1 intervals of $\textbf{x}'\mathbf{\hat{\beta}}$, where $\mathbf{\hat{\beta}}$ is obtained from my fit)
• The grey bars in the background represent number of observations on which the true probabilities are based upon
• The grey areas are where the tail 10% of the data lie (5% each side).

Ideally, any solution would use R.

## Edit

Why am I talking about CDFs? Our GLM equation is:

$$\mathbb{P}(Y = 1) = \mathbb{E}[Y] = g^{-1}(\textbf{x}'\mathbf{\beta})$$

Where $g$ is the link function.

Further, if $g^{-1}$ is a valid probability distribution (i.e. monotonically increasing from 0 to 1, indeed the case with probit, logit, cloglog), then consider a latent (not directly observed) continuous random variable $Y^{*}$ whose distribution (CDF) is given by $g^{-1}$. Then by definition

$$\mathbb{P}(Y^{*} \leq \textbf{x}'\mathbf{\beta}) = g^{-1}(\textbf{x}'\mathbf{\beta})$$

Equating the two equations above, we see the probability of $Y=1$ is exactly equal to the CDF of $Y^{*}$

$$\mathbb{P}(Y = 1) = \mathbb{P}(Y^{*} \leq \textbf{x}'\mathbf{\beta})$$

Hence I talk interchangeably about the expected response $\mathbb{E}[Y]$ and CDF of $Y^{*}$ over linear-predictor ($\textbf{x}'\mathbf{\hat{\beta}}$) space.

• I believe you're confusing the CDF with the relationship between your predictor variable and the probability of a 1 in your binomial data. I don't think any of your lines represent a CDF. The black line definitely can't tell you anything directly about the CDF. – John Sep 3 '14 at 15:00
• You are confused; a binomial random variable cannot have fat tails! (it doesn't have tails at all) – kjetil b halvorsen Sep 3 '14 at 15:17
• As John suggests, you're confounding possible lack of fit with the appearance of the marginal distribution. Investigate it instead as what it is - possible lack of fit. – Glen_b -Reinstate Monica Sep 4 '14 at 3:52
• I see you edit, if your model is equivalent to a CDF then how is the line ever descending? CDF's cannot do that. I'm not saying you're not trying to model a CDF. I'm suggesting your casual equivalence to one isn't warranted so you should word your question more specifically. – John Sep 4 '14 at 12:23

\begin{align} y_i & = \begin{cases} 1 & \text{if } z_i > 0 \\ 0 & \text{if } z_i < 0 \end{cases}\\ z_i & = X_i\beta + \epsilon_i \end{align}
where the latent errors $\epsilon$ are distributed as $$\epsilon_i \sim t_\nu \left(0, \frac{\nu-2}{\nu} \right)$$
with $\nu > 2$ estimated from the data.
This is also called robit regression. In particular see pp.124-125 of Gelman & Hill's Data Analysis Using Regression And Multilevel/Hierarchical Models (what a mouthful); the above is eqn 6.15 in the book. You can fit it in R using package stan (which is also by Andrew Gelman and his collaborators and goes with the book), list and maybe others. You could also write code to maximise the log-likelihood directly using optim or nlminb; I haven't tried it, but it shouldn't be hard.