Tail Behavior Of Alternative Binary Links

Why Endpoint Rates Matter

A binary-response model maps a linear predictor to a probability between zero and one. Different links can agree over a central range of predictor values yet behave quite differently when the predictor is very negative or very positive. Their tail behavior describes how quickly a predicted probability approaches an endpoint, not merely which endpoint it approaches.

Write \(\eta\in\mathbb{R}\) for the linear predictor and \(p(\eta)\) for the fitted probability of the outcome coded as success. This tutorial compares the limits as \(\eta\to-\infty\) and \(\eta\to+\infty\). The comparison is useful when considering how a link represents extreme risks, but it does not by itself determine whether a link is appropriate for a particular study.

Inverse Links and Tail Notation

The inverse link takes a predictor value to a probability. For the probit link, let \(\Phi\) and \(\phi\) denote the standard normal cumulative distribution function and density. We use the following inverse-link conventions:

For a probability approaching zero, its lower tail is \(p(\eta)\); for a probability approaching one, the relevant tail is the failure probability \(1-p(\eta)\). The notation \(a(x)\sim b(x)\) means \(a(x)/b(x)\to1\) in the limit under discussion. We use \(O(r(x))\) for a remainder bounded in absolute value by a constant times \(r(x)\) for sufficiently large \(x\).

The Probit Tail: Gaussian Decay

The standard normal distribution is symmetric, so the two probit endpoint tails reduce to the same upper-tail calculation. For \(x>0\), define \(Q(x)=1-\Phi(x)=\Phi(-x)\). Its tail is asymptotic to the normal density divided by \(x\), a result often called Mills’ ratio.

Proof. Since \(\phi'(t)=-t\phi(t)\), for \(t\geq x>0\) we have \(1\leq t/x\). Integrating gives \(Q(x)=\int_x^\infty\phi(t)\,dt\leq x^{-1}\int_x^\infty t\phi(t)\,dt=\phi(x)/x\). For the lower bound, integration by parts yields \(Q(x)=\phi(x)/x-\int_x^\infty \phi(t)/t^2\,dt\). The remaining integral is at most \(Q(x)/x^2\), because \(t^{-2}\leq x^{-2}\) for \(t\geq x\). Therefore \(Q(x)\geq\phi(x)/x-Q(x)/x^2\), which rearranges to the stated lower bound. The ratio of that lower bound to \(\phi(x)/x\) is \(x^2/(x^2+1)\to1\), while the upper-bound ratio is \(1\). The squeeze theorem proves the asymptotic equivalence. \(\square\)

Taking \(x=|\eta|\), this result gives both probit tails: \(p_{\mathrm{probit}}(\eta)\sim\phi(|\eta|)/|\eta|\) as \(\eta\to-\infty\), and \(1-p_{\mathrm{probit}}(\eta)\sim\phi(\eta)/\eta\) as \(\eta\to+\infty\). Since \(\phi(x)=\exp(-x^2/2)/\sqrt{2\pi}\), the leading tail has Gaussian, or quadratic-exponential, decay.

Complementary Log-Log and Log-Log: Reversed Asymmetry

For the complementary log-log inverse link, let \(u=\exp(\eta)\). As \(\eta\to-\infty\), \(u\to0\), and \(1-\exp(-u)\sim u\). As \(\eta\to+\infty\), the failure probability is exactly \(\exp(-u)\). The log-log inverse link reverses these endpoint behaviors.

Derivation. For \(u>0\), the identity \(1-\exp(-u)=\int_0^u\exp(-t)\,dt\) implies \(\exp(-u)\leq[1-\exp(-u)]/u\leq1\), since \(\exp(-u)\leq\exp(-t)\leq1\) for \(0\leq t\leq u\). As \(u\downarrow0\), both bounds tend to one, proving \(1-\exp(-u)\sim u\). Substituting \(u=\exp(\eta)\) proves the lower-tail complementary log-log result. Its upper-tail failure probability follows directly from \(1-p_{\mathrm{cloglog}}(\eta)=\exp\{-\exp(\eta)\}\). The log-log inverse link obeys \(p_{\mathrm{loglog}}(\eta)=1-p_{\mathrm{cloglog}}(-\eta)\); substituting \(-\eta\) gives both log-log tails. \(\square\)

Thus the complementary log-log link has an ordinary exponential lower tail and a double-exponential upper failure tail. The log-log link has the reverse arrangement: a double-exponential lower tail and an ordinary exponential upper failure tail. “Double-exponential” here refers to expressions such as \(\exp\{-\exp(\eta)\}\), where the predictor is exponentiated inside the negative exponential.

The Cauchit Tail: Algebraic Decay

The cauchit inverse link is the cumulative distribution function of a standard Cauchy random variable evaluated at \(\eta\). Its tails decay much more slowly than the tails just considered. For \(\eta>0\), the failure probability can be written using the identity \(\arctan(\eta)+\arctan(1/\eta)=\pi/2\): \(1-p_{\mathrm{cauchit}}(\eta)=\arctan(1/\eta)/\pi\). By symmetry, for \(\eta<0\), \(p_{\mathrm{cauchit}}(\eta)=\arctan(1/|\eta|)/\pi\).

Derivation. For \(z>0\), \(\arctan(z)=\int_0^z(1+t^2)^{-1}\,dt\). For \(0\leq t\leq z\), \(1/(1+z^2)\leq1/(1+t^2)\leq1\). Integration gives \(z/(1+z^2)\leq\arctan(z)\leq z\). Dividing by \(z\), both bounds tend to one as \(z\downarrow0\), so \(\arctan(z)\sim z\). Set \(z=1/|\eta|\) in the lower tail and \(z=1/\eta\) in the upper failure tail to obtain the stated results. \(\square\)

Comparing the Four Tail Patterns

The comparison below records the small probability at each extreme. The entries are asymptotic equivalents, so their ratios to the corresponding tail probabilities tend to one.

The probit tails have Gaussian decay; the cauchit tails are algebraic. The complementary log-log and log-log links each combine an exponential tail at one end with a double-exponential tail at the other, in opposite orientations. These differences describe endpoint probabilities as functions of \(\eta\); they do not rank the links’ probability-scale slopes at finite predictor values.

Common Mistake and Practical Interpretation

A common mistake is to say that the complementary log-log and log-log links have the same tail behavior because each uses exponentials. That reasoning overlooks where the predictor appears: \(\exp(\eta)\) in one inverse link and \(\exp(-\eta)\) in the other. As a result, their fast and slow tails occur at opposite endpoints. The correction is to write both the success and failure probabilities explicitly, then take each limit separately.

Tail behavior matters when a fitted predictor can be far from the center of its range. A cauchit model retains comparatively more probability in its extreme tails, whereas a double-exponential tail approaches its endpoint exceptionally quickly. These are mathematical properties of the chosen inverse link, not guarantees about how observed outcomes behave. A link should be interpreted in the context of the model and data rather than selected from a tail comparison alone.