Logistic Quantiles and Exponential Tail Decay

From probabilities to thresholds

For a continuous distribution, a quantile answers an inverse question: what value is exceeded with a specified probability, or what value is reached with a specified cumulative probability? For the logistic distribution, both questions have explicit answers. Those formulas are especially useful in the tails, where a small change in the target probability can move the quantile substantially.

We use the location-scale logistic distribution with location \(\mu\in\mathbb{R}\) and scale \(s>0\). Its cumulative distribution function is \(F(x)=1/(1+\exp(-(x-\mu)/s))\). The survival function is \(S(x)=P(X>x)=1-F(x)\); continuity means \(P(X>x)=P(X\geq x)\).

The ratio \(p/(1-p)\) is the odds corresponding to cumulative probability \(p\). Thus the quantile is the location plus the scale times the log-odds. For an upper-tail probability \(\alpha\), where \(0<\alpha<1\), the threshold exceeded with probability \(\alpha\) is

This form is often convenient for rare events: specify the probability of exceedance directly, then calculate the threshold. The survival function also gives the probability of exceeding any specified threshold without first calculating a CDF.

Deriving and checking the quantile formula

The quantile expression follows by solving the CDF equation, and strict monotonicity guarantees that the solution is unique. We show both points so that the inversion is clear.

The survival formula follows directly by subtraction: \(S(x)=1-F(x)=1/(1+\exp((x-\mu)/s))\). As a check, substituting \(x=Q(p)\) gives \(\exp((x-\mu)/s)=p/(1-p)\), and therefore \(S(Q(p))=1-p\). This agrees with the interpretation of \(Q(p)\) as the threshold exceeded with probability \(1-p\).

Worked quantile and tail calculations

Exponential decay in both tails

The logistic tails decay exponentially in distance from the location. The scale \(s\) determines the rate: increasing \(s\) makes the exponential factor decrease more slowly as a function of the distance measured in the original units.

For the right tail, put \(z=(x-\mu)/s\). When \(x\) tends to \(+\infty\), \(z\) tends to \(+\infty\), and

For the left tail, \(x\to-\infty\) means \(z\to-\infty\). Using \(F(x)=1/(1+e^{-z})\),

Both limits are obtained from exact algebraic identities and the fact that \(e^{-z}\to0\) as \(z\to+\infty\), and \(e^z\to0\) as \(z\to-\infty\). No interchange of a limit and an integral is needed. \(\square\)

The limits also yield useful bounds at finite thresholds. For \(z\geq0\), \(1\leq1+e^{-z}\leq2\), so

Thus, whenever \(x\geq\mu\), the right-tail probability lies between half its leading exponential term and that term. By symmetry, whenever \(x\leq\mu\),

These bounds show why the exponential expression is a useful approximation far into a tail, while also making clear that it is a bound rather than an identity at finite \(x\).

Extreme quantiles and the size of the tail

For an upper-tail probability \(\alpha\), the exact threshold is

As \(\alpha\) decreases towards zero, the last term \(s\log(1-\alpha)\) tends to zero. Consequently,

Here the additive approximation error is exactly \(s\log(1-\alpha)\), not an unspecified correction. By symmetry around \(\mu\), the lower-tail quantile satisfies \(Q(\alpha)=\mu-s\log((1-\alpha)/\alpha)\), and tends to \(-\infty\) at the same rate. In particular, an extreme logistic quantile grows with \(\log(1/\alpha)\), rather than with a power of \(1/\alpha\).

This relationship makes the quantile formula practically valuable. If a model specifies an allowable exceedance probability, the formula translates it directly into a threshold on the measurement scale. Conversely, the survival function returns the probability associated with a proposed threshold. The two calculations are inverse checks on one another.

Common pitfall: replacing the exact tail too soon

A tempting calculation is to replace \(S(x)\) by \(\exp(-(x-\mu)/s)\) as soon as \(x>\mu\), then report the result as the exact probability. The theorem only says that the ratio approaches 1 as \(x\to\infty\); for a finite threshold, the exact survival probability is \(e^{-z}/(1+e^{-z})\), not \(e^{-z}\). The bound \(S(x)\leq e^{-z}\) also shows that the approximation is an upper bound on the right tail.

For an exact quantile, retain the factor \(1-\alpha\) in \(\log((1-\alpha)/\alpha)\). Dropping it is appropriate only when explicitly reporting the extreme-tail approximation \(\mu+s\log(1/\alpha)\). These formulas allow the error to be checked directly: the difference between the exact and approximate quantiles is \(s\log(1-\alpha)\).