Canonical Negative-Inverse Link

From the Gamma Density to Its Natural Parameter

For a gamma response, the mean \(\mu\) must be positive. The canonical link is determined by the distribution’s exponential-family representation, not chosen simply because a reciprocal transformation seems convenient. With the density convention used here, the resulting canonical link is the negative reciprocal of the mean.

Write a gamma response with mean \(\mu>0\) and dispersion \(\phi>0\) as having shape \(1/\phi\) and scale \(\phi\mu\). Its density, for \(y>0\), is

The exponential-dispersion form is

Here the response enters through the statistic \(y\), \(\theta\) is the natural parameter, \(b\) is the cumulant function, and \(\phi\) is the dispersion. To identify these quantities, take the logarithm of the gamma density and collect the terms involving \(y\) and \(\mu\):

The term involving \(y\) linearly is \(-y/(\phi\mu)\), so matching it to \(y\theta/\phi\) gives \(\theta=-1/\mu\). The mean-dependent term is \(-\log\mu/\phi\). Since \(-\theta=1/\mu\), we have \(\log(-\theta)=-\log\mu\), and therefore this term equals \(\log(-\theta)/\phi\). In the exponential-dispersion form it is \(-b(\theta)/\phi\), which identifies \(b(\theta)=-\log(-\theta)\).

The remaining terms give \(c(y,\phi)=(1/\phi-1)\log y-(\log\phi)/\phi-\log\Gamma(1/\phi)\). These terms do not depend on \(\mu\). The domain \(\theta<0\) is necessary: the logarithm in \(b(\theta)\) requires \(-\theta>0\), and the corresponding mean \(\mu=-1/\theta\) is positive.

Why the Canonical Link Is Negative Reciprocal

In a generalized linear model, the linear predictor is \(\eta=x^\mathsf{T}\beta\), and the link specifies \(\eta=g(\mu)\). For a canonical link, the link maps the mean to the natural parameter in the exponential-family representation. Since the gamma natural parameter just derived is \(-1/\mu\), the canonical choice is \(g(\mu)=-1/\mu\), not \(1/\mu\).

Derivation. The exponential-family form contains the response term \(y\theta/\phi\). The gamma log density contains \(-y/(\phi\mu)\). Matching these coefficients for every \(y>0\) forces \(\theta=-1/\mu\). By definition, a canonical link sets \(g(\mu)=\theta\), giving \(g(\mu)=-1/\mu\). Because \(\mu>0\), this link always yields a negative value. Conversely, if \(\eta<0\), solving \(\eta=-1/\mu\) gives \(\mu=-1/\eta>0\). Hence the link and inverse have the stated domains. \(\square\)

The cumulant function also confirms the mean and variance encoded by this representation. Differentiating on \(\theta<0\) gives \(b'(\theta)=-1/\theta\) and \(b''(\theta)=1/\theta^2\). Substituting \(\theta=-1/\mu\) yields \(b'(\theta)=\mu\) and \(b''(\theta)=\mu^2\).

For completeness, the derivative identities follow by differentiating the normalizing integral. At fixed \(\phi>0\), the part of the density involving \(y\) is proportional to \(y^{1/\phi-1}\exp(\theta y/\phi)\), for \(y>0\). On any compact interval of negative \(\theta\) values, this integrand and its first two derivatives with respect to \(\theta\) are bounded by integrable functions: near zero they are bounded by constants times \(y^{1/\phi-1}\), and at infinity by a power of \(y\) times a decaying exponential. Differentiation under the integral is therefore valid locally. Differentiating the normalizing identity once gives \(E[Y]=b'(\theta)\); differentiating a second time gives \(\operatorname{Var}(Y)=\phi b''(\theta)\). The formulas for \(b'\) and \(b''\) then give the stated mean and variance.

Connection to the Positive Reciprocal Link

A positive reciprocal link is also used for gamma means: \(g_+(\mu)=1/\mu\). It is not the canonical link under the density convention used here. The two links differ by a minus sign, so their linear predictors and regression coefficients must be compared with the same sign change.

This relationship is algebraic. For any fixed covariate vector \(x\), if \(x^\mathsf{T}\beta_+=1/\mu>0\), then setting \(\beta_{\mathrm{can}}=-\beta_+\) gives \(x^\mathsf{T}\beta_{\mathrm{can}}=-x^\mathsf{T}\beta_+=-1/\mu\). Conversely, negating a canonical coefficient vector gives the positive reciprocal coefficient vector. Thus, when coefficient spaces and predictor constraints are transformed together, both forms describe the same set of positive fitted means. They do not use the same numerical coefficient signs or predictor domain.

Domain Checks and a Common Mistake

The link domain is not a minor technicality. Under the canonical link, every fitted linear predictor must be negative if it is to correspond to a finite positive mean. Under the positive reciprocal convention, the predictor must instead be positive. A coefficient vector that produces a predictor outside the required range cannot be inverted into an admissible mean under that link.

Common mistake: It is tempting to say that the canonical link is \(1/\mu\), because a reciprocal appears when solving for the mean. That reasoning drops the sign in the natural parameter. The gamma log density has the linear response term \(-y/(\phi\mu)\), so its natural parameter is \(-1/\mu\), not \(1/\mu\). The positive reciprocal link is a separate convention related by negating the predictor and coefficients.

A second sign error can occur while identifying the cumulant function. Since \(-\theta=1/\mu\), the correct identity is \(\log(-\theta)=-\log\mu\). Thus \(-\log\mu=\log(-\theta)\), whereas \(b(\theta)=-\log(-\theta)=\log\mu\). Keeping the cumulant \(b\) distinct from the term \(-b\) in the exponent avoids reversing this sign.