The Delta Method for Ratios

Why Ratios Need Joint Asymptotics

Ratios occur throughout statistics: an estimated rate divides an event count by an exposure, a conditional proportion divides one count by another, and a ratio of means compares two population quantities. In each case, uncertainty in both the numerator and the denominator matters. If the two quantities are computed from the same observations, their dependence matters too.

The ratio delta method gives a first-order limit when the numerator and denominator fluctuate jointly around fixed limits and the denominator limit is nonzero. Its central calculation is simple but important: the numerator contributes positively to the first-order error, while denominator error contributes with the opposite sign. The resulting variance therefore includes a covariance term.

Definition: Let \(A_n\) and \(B_n\) be random variables, and let \(a,b\) be constants with \(b\ne 0\). If \[ \sqrt{n} \begin{pmatrix} A_n-a\\ B_n-b \end{pmatrix} \xrightarrow{d} \begin{pmatrix} Z\\ W \end{pmatrix}, \] then the ratio delta method describes the limiting distribution of \(A_n/B_n\), centered at \(a/b\) and scaled by \(\sqrt n\). The ratio must be defined with probability tending to one; its value on any event where \(B_n=0\) can be assigned arbitrarily if that event has probability tending to zero.

First, the Unscaled Ratio Limit

A ratio can also be studied without a normal approximation. If \((A_n,B_n)\xrightarrow{d}(A,B)\) and \(P(B=0)=0\), then the continuous mapping theorem gives \(A_n/B_n\xrightarrow{d}A/B\), provided the ratios are defined where needed. The function \((x,y)\mapsto x/y\) is continuous at every point with \(y\ne0\); the condition \(P(B=0)=0\) ensures that the limiting random vector is in this continuity set with probability one.

A particularly common case is \(A_n\xrightarrow{p}a\) and \(B_n\xrightarrow{p}b\ne0\). The pair then converges in probability to \((a,b)\), so continuity of division at \((a,b)\) yields \(A_n/B_n\xrightarrow{p}a/b\). This gives consistency of the ratio. To describe its sampling fluctuations, more information is needed: a joint limit for the scaled numerator and denominator errors.

The First-Order Ratio Expansion

Write \(Z_n=\sqrt n(A_n-a)\) and \(W_n=\sqrt n(B_n-b)\). The following identity exposes both sources of error and is the key step in the proof.

\sqrt n\left(\frac{A_n}{B_n}-\frac{a}{b}\right) = \frac{bZ_n-aW_n}{b\left(b+W_n/\sqrt n\right)}.

To check the identity, substitute \(A_n=a+Z_n/\sqrt n\) and \(B_n=b+W_n/\sqrt n\). Then \[ \frac{A_n}{B_n}-\frac{a}{b} = \frac{bA_n-aB_n}{bB_n} = \frac{bZ_n-aW_n}{\sqrt n\,bB_n}, \] and \(B_n=b+W_n/\sqrt n\). This calculation requires \(B_n\ne0\); under the theorem’s assumptions, that holds with probability tending to one.

Theorem: Suppose \(b\ne0\) and \[ \sqrt n \begin{pmatrix} A_n-a\\ B_n-b \end{pmatrix} \xrightarrow{d} \begin{pmatrix} Z\\ W \end{pmatrix}, \] where \((Z,W)\) is a finite-valued random vector. If \(A_n/B_n\) is defined with probability tending to one, then \[ \sqrt n\left(\frac{A_n}{B_n}-\frac{a}{b}\right) \xrightarrow{d} \frac{Z}{b}-\frac{aW}{b^2}. \]

Proof. Joint convergence in distribution implies that \(Z_n\) and \(W_n\) are tight. Consequently, \(W_n/\sqrt n\xrightarrow{p}0\), and therefore \(b+W_n/\sqrt n\xrightarrow{p}b\). Since \(b\ne0\), the denominator \(b(b+W_n/\sqrt n)\) converges in probability to \(b^2\ne0\). By the identity above and Slutsky’s theorem, the scaled ratio has the same limit as \[ \frac{bZ_n-aW_n}{b^2}. \] The continuous mapping theorem applied to the linear function \((z,w)\mapsto (bz-aw)/b^2\) gives the limit \((bZ-aW)/b^2=Z/b-aW/b^2\). Also, \(B_n=b+W_n/\sqrt n\xrightarrow{p}b\ne0\), so \(P(B_n=0)\to0\). This justifies ignoring any event on which the ratio is undefined, or assigning the ratio an arbitrary value there. \(\square\)

Asymptotic Normality and the Covariance Term

A frequent application is when the joint limit \((Z,W)\) is mean-zero bivariate normal. Suppose its variances are \(v_A\) and \(v_B\), and its covariance is \(c\). Since a linear combination of jointly normal variables is normal, the theorem gives a normal limit. Its variance follows by expanding the variance of the linear combination:

\operatorname{Var}\left(\frac{Z}{b}-\frac{aW}{b^2}\right) = \frac{v_A}{b^2} + \frac{a^2v_B}{b^4} - \frac{2ac}{b^3}.

The minus sign in the covariance term comes from the negative coefficient of \(W\) in the ratio expansion. If the numerator and denominator errors are positively correlated, their effects on the ratio can partly offset one another. If they are negatively correlated, they can reinforce one another. Independence is not a default assumption: when the two quantities use the same data, their covariance generally needs to be calculated.

Corollary: If \[ \sqrt n \begin{pmatrix} A_n-a\\ B_n-b \end{pmatrix} \xrightarrow{d} N\left( \begin{pmatrix}0\\0\end{pmatrix}, \begin{pmatrix}v_A&c\\c&v_B\end{pmatrix} \right), \qquad b\ne0, \] then \[ \sqrt n\left(\frac{A_n}{B_n}-\frac{a}{b}\right) \xrightarrow{d} N\left(0,\frac{v_A}{b^2}+\frac{a^2v_B}{b^4}-\frac{2ac}{b^3}\right). \] The displayed variance is nonnegative because it is the variance of the limiting linear combination.

Worked Applications

Worked Example: A Ratio of Paired Sample Means

Suppose \((X_i,Y_i)\), \(i=1,\ldots,n\), are independent and identically distributed pairs with \(X_i=4+3R_i\) and \(Y_i=2+S_i\). Each sign is either \(-1\) or \(1\), with \(P(R_i=S_i)=3/4\) and \(P(R_i\ne S_i)=1/4\); each sign has marginal probabilities \(1/2,1/2\). Find the asymptotic distribution of \(\bar X/\bar Y\).

First, \(E[R_i]=E[S_i]=0\), so \(E[X_i]=4\) and \(E[Y_i]=2\). Each sign has variance \(1\). Moreover, \(E[R_iS_i]=P(R_i=S_i)-P(R_i\ne S_i)=3/4-1/4=1/2\). Thus \[ \operatorname{Var}(X_i)=9,\qquad \operatorname{Var}(Y_i)=1,\qquad \operatorname{Cov}(X_i,Y_i)=3E[R_iS_i]=3/2. \]

The multivariate central limit theorem for iid pairs with finite second moments gives a joint normal limit for \(\sqrt n(\bar X-4,\bar Y-2)\) with these variances and covariance. Here \(a=4\), \(b=2\), \(v_A=9\), \(v_B=1\), and \(c=3/2\). The ratio limit is centered at \(4/2=2\), and its asymptotic variance is \[ \frac{9}{2^2}+\frac{4^2(1)}{2^4}-\frac{2(4)(3/2)}{2^3} = \frac94+1-\frac32 = \frac74. \] Therefore \(\sqrt n(\bar X/\bar Y-2)\xrightarrow{d}N(0,7/4)\). Since every \(Y_i\) is either \(1\) or \(3\), \(\bar Y\) is always positive and the ratio is defined for every sample.

Worked Example: A Conditional-Proportion Estimator

Let each observation belong to exactly one of three categories: both \(A\) and \(B\), \(B\) but not \(A\), or not \(B\), with probabilities \(0.3\), \(0.2\), and \(0.5\), respectively. From \(n\) iid observations, let \(\hat p\) be the count in the first category divided by the count in either of the first two categories. Find its asymptotic variance.

For one observation, set \(U=1\) if it is in both \(A\) and \(B\), and \(V=1\) if it is in \(B\). Then \(\hat p=\bar U/\bar V\) when \(\bar V>0\). The means are \(a=E[U]=0.3\) and \(b=E[V]=0.5\), so \(a/b=0.6\). Because these are indicators, \[ v_A=\operatorname{Var}(U)=0.3(0.7)=0.21,\qquad v_B=\operatorname{Var}(V)=0.5(0.5)=0.25. \] Since \(UV=U\), the covariance is \[ c=E[UV]-E[U]E[V]=0.3-(0.3)(0.5)=0.15. \]

The iid multivariate central limit theorem and the ratio formula give \[ \frac{0.21}{0.5^2} +\frac{0.3^2(0.25)}{0.5^4} -\frac{2(0.3)(0.15)}{0.5^3} =0.84+0.36-0.72 =0.48. \] Thus \(\sqrt n(\hat p-0.6)\xrightarrow{d}N(0,0.48)\). The denominator is zero only if no observation belongs to \(B\), an event with probability \(0.5^n\to0\). Defining \(\hat p\) arbitrarily on that event does not change this limit.

Worked Example: Independent Sample-Mean Errors

Let \(X_i=5+R_i\) and \(Y_i=3+S_i\), where \(R_i\) and \(S_i\) are independent signs taking values \(-1\) and \(1\) with equal probability. Suppose the pairs are iid across \(i\). Find the limit for \(\bar X/\bar Y\).

The means are \(a=5\) and \(b=3\), and both variances are \(1\). Independence gives covariance \(c=0\). The joint central limit theorem and the ratio formula yield asymptotic variance \[ \frac{1}{3^2}+\frac{5^2}{3^4} = \frac19+\frac{25}{81} = \frac{34}{81}. \] Consequently, \(\sqrt n(\bar X/\bar Y-5/3)\xrightarrow{d}N(0,34/81)\). Each \(Y_i\) is either \(2\) or \(4\), so the denominator sample mean is strictly positive.

Common Pitfall: Dropping the Covariance

A tempting but generally wrong calculation is to treat numerator and denominator errors as independent and keep only the two variance terms. In the paired-means example, that would give \(9/4+1=13/4\), rather than the correct \(7/4\). The covariance is \(3/2\), and the missing term is \(-2(4)(3/2)/2^3=-3/2\). The discrepancy is substantial because the two sample means are calculated from the same paired observations.

The condition \(b\ne0\) is equally essential. If the denominator’s limit is zero, division is not continuous at the limiting point, and the expansion used here has a denominator tending to zero rather than to a nonzero constant. The limiting behavior then requires separate analysis; the ordinary ratio delta method does not apply. Likewise, a nonzero but very small \(b\) can make the asymptotic variance large, even though the theorem remains valid.

Key takeaway: For a ratio with numerator limit \(a\) and nonzero denominator limit \(b\), the first-order error is numerator error divided by \(b\), minus \(a\) times denominator error divided by \(b^2\). Use the joint limit, not separate marginal limits, to retain the covariance that determines the correct variance.

Check Your Understanding

Use the expansion and conditions above to answer these questions.

If the scaled numerator and denominator errors converge jointly to \((Z,W)\), what is the limiting scaled error of their ratio when the denominator limit is \(b\ne0\)?
For a mean-zero joint normal limit with variances \(v_A,v_B\) and covariance \(c\), identify the covariance contribution to the ratio’s asymptotic variance.
In the conditional-proportion example, why does the event of a zero denominator not prevent the stated asymptotic limit?
What fails in the ratio expansion if the denominator limit is zero?
When might omitting the covariance term give a particularly misleading approximation?

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