Independent two-sample PRS asymptotics

This note completes the first mathematical research task in #44.

The result below concerns two independent multinomial samples only. It does not yet define a two-sample resemblance decision procedure or critical values for nested \(\delta\)-resemblance hypotheses.

The source PRS paper explicitly keeps \(p_0\) fixed and notes a two-sample formulation as future research. The derivation here is therefore an extension beyond the published one-sample method.

Setup and assumptions

For a fixed number of categories \(B\), let

\[ X_n \sim \mathrm{Multinomial}(n,p_n), \qquad Y_m \sim \mathrm{Multinomial}(m,q_m), \]

with the two samples independent. Define empirical proportions

\[ \widehat p = X_n/n, \qquad \widehat q = Y_m/m. \]

Assume:

1. \(n\to\infty\) and \(m\to\infty\);
2. \(n/(n+m)\to\rho\in(0,1)\);
3. \(p_n\to r\) and \(q_m\to r\) for an interior probability vector \(r\), with \(r_j>0\) and \(\sum_j r_j=1\);
4. under local alternatives,

[ \sqrt{n_{\mathrm{eff}}}(p_n-q_m)\to\xi, \qquad \mathbf 1^\top\xi=0, ]

where

[ n_{\mathrm{eff}}=\frac{nm}{n+m}. ]

The interior condition is needed because the Pearson weights contain reciprocals of the limiting probabilities.

Difference-of-proportions CLT

Write

\[ Z_{n,m} = \sqrt{n_{\mathrm{eff}}} \left( \widehat p-\widehat q \right). \]

Using

\[ \sqrt{n_{\mathrm{eff}}} = \sqrt{\frac{m}{n+m}}\sqrt n = \sqrt{\frac{n}{n+m}}\sqrt m, \]

we obtain

\[ Z_{n,m} = \sqrt{\frac{m}{n+m}} \sqrt n(\widehat p-p_n) - \sqrt{\frac{n}{n+m}} \sqrt m(\widehat q-q_m) + \sqrt{n_{\mathrm{eff}}}(p_n-q_m). \]

The multinomial central limit theorem gives

\[ \sqrt n(\widehat p-p_n) \xrightarrow{d} N(0,\Sigma(r)), \]

and

\[ \sqrt m(\widehat q-q_m) \xrightarrow{d} N(0,\Sigma(r)), \]

where

\[ \Sigma(r)=D(r)-rr^\top, \qquad D(r)=\mathrm{diag}(r). \]

Independence of the samples and \(m/(n+m)\to 1-\rho\), \(n/(n+m)\to\rho\) imply

\[ Z_{n,m} \xrightarrow{d} N(\xi,\Sigma(r)). \]

Under exact equality, \(p_n=q_m=r\), so \(\xi=0\).

Rank and degrees of freedom

Let

\[ u=(\sqrt{r_1},\ldots,\sqrt{r_B})^\top. \]

Then

\[ D(r)^{-1/2}\Sigma(r)D(r)^{-1/2} = I-uu^\top. \]

Because \(u^\top u=1\), the matrix

\[ P=I-uu^\top \]

is symmetric and idempotent:

\[ P^2=P. \]

It is the orthogonal projector onto the \((B-1)\)-dimensional subspace orthogonal to \(u\). Hence

\[ \mathrm{rank}\{\Sigma(r)\}=B-1. \]

This is the same loss of one degree of freedom caused by the simplex constraint in the one-sample multinomial problem.

Known-weight quadratic form

Suppose temporarily that the limiting common probability vector \(r\) is known. Define

\[ Q_{n,m}(r) = n_{\mathrm{eff}} \sum_{j=1}^{B} \frac{(\widehat p_j-\widehat q_j)^2}{r_j}. \]

Equivalently,

\[ Q_{n,m}(r) = Z_{n,m}^\top D(r)^{-1}Z_{n,m}. \]

Let

\[ W=D(r)^{-1/2}Z, \qquad Z\sim N(\xi,\Sigma(r)). \]

Then

\[ W\sim N(\mu,P), \qquad \mu=D(r)^{-1/2}\xi. \]

Since \(\mathbf 1^\top\xi=0\),

\[ u^\top\mu = \sum_{j=1}^{B}\xi_j = 0, \]

so the mean vector lies in the range of \(P\). Therefore

\[ W^\top W \sim \chi^2_{B-1}(\lambda), \]

with

\[ \lambda = \mu^\top\mu = \xi^\top D(r)^{-1}\xi = \sum_{j=1}^{B}\frac{\xi_j^2}{r_j}. \]

Hence

\[ Q_{n,m}(r) \xrightarrow{d} \chi^2_{B-1}(\lambda). \]

Under exact equality, \(\lambda=0\), giving the central \(\chi^2_{B-1}\) limit.

For a local sequence satisfying \(\sqrt{n_{\mathrm{eff}}}(p_n-q_m)\to\xi\), the non-centrality may also be written as

\[ n_{\mathrm{eff}} \sum_{j=1}^{B} \frac{(p_{n,j}-q_{m,j})^2}{r_j} \to \lambda. \]

Unknown common probabilities and pooled plug-in

A genuine two-sample problem does not know \(r\). Define the pooled empirical distribution

\[ \widehat r_j = \frac{X_{n,j}+Y_{m,j}}{n+m} = \frac{n}{n+m}\widehat p_j + \frac{m}{n+m}\widehat q_j. \]

Under the assumptions above,

\[ \widehat r\xrightarrow{p}r. \]

Consider the plug-in statistic

\[ Q_{n,m}(\widehat r) = n_{\mathrm{eff}} \sum_{j=1}^{B} \frac{(\widehat p_j-\widehat q_j)^2}{\widehat r_j}. \]

For each category,

\[ n_{\mathrm{eff}} (\widehat p_j-\widehat q_j)^2 = O_p(1), \]

while

\[ \frac{1}{\widehat r_j} - \frac{1}{r_j} = o_p(1). \]

Because \(B\) is fixed and every \(r_j>0\),

\[ Q_{n,m}(\widehat r)-Q_{n,m}(r) = o_p(1). \]

By Slutsky's theorem,

\[ Q_{n,m}(\widehat r) \xrightarrow{d} \chi^2_{B-1}(\lambda). \]

Thus the pooled plug-in preserves the same limiting distribution under both equality and the stated local alternatives.

Exact connection to Pearson's homogeneity statistic

The plug-in statistic is not merely analogous to the classical Pearson two-sample homogeneity statistic: it is exactly the same statistic.

The Pearson statistic for the \(2\times B\) contingency table is

\[ X^2 = \sum_{j=1}^{B} \frac{(X_{n,j}-n\widehat r_j)^2}{n\widehat r_j} + \sum_{j=1}^{B} \frac{(Y_{m,j}-m\widehat r_j)^2}{m\widehat r_j}. \]

Using

\[ X_{n,j}-n\widehat r_j = n_{\mathrm{eff}} (\widehat p_j-\widehat q_j) \]

and

\[ Y_{m,j}-m\widehat r_j = - n_{\mathrm{eff}} (\widehat p_j-\widehat q_j), \]

the two terms combine to

\[ X^2 = n_{\mathrm{eff}} \sum_{j=1}^{B} \frac{(\widehat p_j-\widehat q_j)^2}{\widehat r_j} = Q_{n,m}(\widehat r). \]

So the independent two-sample candidate has a classical foundation under the point null.

What remains novel, and unresolved, is the resemblance extension: replacing exact equality with nested composite tolerance regions and deriving valid least-favourable critical values.

What #44 establishes

For independent samples and fixed \(B\), with both sample proportions converging to an interior common limit:

• the effective sample size is \(nm/(n+m)\);
• the normalized difference has limiting covariance \(\Sigma(r)\);
• the covariance has rank \(B-1\);
• the known-weight quadratic form has a non-central \(\chi^2_{B-1}\) limit under local alternatives;
• pooled empirical weights are asymptotically valid;
• the pooled quadratic form is exactly Pearson's two-sample homogeneity statistic.

What #44 does not establish

This derivation does not yet establish:

• a two-sample \(\delta\)-resemblance hypothesis;
• a recommended two-sample tolerance;
• a least-favourable non-centrality over that tolerance set;
• two nested decision boundaries;
• finite-sample calibration;
• validity under overlapping or dependent samples.

Those questions remain in #45–#47.