Plug-in SPMLE estimates survive non-uniqueness and nonresponse

Referee: Theorem 2 proof gap: |h(f_{Ĝ_n}) - h_m(f_{Ĝ_n})| is uncontrolled. Assumption 1(ii) guarantees approximation only at the true density f_G, not at the SPMLE density f_{Ĝ_n}.

Authors: The referee is correct that there is a gap in the proof of Theorem 2 as written. The triangle inequality decomposition makes the issue precise: the first term |h(f_{Ĝ_n}) - h_m(f_{Ĝ_n})| is indeed not controlled by Assumption 1 as currently stated. Assumption 1(ii) provides approximation of h by h_m only at the true density f_G, and Assumption 1(iii) provides pointwise convergence of f_{Ĝ_n} to f_G at finitely many points. Neither directly controls the approximation error of h_m for h when both are evaluated at f_{Ĝ_n}. revision: yes

Referee: Theorem 2 proof: typo (convergence to h(f_{Ĝ_n}) should read h(f_G)), and proof should use triangle inequality with h_m, address the first term, and clarify roles of h and h_m.

Authors: The referee is correct on both counts. The statement 'h(f_{Ĝ_n}(Y_{1:m})) converges in probability to h(f_{Ĝ_n}(Y_{1:m}))' is indeed a typo; it should read convergence to h(f_G(Y_{1:m})). More substantively, the proof as written does not properly distinguish between h and h_m and does not invoke the triangle inequality with the intermediate approximation h_m. We will rewrite the proof to explicitly use the triangle inequality with h_m and to address the first term. To close the gap, we will strengthen Assumption 1 by requiring that h_m approximates h uniformly over a neighborhood of f_G that contains f_{Ĝ_n} with probability tending to 1. Specifically, we will add the condition that for any ε > 0, there exists m and δ > 0 such that sup_{g: ||g - f_G||_{TV} < δ} |h(g) - h_m(g)| < ε with probability tending to 1. This is satisfied in our examples: in the Poisson case (Example 6), the tail of f_{Ĝ_n} is controlled by Theorem 1's TV consistency combined with the sub-geometric tail condition (5), which ensures that the tail mass of f_{Ĝ_n} beyond m is small uniformly. We will verify this for Examples 5-9 and state the strengthened assumption clearly. revision: yes

Referee: Theorem 1 proof: the bound f̃_n/f_G = O_p(√(f_G/n)) is used without justification; Taylor expansion step with α* not fully derived; connection between sub-geometric tail condition (5) and rates ν(ε)=O(|log ε|), ξ(ε)=O(ε) should be made explicit.

Authors: We agree that the proof sketch of Theorem 1 is dense and that several intermediate steps need expansion. The bound f̃_n/f_G = O_p(√(f_G/n)) on its support follows from standard empirical process arguments for multinomial proportions: for cells with f_G(y_j) ≥ L_n/n, the empirical frequency f̃_n(y_j) satisfies a CLT-type bound f̃_n(y_j) = f_G(y_j) + O_p(√(f_G(y_j)/n)), which gives the stated ratio. We will add this justification explicitly. The Taylor expansion step with α* is a second-order expansion of log(1 + x) around x = (f_{Ĝ_n} - f̃_n)/f̃_n, where α* ∈ [0,1] is the intermediate value from the mean-value theorem applied to the function log(1+·). We will spell out this derivation. Regarding the connection between condition (5) and the rates: condition (5) states that inf_m f_G(y_m) · Σ_{j≥m} f_G(y_j) > 0, which means the tail mass Σ_{j≥m} f_G(y_j) decays at most as fast as 1/f_G(y_m). For sub-geometric tails (e.g., Poisson with bounded second moment), f_G(y_m) decays roughly as e^{-c m log m}, so the number of cells with f_G ≥ ε is O(|log ε|) (giving ν(ε) = O(|log ε|)), and the tail mass below ε is O(ε) (giving ξ(ε) = O(ε)). We will make this connection explicit in the revised manuscript. revision: yes

Referee: Assumption 1(iii): clarify conditions under which it holds, whether Theorem 1 conditions are intended to be sufficient, and how the negative examples in Example 4 relate to the applicability of Theorem 2.

Authors: We agree that the relationship between Assumption 1(iii), Theorem 1, and Example 4 needs clarification. The conditions of Theorem 1 (specifically, the sub-geometric tail condition (5)) are intended to be sufficient for Assumption 1(iii): TV consistency implies pointwise convergence of densities at any fixed set of points, which is what Assumption 1(iii) requires. The negative examples in Example 4 are precisely cases where the conditions of Theorem 1 fail. In the first Example 4 scenario (uniform mixing on (0, m_n) with m_n > n^2), the tail condition (5) fails because the support grows with n in a triangular array, and the SPMLE does not converge in TV. In the second scenario (star-shaped density), the SPMLE converges to the wrong distribution, again because the conditions of Theorem 1 are violated. These examples serve to delineate the boundary of applicability: Theorem 2 applies when Theorem 1's conditions hold (ensuring Assumption 1(iii)), and Example 4 illustrates what goes wrong when they do not. We will add a remark after Assumption 1 making this relationship explicit: Assumption 1(iii) is verified by the conditions of Theorem 1, and the examples in Example 4 show that these conditions cannot simply be dropped. revision: yes