samples/texts/2675357/page_13.md

$D_0 > 0$ such that whenever the constant $D'$ in the definition of $S_i(\mathbf{p}^*)$ is larger than $D_0$,

$$r_{i,S_i(p^{\ast })}^{{\mathrm{\ell }}_u^u}(\mathbf{Q})\ge C_u{\left(\frac{n{k}^2}{(k-1{)}^2}\sum _{j=1}^L\frac{(q_{ji}-q_j{)}^2}{q_j}\right)}^{-u\mathrm{/}2}-\delta n^{-u\mathrm{/}2}\text{for all }u\ge 1.(23)r\_\{i,S\_i(p^*)\}^\{\backslash ell\_u^u\}(\backslash mathbf\{Q\})\; \backslash ge\; C\_u\; \backslash left(\; \backslash frac\{nk^2\}\{(k-1)^2\}\; \backslash sum\_\{j=1\}^\{L\}\; \backslash frac\{(q\_\{ji\}-q\_j)^2\}\{q\_j\}\; \backslash right)^\{-u/2\}\; -\; \backslash delta\; n^\{-u/2\}\; \backslash quad\; \backslash text\{for\; all\; \}\; u\; \backslash ge\; 1.\; \backslash quad\; (23)$$

The constant $D'$ is required to be large for the local asymptotic normality arguments to hold (refer again to [15, Chapter 2, Theorem 1.1] and [24, Ch. 7]).

Proposition 4.1. Let $\mathcal{P}$ be the Euclidean ball around $\mathbf{p}_U$ defined in (13). For a sufficiently large constant $D$ and any $u \ge 1$ we have

$$r_{i,\text{Bayes}}^{{\mathrm{\ell }}_u^u}(\mathbf{Q})\ge C_u{\left(\frac{n{k}^2}{(k-1{)}^2}\sum _{j=1}^L\frac{(q_{ji}-q_j{)}^2}{q_j}\right)}^{-u\mathrm{/}2}-o(n^{-u\mathrm{/}2})\mathrm{.}(24)r\_\{i,\; \backslash text\{Bayes\}\}^\{\backslash ell\_u^u\}(\backslash mathbf\{Q\})\; \backslash ge\; C\_u\; \backslash left(\; \backslash frac\{nk^2\}\{(k-1)^2\}\; \backslash sum\_\{j=1\}^\{L\}\; \backslash frac\{(q\_\{ji\}-q\_j)^2\}\{q\_j\}\; \backslash right)^\{-u/2\}\; -\; o(n^\{-u/2\}).\; \backslash quad\; (24)$$

Proof. We can view $\mathcal{P}$ as a union of (uncountably many) parallel line segments with direction vector $\mathbf{v}_i$ defined in (15). Each of these line segments can be written as $S_i(\mathbf{p}^*)$ (see (16)), with a suitably chosen midpoint $\mathbf{p}^* \in \mathcal{P}$. Since the midpoints of all the line segments lie inside $\mathcal{P}$, which is a neighborhood of the uniform distribution, by (23) we have that for any estimator $\hat{\mathbf{p}}i$, the average $\ell_u^u$ estimation loss $r{i,S_i(p^*)}^{\ell_u^u}(\mathbf{Q}, \hat{\mathbf{p}}_i)$ on any of these line segments $S_i(\mathbf{p}^*)$ with $D' \ge D_0$ is lower bounded by

$$r_{i,S_i(p^{\ast })}^{{\mathrm{\ell }}_u^u}(\mathbf{Q},{\widehat{\mathbf{p}}}_i)\ge r_{i,S_i(p^{\ast })}^{{\mathrm{\ell }}_u^u}(\mathbf{Q})\ge C_u{\left(\frac{n{k}^2}{(k-1{)}^2}\sum _{j=1}^L\frac{(q_{ji}-q_j{)}^2}{q_j}\right)}^{-u\mathrm{/}2}-\delta n^{-u\mathrm{/}2}r\_\{i,S\_i(p^*)\}^\{\backslash ell\_u^u\}(\backslash mathbf\{Q\},\; \backslash hat\{\backslash mathbf\{p\}\}\_i)\; \backslash ge\; r\_\{i,S\_i(p^*)\}^\{\backslash ell\_u^u\}(\backslash mathbf\{Q\})\; \backslash ge\; C\_u\; \backslash left(\; \backslash frac\{nk^2\}\{(k-1)^2\}\; \backslash sum\_\{j=1\}^\{L\}\; \backslash frac\{(q\_\{ji\}-q\_j)^2\}\{q\_j\}\; \backslash right)^\{-u/2\}\; -\; \backslash delta\; n^\{-u/2\}$$

for $u \ge 1$. To compute the average estimation loss $r_{i,\text{Bayes}}^{\ell_u^u}(\mathbf{Q}, \hat{\mathbf{p}}_i)$ on $\mathcal{P}$ we need to average over all the segments with weight proportional to the length of the segment. Given $D_0$, we can choose $D$ in (13) large enough so that the proportion of the segments $S_i(\mathbf{p}^*)$ with $D' \ge D_0$ out of all the segments in $\mathcal{P}$ is arbitrarily close to one (formally, denote the union of such segments as $\mathcal{P}_0$, then $\text{Vol}(\mathcal{P}_0)/\text{Vol}(\mathcal{P})$ can be made arbitrarily close to 1 as long as we set $D/D_0$ to be large enough). The average estimation loss along each of these segments is uniformly bounded below as in (23), and thus the average loss on $\mathcal{P}_0$ is lower bounded by the same quantity. Combining the fact that $\text{Vol}(\mathcal{P}_0)/\text{Vol}(\mathcal{P}) = 1 - o(1)$, we have

$$r_{i,\text{Bayes}}^{{\mathrm{\ell }}_u^u}(\mathbf{Q},{\widehat{\mathbf{p}}}_i)\ge C_u{\left(\frac{n{k}^2}{(k-1{)}^2}\sum _{j=1}^L\frac{(q_{ji}-q_j{)}^2}{q_j}\right)}^{-u\mathrm{/}2}-o(n^{-u\mathrm{/}2})\text{for all }u\ge 1.r\_\{i,\; \backslash text\{Bayes\}\}^\{\backslash ell\_u^u\}(\backslash mathbf\{Q\},\; \backslash hat\{\backslash mathbf\{p\}\}\_i)\; \backslash ge\; C\_u\; \backslash left(\; \backslash frac\{nk^2\}\{(k-1)^2\}\; \backslash sum\_\{j=1\}^\{L\}\; \backslash frac\{(q\_\{ji\}-q\_j)^2\}\{q\_j\}\; \backslash right)^\{-u/2\}\; -\; o(n^\{-u/2\})\; \backslash quad\; \backslash text\{for\; all\; \}\; u\; \backslash ge\; 1.$$

This lower bound holds for any estimator $\hat{\mathbf{p}}_i$, and this implies the claimed lower bound (24). □