Semiparametric Efficiency — Examples

The masses sum to one. The geometric series gives $Pf={\sum }_{j\ge 1}{2}^{-j\mathrm{/}2}<\mathrm{\infty }$, while $P[f^2]={\sum }_{j\ge 1}1=\mathrm{\infty }$. Thus $\mu =Pf$ is finite and $u=f-\mu$ is integrable, centered, but not square-integrable.

For any bounded centered $s$, choose $t\ge 0$ so small that $\mathrm{\mid }ts\mathrm{\mid }\le 1\mathrm{/}2$ everywhere. Then $1+ts\ge 0$ and $P(1+ts)=1$. On $[-1\mathrm{/}2,1\mathrm{/}2]$, $g(x)=\sqrt{1+x}$ has $g^{\mathrm{\prime }\mathrm{\prime }}(x)=-1\mathrm{/}(4(1+x{)}^{3\mathrm{/}2})$, so $\mathrm{\mid }{g}^{\mathrm{\prime }\mathrm{\prime }}(x)\mathrm{\mid }\le 1$. [F4] with $n=1$ gives $\sqrt{1+ts}=1+\frac{t}{2}s+O(t^2\mathrm{\parallel }s{\mathrm{\parallel }}_{\mathrm{\infty }}^2)$ uniformly; thus the tilt is DQM with score $s$. Since $f\in L^1(P)$ and $s$ is bounded, $\psi (P_{t,s})=Pf+tP[fs]$ and ${\dot{\psi }}_P(s)=P[us]$.

For $M>0$, put $u_M=u{\mathbf{1}}_{\{\mathrm{\mid }u\mathrm{\mid }\le M\}}$, $c_M=P{u}_M$, and $s_M=u_M-c_M$. Then $s_M$ is bounded and centered, $D_M:=P[u^2{\mathbf{1}}_{\{\mathrm{\mid }u\mathrm{\mid }\le M\}}]=P[u{s}_M]$, and $\mathrm{\parallel }{s}_M{\mathrm{\parallel }}_2^2=D_M-c_M^2\le D_M$. For all sufficiently large $j$, $u(j)\ge 2^{j\mathrm{/}2}\mathrm{/}2$, so each such included atom contributes at least $1\mathrm{/}4$ to $D_M$; as $M\to \mathrm{\infty }$, arbitrarily many atoms are included and $D_M\to \mathrm{\infty }$. For large $M$, $s_M\mathrm{\ne }0$ because the truncation vanishes on a positive-mass tail and is nonzero on included atoms. Hence $\mathrm{\mid }{\dot{\psi }}_P(s_M)\mathrm{\mid }\mathrm{/}\mathrm{\parallel }{s}_M{\mathrm{\parallel }}_2\ge \sqrt{D_M}\to \mathrm{\infty }$.

If a finite-variance influence curve $\varphi \in L_0^2(P)$ represented these derivatives, [F3] would give $\mathrm{\mid }{\dot{\psi }}_P(s_M)\mathrm{\mid }=\mathrm{\mid }P[\varphi s_M]\mathrm{\mid }\le \mathrm{\parallel }\varphi {\mathrm{\parallel }}_2\mathrm{\parallel }{s}_M{\mathrm{\parallel }}_2$ for every $M$, contradicting step 1.3. Thus the mean functional has no square-integrable influence curve in this model, refuting the statement.

For each bounded $s$ with $Ps=0$, choose $t\ge 0$ so small that $\mathrm{\mid }ts\mathrm{\mid }\le 1\mathrm{/}2$ everywhere. On $[-1\mathrm{/}2,1\mathrm{/}2]$, $g(x)=\sqrt{1+x}$ has $g^{\mathrm{\prime }\mathrm{\prime }}(x)=-1\mathrm{/}(4(1+x{)}^{3\mathrm{/}2})$, so $\mathrm{\mid }{g}^{\mathrm{\prime }\mathrm{\prime }}(x)\mathrm{\mid }\le 1$. [F12] with $n=1$ gives $\sqrt{1+ts}=1+\frac{t}{2}s+O(t^2\mathrm{\parallel }s{\mathrm{\parallel }}_{\mathrm{\infty }}^2)$ uniformly. The remainder is $o(t)$ in $L^2(P)$, so this is a DQM path with score $s$. Its mean is $\psi (P_{t,s})=\mu +tP[Xs]$, hence ${\dot{\psi }}_P(s)=P[(X-\mu )s]$.

Bounded centered functions are dense in $L_0^2(P)$: for $g\in L_0^2(P)$, the bounded functions $g_M=g{\mathbf{1}}_{\{\mathrm{\mid }g\mathrm{\mid }\le M\}}-P[g{\mathbf{1}}_{\{\mathrm{\mid }g\mathrm{\mid }\le M\}}]$ are centered and converge to $g$ in $L^2(P)$ by truncation and $Pg=0$. Thus $\mathcal{T}=L_0^2(P)$. By [F10], ${\dot{\psi }}_P$ is continuous and is represented by $\varphi =X-\mu \in \mathcal{T}$; [F9] makes this the canonical gradient. Its covariance is $V_{\mathrm{c}\mathrm{a}\mathrm{n}}=P[(X-\mu {)}^2]={\sigma }^2$.

The identity ${\bar{X}}_n-\mu =(P_n-P)(X-\mu )$ proves exact asymptotic linearity with influence $\phi =X-\mu$. Therefore [F7] gives the base limit $N(0,{\sigma }^2)$.

Fix a bounded centered score $s$ and $h\ge 0$. Under the base law, the iid vectors $(X_i-\mu ,s(X_i))$ have finite second moment. By [F4], their normalized sums converge jointly to a centered Gaussian with covariance entries ${\sigma }^2$, $P[(X-\mu )s]$, and $P[s^2]$. The lemma [F5] also gives $P(L_n=0)\to 0$; set ${\mathrm{\ell }}_n=\log L_n$ when $L_n>0$ and ${\mathrm{\ell }}_n=0$ otherwise, which changes the joint law only on an $o(1)$-probability event. Scalar Slutsky and Cramér–Wold [F11] give the joint Gaussian limit with ${\mathrm{\ell }}_n$; exponentiation gives the likelihood-ratio limit required by [F6]. The proxy limit of $\sqrt{n}({\bar{X}}_n-\mu )$ is therefore $N(hP[(X-\mu )s],{\sigma }^2)$, and total-variation closeness [F5] transfers it to the actual local law.

The target shift is exact: $\sqrt{n}\{\psi (P_{h\mathrm{/}\sqrt{n},s})-\mu \}=hP[(X-\mu )s]$. Subtracting it from step 1.4 gives $N(0,{\sigma }^2)$ for every fixed $h\ge 0$, so the sample mean is regular on every allowed path. Its influence covariance is ${\sigma }^2=V_{\mathrm{c}\mathrm{a}\mathrm{n}}$ by step 1.2; [F8] shows it attains the regular asymptotically linear lower bound. This includes ${\sigma }^2=0$, when both covariance matrices vanish.