Sign Test

Definition

Let $X_i=\theta +{ϵ}_i$, where ${ϵ}_i$‘s are i.i.d. with CDF $F_X$ and PDF $f_x$ with median $0$. i.e. median of $X_i$‘s are $\theta$.

We want to test $H_0:\theta ={\theta }_0,H_1:\theta >{\theta }_0$, which is called sign test Consider a statistic $S({\theta }_0)=\sum _{i=1}^{n}I(X_i>{\theta }_0)$, which is called sign statistic We reject $H_0$ if $S({\theta }_0)\ge c$.

Distribution of Sign Statistic

$I(X_i>{\theta }_0)\sim B(1,\frac{1}{2})$ under $H_0$. Therefore, $S({\theta }_0)\sim B(n,\frac{1}{2})$. Also, by CLT, $\frac{S({\theta }_0)-\frac{n}{2}}{\sqrt{n}/2}\sim N(0,1)$

Consider a sequence of local alternatives $H_0:\theta =0,H_1:{\theta }_n=\delta /\sqrt{n},\delta >0$ The efficacy of the sign statistic $S({\theta }_0)$ is $c_S={\tau }_S^{-1}=2f(0)$

Composite Sign Test

Consider a composite hypothesis $H_0:\theta \le {\theta }_0,H_1:\theta >{\theta }_0$ Since the power function $\gamma (\theta )$ is non-decreasing, the sign test of size $\alpha$ is $\underset{\theta \le {\theta }_0}{\max }\gamma (\theta )$

Facts

Consider a test $H_0:\theta ={\theta }_0,H_1:\theta >{\theta }_0$ and the test statistic $S({\theta }_0)=\sum _{i=1}^{n}I(X_i>{\theta }_0)$, then $\forall k,P_{\theta }(S(0)\ge k)=P_0(S(-\theta )>k)$ Also, the power function $\gamma (\theta )$ of the test is a non-decreasing function of $\theta$

Consider a sequence of hypothesis $H_0:\theta =0,H_1:{\theta }_n=\delta /\sqrt{n},\delta >0$, where $H_1$ is called a local alternative, then ${\lim }_{n\to \infty }\gamma ({\theta }_n)=1-\Phi (z_a-\delta {\tau }_S^{-1})$ where $\Phi$ is a CDF of standard normal distribution, and ${\tau }_S=c_S^{-1}=\frac{1}{2f(0)}$

Let $X_i=\theta +{ϵ}_i$ be random variables from location model, where ${ϵ}_i$‘s are i.i.d. with CDF $F(x)$, PDF $f(x)$, and median 0, and $Q_2$ be the sample median of $X_i$, then $\sqrt{n}(Q_2-\theta )\stackrel{D}{\to }N(0,{\tau }_s^2)$ where ${\tau }_s=\frac{1}{2f(0)}$