Smoothed nonparametric tests and approximations of p-values

Abstract

We propose new smoothed sign and Wilcoxon’s signed rank tests that are based on kernel estimators of the underlying distribution function of the data. We discuss the approximations of the p-values and asymptotic properties of these tests. The new smoothed tests are equivalent to the ordinary sign and Wilcoxon’s tests in the sense of Pitman’s asymptotic relative efficiency, and the differences between the ordinary and new tests converge to zero in probability. Under the null hypothesis, the main terms of the asymptotic expectations and variances of the tests do not depend on the underlying distribution. Although the smoothed tests are not distribution-free, making use of the specific kernel enables us to obtain the Edgeworth expansions, being free of the underlying distribution.

Appendix

Appendix gives brief proofs of Theorems 1 and 3. The complete proofs are found in Maesono et al. (2016).

Proof of Theorem 1

For the ordinary sign test S, we have

$$\begin{aligned} V_{\theta }(S)=nF(\theta )[1-F(\theta )]. \end{aligned}$$

Then, it is sufficient to show that

$$\begin{aligned} E_{\theta }\left[ \left\{ S-F(\theta )\right\} \left\{ \widetilde{S}-E_{\theta }(\widetilde{S})\right\} \right] =n\left\{ F(\theta )[1-F(\theta )]+O(h_n)\right\} . \end{aligned}$$

Since S and $\widetilde{S}$ are sums of i.i.d. random variables, we have

$$\begin{aligned}&E_{\theta }\left[ \left\{ S-E_{\theta }(S)\right\} \left\{ \widetilde{S}-E_{\theta }(\widetilde{S})\right\} \right] \\&\quad =nE_{\theta }\left[ \{I(X_1 \ge 0)-E_{\theta }(I(X_1 \ge 0))\}\left\{ 1-K\left( -\frac{X_1}{h_n}\right) -e_1(\theta )\right\} \right] . \end{aligned}$$

Using the transformation $u=x/h_n$, integration by parts, and a Taylor expansion, we get

$$\begin{aligned} \int _{-\infty }^{\infty }I(x \ge 0)\left[ 1-K\left( -\frac{x}{h_n}\right) \right] f(x-\theta )\mathrm{d}x=F(\theta )+O(h_n). \end{aligned}$$

Since $E_{\theta }(I(x \ge 0))=F(\theta )$ and $E_{\theta }(1-K)=F(\theta )+O(h_n^2)$, we have

$$\begin{aligned} E_{\theta }\left[ \left\{ S-E_{\theta }(S)\right\} \left\{ \widetilde{S}-E_{\theta }(\widetilde{S})\right\} \right] =n\{F(\theta )-[F(\theta )]^2+O(h_n)\}. \end{aligned}$$

Thus, we get the desired result. $\square$

Similarly, we can show that the difference between W and $\widetilde{W}$ goes to zero.

Proof of Theorem 3

Assuming that the density f is differentiable, we have

$$\begin{aligned} \frac{1}{n}E_0(\widetilde{S})= & {} 1-\int _{-\infty }^{\infty } K\left( -\frac{x}{h_n}\right) f(x)\mathrm{d}x=1-\int _{-\infty }^{\infty } k(u)F(-h_nu)\mathrm{d}u\\= & {} 1-F(0)+h_nf(0)A_{0,1}-\frac{h_n^2}{2}f'(0)A_{0,2}+\frac{h_n^3}{6}f''(0)A_{0,3}\\&-\frac{h_n^4}{24}f^{(3)}(0)A_{0,4}+\frac{h_n^5}{120}f^{(4)}(0)A_{0,5}+O(h_n^6). \end{aligned}$$

Similarly, we can show that

$$\begin{aligned} E_0\left\{ K^2\left( -\frac{X_1}{h_n}\right) \right\} =F(0)-2h_nf(0)A_{1,1}+h_n^2f'(0)A_{1,2}-\frac{h_n^3}{3}f''(0)A_{1,3}+O(h_n^4) \end{aligned}$$

and

$$\begin{aligned} \frac{1}{n}V_0(\widetilde{S})= & {} F(0)\{1-F(0)\}-2h_nf(0)A_{1,1}+h_n^2f'(0)\{A_{1,2}-F(0)A_{0,2}\}\\&-\frac{h_n^3}{3}f''(0)\{A_{1,3}-F(0)A_{0,3}\}+O(h_n^4). \end{aligned}$$

Note that $k(-u)=k(u)$ yields $A_{0,1}=A_{0,3}=A_{0,5}=0$. Furthermore, since $f(-x)=f(x)$, we get

$$\begin{aligned} f'(0)=0,\quad f''(-x)=f''(x), \quad f^{(3)}(-x)=-f^{(3)}(x), \quad \mathrm{and}\quad f^{(3)}(0)=0. \end{aligned}$$

We can derive equations (1) and (2) in a similar manner (see Maesono et al. 2016). $\square$