Bahadur efficiency for certain goodness-of-fit tests based on the empirical characteristic function

Abstract

We study the Bahadur efficiency of several weighted L2-type goodness-of-fit tests based on the empirical characteristic function. The methods considered are for normality and exponentiality testing, and for testing goodness-of-fit to the logistic distribution. Our results are helpful in deciding which specific test a potential practitioner should apply. For the celebrated BHEP and energy tests for normality we obtain novel efficiency results, with some of them in the multivariate case, while in the case of the logistic distribution this is the first time that efficiencies are computed for any composite goodness-of-fit test.

Appendix

1.1 Regularity conditions A

There exists $\delta >0$ such that for $|\theta |\le \delta$ it holds:

• A1 Let MLEs ${\hat{\mu }}$ and ${\hat{\sigma }}$ of $\mu$ and $\sigma$ exist (under the null model) such that ${\hat{\mu }}\overset{P_{\theta }}{\rightarrow }\mu (\theta )$ and ${\hat{\sigma }}\overset{P_{\theta }}{\rightarrow }\sigma (\theta )$;

• A2 Functions $\mu (\theta )$ and $\sigma (\theta )$ are three times continuously differentiable;

• A3 Functions $L_1(x;\theta )=\log g_{\theta }(x)$ and $L_2(x;\theta )=\log \Big (\frac{1}{\sigma (\theta )}f_0\Big (\frac{x-\mu (\theta )}{\sigma (\theta )}\Big )\Big )$ are three times differentiable;

• A4 $\Big |\frac{\partial ^{i}L_k(x;\theta )}{\partial \theta ^i}\frac{\partial ^{j}g_{\theta }(x)}{\partial \theta ^j}\Big |<M_{i,j}(x)$ for $i,j=0,1,2$, $i+j\le 3$, $k=1,2$, where $M_{i,j}(x)$ are integrable functions.

Proof of Theorem 3.1

Let $\mu =\mu (\theta )$ and $\sigma =\sigma (\theta )$ be the values that minimize (3.1), their existence following from Condition A1, by considering the sample Kullback–Leibler distance $\frac{1}{n}\sum _{i=1}^n\log \Big (\frac{g_{\theta }(x)}{\frac{1}{\sigma } f_0(\frac{x-\mu }{\sigma })}\Big )$ as in Villa (2016).

Conditions A2–A4 enable differentiation under the integral sign. Hence, differentiating (3.1) with respect to $\theta$ we get

$$\begin{aligned} K'(\theta )&=\int \log g_\theta (x)g_\theta '(x)\textrm{d}x- \int \frac{1}{f_0(\frac{x-\mu (\theta )}{\sigma (\theta )})}f_0'\Big (\frac{x-\mu (\theta )}{\sigma (\theta )}\Big ) \Big (\frac{x-\mu (\theta )}{\sigma (\theta )}\Big )'g_\theta (x)\textrm{d}x \\&\quad +\frac{\sigma '(\theta )}{\sigma (\theta )}-\int \log f_0\Big (\frac{x-\mu (\theta )}{\sigma (\theta )}\Big ) g_\theta '(x)\textrm{d}x. \end{aligned}$$

It is easy to show that $K'(0)=0$. Differentiating (3.1) once more we obtain at $\theta =0$,

$$\begin{aligned} K''(0)&=\int \frac{h^2(x)}{\frac{1}{\sigma _0}f_0(\frac{x-\mu _0}{\sigma _0})}\textrm{d}x+\int \log \frac{1}{\sigma _0}f_0\Big (\frac{x-\mu _0}{\sigma _0}\Big ) u(x)dx+\frac{\sigma ''(0)\sigma _0-(\sigma '(0))^2}{\sigma _0^2} \\&\quad +\int \frac{\Big (f_0'\Big (\frac{x-\mu _0}{\sigma _0}\Big )\Big )^2}{\sigma _0f_0\Big (\frac{x-\mu _0}{\sigma _0}\Big )}\left( \frac{\mu '(0)\sigma _0+(x-\mu _0)\sigma '(0)}{\sigma _0^2}\right) ^2\textrm{d}x \\ {}&\quad -\int \frac{1}{\sigma _0}f_0''\left( \frac{x-\mu _0}{\sigma _0}\right) \left( \frac{\mu '(0)\sigma _0+(x-\mu _0)\sigma '(0)}{\sigma _0^2}\right) ^2\textrm{d}x \\&\quad -\int \frac{1}{\sigma _0}f_0'\left( \frac{x-\mu _0}{\sigma _0}\right) \left( \frac{-\mu ''(0)\sigma _0+2 \mu '(0) \sigma '(0)+(x-\mu _0) \left( 2 \frac{\sigma '(0)^2}{\sigma _0}-\sigma ''(0)\right) }{\sigma _0^2}\right) \textrm{d}x \\&\quad +\int \frac{f_0'\left( \frac{x-\mu _0}{\sigma _0}\right) }{f_0\left( \frac{x-\mu _0}{\sigma _0}\right) }\left( \frac{\mu '(0)\sigma _0+(x-\mu _0)\sigma '(0)}{\sigma _0^2}\right) h(x)\textrm{d}x\\&\quad +\int \frac{f_0'\left( \frac{x-\mu _0}{\sigma _0}\right) }{f_0\left( \frac{x-\mu _0}{\sigma _0}\right) }\left( \frac{\mu '(0)\sigma _0+(x-\mu _0)\sigma '(0)}{\sigma _0^2}\right) h(x)\textrm{d}x-\int \log f_0\left( \frac{x-\mu _0}{\sigma _0}\right) u(x)\textrm{d}x, \end{aligned}$$

where $h(x)=\frac{\partial }{\partial \theta }g_{\theta }(x)|_{\theta =0}$ and $u(x)=\frac{\partial ^2}{\partial \theta ^2}g_{\theta }(x)|_{\theta =0}$. Using the change of variable, rearranging terms in the above expression, and expanding $K(\theta )$ into a Maclaurin expansion completes the proof. $\square$

1.2 Regularity conditions B

Let the test statistic be a V-statistic of the form

$$\begin{aligned} T_n=\frac{1}{n^2}\sum _{i,j}\varPhi \Big (\frac{X_i-{\hat{\mu }}}{{\hat{\sigma }}},\frac{X_j-{\hat{\mu }}}{{\hat{\sigma }}}\Big ) \end{aligned}$$

for some degenerate kernel $\varPhi$. Then $b(\theta )$ has representation

$$\begin{aligned} b(\theta )&=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\varPhi \Big (\frac{x_1-\mu (\theta )}{\sigma (\theta )},\frac{x_2-\mu (\theta )}{\sigma (\theta )}\Big ) g_{\theta }(x_1)g_{\theta }(x_2)\textrm{d}x_1\textrm{d}x_2 \end{aligned}$$

(6.1)

$$\begin{aligned}&=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\varPhi (z_1,z_2) {\widetilde{g}}(z_1;\theta ){\widetilde{g}}(z_2;\theta )\textrm{d}z_1\textrm{d}z_2, \end{aligned}$$

(6.2)

where

$$\begin{aligned} {\widetilde{g}}(z;\theta )=\sigma (\theta )g_{\theta }(z\sigma (\theta )+\mu (\theta )). \end{aligned}$$

To allow a Taylor expansion of $b(\theta )$, suppose the following regularity conditions similar to A1-A4 are satisfied.

• B1 Let ${\hat{\mu }}$ and ${\hat{\sigma }}$ be consistent estimators of $\mu (\theta )$ and $\sigma (\theta )$, i.e. ${\hat{\mu }}\overset{P_{\theta }}{\rightarrow }\mu (\theta )$ and ${\hat{\sigma }}\overset{P_{\theta }}{\rightarrow }\sigma (\theta )$;

• B2 Functions $\mu (\theta )$ and $\sigma (\theta )$ are three times continuously differentiable;

• B3 Function ${\widetilde{g}}(x;\theta )$ is three times differentiable;

• B4 $\Big |\frac{\partial ^{i}{\widetilde{g}}(x;\theta )}{\partial \theta ^i}\frac{\partial ^{j}{\widetilde{g}}(y;\theta )}{\partial \theta ^j}\Big |<{\widetilde{M}}_{i,j}(x,y)$ for $i,j=0,1,2$, $i+j\le 3$, $k=1,2$, where ${\widetilde{M}}_{i,j}(x,y)$ are integrable functions.

Following Nikitin and Peaucelle (2004) and arguments therein the regularity conditions can be expressed in terms of ${\widetilde{g}}(z;\theta ),$ i.e. Assumptions WD:

• WD1
  $$\begin{aligned} \int _{0}^1t^3(1-t)^3\int _{-\infty }^{\infty }\varPhi (z_1,z_2){\widetilde{g}}'''(z_1;t\theta ){\widetilde{g}}(z_2;t\theta )\textrm{d}z_1\textrm{d}z_2\textrm{d}t<\infty ; \end{aligned}$$
• WD2
  $$\begin{aligned} \int _{0}^1t^3(1-t)^3\int _{-\infty }^{\infty }\varPhi (z_1,z_2){\widetilde{g}}''(z_1;t\theta ){\widetilde{g}}'(z_2;t\theta )\textrm{d}z_1\textrm{d}z_2\textrm{d}t<\infty . \end{aligned}$$

Remark

If the kernel function $\varPhi (\cdot )$ is uniformly bounded, then conditions B3 and B4 can be simplified by replacing ${\widetilde{g}}$ with g, while WD1 and WD2 follow from B3 and B4.

By way of example we consider the regularity conditions for the second Ley-Paindaveine alternative to the logistic null distribution. (If we consider the same alternative in the case of the normal distribution the regularity conditions may be shown by analogous arguments). Specifically for regularity conditions A, note that the density is:

$$\begin{aligned} g_\theta (x)=\frac{e^{-x}}{(1+e^{-x})^2}\Big (1-\theta \pi \cos \Big (\frac{\pi }{1+e^{-x}}\Big )\Big ). \end{aligned}$$

(For simplicity and without loss of generality here we can set the null parameters to $\mu =0$ and $\sigma =1$).

Condition A1. Given the MLEs for $\mu$ and $\sigma$ in the logistic model, the consistency of estimators under the considered alternative can be justified using arguments from Theorem 2.6.1. (from Subba Rao (2017)). In particular, the functions $\log g_\theta (x)$ and $\frac{\partial ^2}{\partial \theta ^2}\log g_{\theta }(x)$ are Lipschitz-continuous for $|\theta |\le \delta$, the parameter space is compact ($\{\theta : |\theta |\le \delta \}$), and point-wise convergence of the log-likelihood and its second derivative to their expectations hold due to law of large numbers.

Conditions A2 follows from the implicit function theorem.

Condition A3 is obviously satisfied.

Condition A4. Let $|\theta |\le \delta$. Then functions $\mu (\theta )$ and $\sigma (\theta )$ are bounded due to their continuity. The integrable dominants are available from the following bounds, where all constants C and $C_j$ depend on $\delta$ (and are different in each equation).

From the inequality $|\log \frac{e^{-x}}{(1+e^{-x})^2}|\le |x|+\ln 2$ we get

$$\begin{aligned} |L_1(x;\theta )|&\le |x|+C;\\ |L_2(x;\theta )|&\le C_1 |x|+C_0; \end{aligned}$$

For the derivatives of $L_1$ we have

$$\begin{aligned} \Big |\frac{\partial ^{i}}{\partial \theta ^{i}}L_1(x;\theta )\Big |&=\Bigg |\frac{\pi ^i \cos ^i\left( \frac{\pi }{1+e^{-x}}\right) }{\left( 1-\pi \theta \cos \left( \frac{\pi }{1+e^{-x}}\right) \right) ^i}\Bigg |\le C; \end{aligned}$$

The expressions for the derivatives of $L_2$ are too cumbersome to display. However, using boundness of functions $\frac{e^{-ax}}{(1+e^{-x})^b}$ for $a\le b$ and boundedness of derivatives of $\mu (\theta )$ and $\sigma (\theta )$, we get

$$\begin{aligned} \Big |\frac{\partial }{\partial \theta }L_2(x;\theta )\Big |&\le C_0+C_1x;\\ \Big |\frac{\partial ^{2}}{\partial \theta ^{2}}L_2(x;\theta )\Big |&\le C_0+C_1x+C_2x^2;\\ \Big |\frac{\partial ^{3}}{\partial \theta ^{3}}L_2(x;\theta )\Big |&\le C_0+C_1x+C_2x^2+C_3x^3. \end{aligned}$$

The density $g_\theta (x)$ itself and its first derivative is simply bounded by the standard logistic density as

$$\begin{aligned} |g_\theta (x)|&=\frac{e^{-x} }{\left( 1+e^{-x}\right) ^2} \left| 1-\pi \theta \cos \left( \frac{\pi }{e^{-x}+1}\right) \right| \le C \cdot \frac{e^{-x}}{(1+e^{-x})^2};\\ \Big |\frac{\partial }{\partial \theta }g_\theta (x)\Big |&=\frac{e^{-x} }{\left( 1+e^{-x}\right) ^2} \left| \pi \cos \left( \frac{\pi }{e^{-x}+1}\right) \right| \le C \cdot \frac{e^{-x}}{(1+e^{-x})^2}, \end{aligned}$$

and its higher derivatives are equal to zero.

The integrability of all dominants now follows form the finiteness of moments of the logistic distribution.

Turning to regularity conditions B, Condition B1 in the case of MLEs, coincides with Condition A1, while in the case of moment estimators, which we have in analytic form, it follows from the law of large numbers. Condition B2 in the case of MLEs, again coincides with Condition A2, while in the case on moment estimators it follows from the differentiability of the function $g(\cdot )$ and the finiteness of $\int _{-\infty }^{\infty }x^2|\frac{\partial ^k g_{\theta }(x)}{\partial \theta ^k}| dx, k=0,1,2,3$. Since the kernel $\varPhi$ of the test for logistic distribution is bounded, from the Remark above, conditions B3 and B4 follow from A3 and A4.