New consistent exponentiality tests based on V-empirical Laplace transforms with comparison of efficiencies

Abstract

We present new consistent goodness-of-fit tests for exponential distribution, based on the Desu characterization. The test statistics represent the weighted $L^2$ and $L^{\infty }$ distances between appropriate V-empirical Laplace transforms of random variables that appear in the characterization. In addition, we perform an extensive comparison of Bahadur efficiencies of different recent and classical exponentiality tests. We also present the empirical powers of new tests.

Appendices

Appendix A: Proofs of theorems

Proof (Theorem 2)

Our statistic $M^{{\mathcal {D}}}_{n,a}({\widehat{\lambda }}_n)$ can be rewritten as

$$\begin{aligned} \begin{aligned} M^{{\mathcal {D}}}_{n,a}({\widehat{\lambda }}_n)&=\int _0^{\infty }\left( \frac{1}{n^{2}}\sum _{i_1,i_2=1}^n\xi (X_{i_1},X_{i_2},t;a{\widehat{\lambda }}_n)\right) ^2e^{-at}dt\\&=\int _0^{\infty }V_n(t,{\widehat{\lambda }}_n)^2e^{-at}dt. \end{aligned} \end{aligned}$$

Here $V_n(t;{\widehat{\lambda }}_n)$, for each $t>0$, is a V-statistic of order 2 with an estimated parameter, and kernel

$$\begin{aligned} \xi (x_{1},x_{2},t;a,{\widehat{\lambda }}_n)=\frac{1}{2}\left( e^{-t{\hat{\lambda }}_nx_1}+e^{-t{\hat{\lambda }}_nx_2}-e^{-t{\hat{\lambda }}_n2\min (x_1,x_2)}\right) . \end{aligned}$$

Since the function $\xi (x_{1},x_{2},t;a,\gamma )$ is continuously differentiable with respect to $\gamma$ at the point $\gamma =\lambda$ we may apply the mean value theorem. We have

$$\begin{aligned} V_n(t;{\widehat{\lambda }}_n)=V_n(t;\lambda )+({\widehat{\lambda }}_n-\lambda )\frac{\partial V_n(t;\gamma )}{\partial \gamma }|_{\gamma =\lambda ^*}, \end{aligned}$$

(13)

for some $\lambda ^*$ between $\lambda$ and ${\widehat{\lambda }}_n$. From the Law of large numbers for V-statistics [47, 6.4.2.], the partial derivative $\frac{\partial V_n(t;\gamma )}{\partial \gamma }$ converges to

$$\begin{aligned} E\left( 2t\min \{X_1,X_2\}e^{-2t\min \{X_{1},X_{2}\}\gamma }-tX_{1}e^{-tX_{1}\gamma }\right) =0. \end{aligned}$$

(14)

Since $\sqrt{n}({\widehat{\lambda }}_n-\lambda )$ is stochastically bounded, it follows that statistics $\sqrt{n}V_n(t;{\widehat{\lambda }}_n)$ and $\sqrt{n}V_n(t;1)$ are asymptotically equally distributed. Therefore, $nM^{{\mathcal {D}}}_{n,a}({\widehat{\lambda }}_n)$ and $nM^{{\mathcal {D}}}_{n,a}(\lambda )$ will have the same limiting distribution. Hence we need to derive limiting distribution of $nM^{{\mathcal {D}}}_{n,a}(\lambda )$.

First notice that $M^{{\mathcal {D}}}_{n,a}(\lambda )$ is a V-statistic with symmetric kernel H. Also, since the distribution of $M^{{\mathcal {D}}}_{n,a}(\lambda )$ does not depend on $\lambda$ we may assume that $\lambda =1.$

It is easy to show that its first projection of kernel H on $X_1$ is equal to zero. After some calculations, we obtain that its second projection on $(X_1,X_2)$ is given by

$$\begin{aligned} {\widetilde{h}}_2(u,v;a)= & {} E(H(X_1,X_2,X_3,X_4;a,1)|X_1=u,X_2=v)\\= & {} \frac{1}{6}\bigg (3+\frac{1}{a+u+v}-\frac{2e^{-u}}{a+2u+v}-\frac{2e^{-v}}{a+u+2v}-(4-a)e^aEi(-a)\\&+e^\frac{a+v}{2}\Big (Ei\big (-\frac{a+v}{2}\big )-Ei\big (-\frac{a+2u+v}{2}\big )\Big )+e^{a+u}\Big (4Ei(-a-2u)\\&\quad -Ei(-a-u)\Big ) +e^\frac{a+u}{2}\Big (Ei\big (-\frac{a+u}{2}\big )-Ei\big (-\frac{a+u+2v}{2}\big )\Big ) \\&\quad +e^{a+v}\Big (4Ei(-a-2v)-Ei(-a-v)\Big ) -2(e^{-u}+e^{-v})\\&\quad +e^{\frac{a}{2}}\Big (-(4+a+2u)Ei(-\frac{a}{2}-u)+(a+4)Ei(-\frac{a}{2})\\&\quad +(a+2(2+u+v))Ei(-\frac{a}{2}-u-v) -(4+a+2v)Ei(-\frac{a}{2}-v)\Big )\\&\quad +\frac{e^{-u-v}}{a+2(u+v)}(2a+4(1+u+v))\bigg ), \end{aligned}$$

where $\text {Ei}(x)=-\int _{-x}^\infty \frac{e^{-t}}{t}dt$ is the exponential integral. The function ${\widetilde{h}}_2$ is non-constant for any $a>0$. Hence, kernel h is degenerate with degree 2.

Since the kernel H is bounded and degenerate, from the theorem on asymptotic distribution of U-statistics with degenerate kernels [26, Corollary 4.4.2], and the Hoeffding representation of V-statistics, we get that, $M^{{\mathcal {D}}}_{n,a}(1)$, being a V-statistic of degree 2, has the following asymptotic distribution

$$\begin{aligned} nM^{{\mathcal {D}}}_{n,a}(1)\overset{d}{\rightarrow }6\sum _{k=1}^\infty \delta _kW^2_k, \end{aligned}$$

(15)

where $\{\delta _k\}$ are the eigenvalues of the integral operator ${\mathcal {M}}_a$ defined by

$$\begin{aligned} {\mathcal {M}}_{a}q(x)=\int _{0}^{+\infty }{\widetilde{h}}_2(x,y;a)q(y)dF(y), \end{aligned}$$

(16)

and $\{W_{k}\}$ is the sequence of i.i.d. standard Gaussian random variables. $\square$

Proof (Theorem 3)

The test statistic can be represented as

$$\begin{aligned} L^{{\mathcal {D}}}_{n,a}= \sup \limits _{t\ge 0}|V_n(t;{\widehat{\lambda }}_n)e^{-at}|, \end{aligned}$$

(17)

where $\{V_n(t;{\widehat{\lambda }}_n)\}$ is a V-empirical process introduced in the proof of Theorem 2.

Substituting $s=e^{-t}$ in (17) we can express our statistic as

$$\begin{aligned} L^{{\mathcal {D}}}_{n,a}=\sup _{s\in (0,1)}|V_{n}(-\ln s;{\hat{\lambda }}_n)s^a|, \end{aligned}$$

thus obtaining, as a core of the statistic, the process $V_{n}(-\ln s;{\hat{\lambda }}_n)$, $s\in (0,1)$, defined on C[0, 1] equipped with supremum norm.

Next we show that the difference between $\sqrt{n}L^{{\mathcal {D}}}_{n,a}$ and $\sqrt{n}\sup _{s\in (0,1)}|V_{n}(-\ln s;\lambda )s^a|$ (for a fixed $\lambda$) tends uniformly to zero and proceed finding the limiting distribution of the latter.

This is, taking into account (13), equivalent to $\lim _{n\rightarrow \infty }\sup _{s\in (0,1)}R_n(s)=0$, where

$$\begin{aligned} R_n(s)&=s^a\Big |\frac{1}{n^2}\sum _{i,j}\Big (-2\ln s\min \{X_1,X_2\}s^{2\min \{X_{1},X_{2}\}\gamma }+\ln s X_{1}s^{X_{1}\gamma }\Big )\Big |\\&= -\ln s\cdot s^a \Big |\frac{1}{n^2}\sum _{i,j}\Big (2\min \{X_1,X_2\}s^{2\min \{X_{1},X_{2}\}\gamma }- X_{1}s^{X_{1}\gamma }\Big )\Big |. \end{aligned}$$

From (14) we know that the expression in the absolute parentheses tends to zero for each s, and using [44, Lemma 1] we get that its supremum also tends to zero. Since $-\ln s\cdot s^a$ is bounded function, $\sup _s R_n(s)$ tends also to zero.

Since, for a fixed s, $\sqrt{n}V_n(-\ln s;\lambda )$ is a non-degenerate V-statistic, $\sqrt{n}V_n(-\ln s;\lambda )$ is asymptotically normally distributed. The same holds for finite-dimensional distributions of the process $\sqrt{n}V_n(-\ln s;\lambda )$. In addition, it can be shown that $\sqrt{n}V_n(-\ln s;\lambda )$ satisfies conditions from Billingsley [8, Theorem 12.3], and is, therefore, tight in C[0, 1]. Therefore it converges weakly to a zero mean Gaussian process $\{\eta ^\star (s)\}$ with covariance function

$$\begin{aligned} \begin{aligned} K^\star (u,v)=K(-\ln {u},-\ln {v}), \end{aligned} \end{aligned}$$

where K is defined in (3).

Since the supremum is continuous on C[0, 1], using the continuous mapping theorem we get that $\sqrt{n}\sup _{s\in (0,1)}|V_{n}(-\ln s;\lambda )s^a|$ converges to $\sup _{s\in (0,1)}|\eta ^\star (s)s^a|$, which is the same as $\sup _{t>0}|\eta (t)e^{-at}|$. This completes the proof. $\square$

Proof (Lemma 4)

Using the result of Zolotarev [55], the logarithmic tail behavior of limiting distribution function of ${\widetilde{M}}_{n,a}^{{\mathcal {D}}}({\widehat{\lambda }}_n)=\sqrt{nM^{{\mathcal {D}}}_{n,a}({\widehat{\lambda }}_n)}$ is

$$\begin{aligned} \ln (1-F_{{\widetilde{M}}_a}(t))=-\frac{t^2}{12\delta _1}+o(t^2),\;\;t\rightarrow \infty . \end{aligned}$$

Therefore, $a_{{\widetilde{M}}_a}=\frac{1}{6\delta _1}.$ The limit in probability $P_{\theta }$ of ${\widetilde{M}}_{n,a}({\widehat{\lambda }}_n)/\sqrt{n}$ is

$$\begin{aligned} b_{{\widetilde{M}}_{a}}=\sqrt{b_{M}(\theta )}. \end{aligned}$$

The expression for $b_M(\theta )$ is derived in the following lemma.

Lemma A1

For a given alternative density $g(x;\theta )$ whose distribution belongs to ${\mathcal {G}}$, we have that the limit in probability of the statistic $M^{{\mathcal {D}}}_{n,a}({\widehat{\lambda }}_n)$ is

$$\begin{aligned} b_M(\theta )=6\int \limits _{0}^{\infty }\int \limits _{0}^{\infty }{\widetilde{h}}_2(x,y;a)g'_{\theta }(x;0)g'_{\theta }(y;0)dxdy\cdot \theta ^2+o(\theta ^2), \theta \rightarrow 0. \end{aligned}$$

Proof

For brevity, denote ${\varvec{x}}=(x_1,x_2,x_3,x_{4})$ and ${\varvec{G}}({\varvec{x}};\theta )=\prod _{i=1}^{4}G(x_i;\theta )$. Since ${\overline{X}}_n$ converges almost surely to its expected value $\mu (\theta )$, using the Law of large numbers for V-statistics with estimated parameters (see [22]), $M^{{\mathcal {D}}}_{n,a}({\widehat{\lambda }}_n)$ converges to

$$\begin{aligned} \begin{aligned} b_M(\theta )\!&=\!E_{\theta }(H({\varvec{X}},a;\mu (\theta )))\\&\!=\!\int \limits _{{(R^+)}^{4}}\!\Big (\!\frac{\mu (\theta )}{2\min \{x_{1},x_{2}\}\!+\!2\min \{x_{3},x_{4}\}\!+\!a\mu (\theta )}-\!\frac{\mu (\theta )}{x_{1}\!+\!2\min \{x_{3},x_{4}\}\!+\!a\mu (\theta )}\\&\quad \!-\!\frac{\mu (\theta )}{x_{3}+2\min \{x_{1},x_{2}\}+a\mu (\theta )}\!+\!\frac{\mu (\theta )}{x_1+x_{3}+a\mu (\theta )}\Big )d{\varvec{G}}({\varvec{x}};\theta ). \end{aligned} \end{aligned}$$

We may assume that $\mu (0)=1$ since the test statistic is ancillary for $\lambda$ under the null hypothesis. After some calculations we get that $b'_M(0)=0$ and that

$$\begin{aligned} \begin{aligned} b''(0)&=\int _{(R^+)^{4}}H({\varvec{x}},a;1)\frac{\partial ^2}{\partial \theta ^2}d{\varvec{G}}({\varvec{x}},0) =6\int _{(R^+)^2}{\widetilde{h}}_2(x,y)g'_{\theta }(x;0)g'_{\theta }(y;0)dxdy. \end{aligned} \end{aligned}$$

Expanding $b_M(\theta )$ into the Maclaurin series

$$\begin{aligned} b_M(\theta )= b_M(0)+b'_M(0)\cdot \theta +b''_M(0)\theta ^2+o(\theta ^2),\;\theta \rightarrow 0, \end{aligned}$$

we complete the proof. $\square$

Now we pass to the statistic $L^{{\mathcal {D}}}_n.$ The tail behaviour of the random variable $\sup _{t>0}|n_{t}|$ is equal to the inverse of supremum of its covariance function, i.e. the $a_L=\frac{1}{\sup _{t>0}K(t,t)}$ (see [27, 36]).

Similarly like before, since ${\overline{X}}_n$ converges almost surely to its expected value $\mu (\theta )$, using the Law of large numbers for V-statistics with estimated parameters [22], $V_n(t,a;{\hat{\lambda }})e^{-at}$ converges to

$$\begin{aligned} \begin{aligned} b_L(\theta ;t)&=E_{\theta }(\varPhi (X_1,X_2;t,a,\mu (\theta ))). \end{aligned} \end{aligned}$$

Expanding $b_L(\theta ;t)$ in the Maclaurin series we obtain

$$\begin{aligned} b_L(\theta ;t)=2\int _{0}^{\infty }{\widetilde{\varphi }}_1(x,t;a)g'_{\theta }(x;0)dx\cdot \theta +o(\theta ), \end{aligned}$$

where ${\widetilde{\varphi }}_1(x,t;a)=E(\varPhi (X_1,X_2,t;a,1)|X_1=x_1).$

Since V-empirical Laplace transforms are monotonous functions, a Glivenko-Cantelli-type theorem holds, see Novoa-Muñoz and Jiménez-Gamero [44, Lemma 1]. Hence, the limit in probability under the alternative for statistics $L_{n,a}^{{\mathcal {D}}}$ is equal to $\sup _{t\ge 0}|b_L(\theta ;t)|$. Inserting this into the expression for the Bahadur slope completes the proof. $\square$

Appendix B: Bahadur approximate slopes

Proof

Approximate local Bahadur slope of statistics EP and CO

Those statistics can be represented as

$$\begin{aligned} T_n=\frac{1}{n}\sum _{i=1}^n\varPhi (X;{\hat{\mu }}), \end{aligned}$$

where $\varPhi (x;\gamma )$ is continuously differentiable with respect to $\gamma$ at point $\gamma =\mu .$ It was shown that the limiting distribution of $\sqrt{n}T_n$ is zero mean normal with variance $\sigma ^2_{\varPhi }$ (see [10, 13]). Hence, the coefficient $a_T$ is equal to $\frac{1}{\sigma ^2_{\varPhi }}.$

Further, we have

$$\begin{aligned} b(\theta )= & {} E_{\theta }(\varPhi (X;\mu (\theta )))=\int \limits _0^\infty \varPhi (x;\mu (\theta ))dG(x;\theta ) \\ b'(\theta )= & {} \int \limits _0^\infty \frac{\partial }{\partial \mu }\varPhi (x;\mu (\theta ))\frac{\partial }{\partial \theta }\mu (\theta )dG(x;\theta )+\int \limits _0^\infty \varPhi (x;\mu (\theta ))\frac{\partial }{\partial \theta }dG(x;\theta ). \end{aligned}$$

Then it holds that

$$\begin{aligned} \begin{aligned} b(\theta )&=b(0)+b'(0)\theta +o(\theta )\\&=\Bigg (\mu '(0)\int \limits _0^\infty \varPhi '(x;1)g(x;0)dx+\int \limits _0^\infty \varPhi (x;1)g'(x;0)dx)\bigg )\theta +o(\theta ). \end{aligned} \end{aligned}$$

From this we obtain the expression for $c_T(\theta ).$ $\square$

Proof

Approximate local Bahadur slope of statistics BH, HE, Wn, HM, $\omega ^2$ and AD

Let T be the one of considered statistics. It was shown that the limiting distribution of $nT_n$ is $\sum _{i=1}^{\infty }\delta _iW_i^2,$ where $\{W_i\}$ is the sequence of i.i.d. standard normal variables and $\{\delta _i\}$ the sequence of eigenvalues of certain covariance operator. Using the result of Zolotarev in [55], we have that the logarithmic tail behavior of limiting distribution function of ${\widetilde{T}}_n=\sqrt{n T_n}$ is

$$\begin{aligned} \ln (1-F_{{\widetilde{T}}}(s))=-\frac{s^2}{2\delta _1}+o(s^2), s\rightarrow \infty . \end{aligned}$$

Next, the limit in probability of ${\widetilde{T}}_n/\sqrt{n}$ is $b_{{\widetilde{T}}}(\theta )=\sqrt{b_T(\theta )}$. Statistic $T_n$ can be represented as

$$\begin{aligned} T_{n}=\frac{1}{n^2}\sum _{k,j=1}^n\varPhi (X_k,X_j;{\hat{\mu }}). \end{aligned}$$

As before, we may assume that $\mu (0)=1$. Since the sample mean converges almost surely to its expected value, by using the Law of large numbers for V-statistics with estimated parameters (see [22]), we can conclude that the limit in the probability of statistic $T_n$ is equal to the one of

$$\begin{aligned} b_T(\theta )=E_{\theta }(\varPhi (X_1,X_2;\mu (\theta )))=\int \limits _{0}^\infty \int \limits _{0}^\infty \varPhi (x,y;\mu (\theta ))g(x;\theta )g(y;\theta )dxdy. \end{aligned}$$

We get that $b_T'(0)=0$ and that

$$\begin{aligned} \begin{aligned} b_T''(0)&\!=\!2\int \limits _{0}^\infty \!\int \limits _{0}^\infty \varPhi (x,y;1)g'_{\theta }(x;0)g'_{\theta }(y;0)dxdy\\&\quad + 4\mu '(0)\int \limits _{0}^\infty \!\int \limits _{0}^\infty \varPhi '(x,y;1)g(x;0)g'_{\theta }(y;0)dxdy\\&\quad +(\mu '(0))^2\int \limits _{0}^\infty \int \limits _{0}^\infty \varPhi ''(x,y;1)g(x;0)g(y;0)dxdy, \end{aligned} \end{aligned}$$

Expanding $b_T(\theta )$ into Maclaurin series we obtain expression for $b_T$. $\square$

Appendix C: Tables of efficiencies

Table 3 Relative Bahadur efficiency with respect to LRT