D. M. CHIBISOV

ON THE ASYMPTOTIC POWER AND EFFICIENCY OF THE CRITERION \(\omega_n^2\)

(Presented by Academician A. N. Kolmogorov, 20 XII 1960)

§ 1.

Let \(x_1, x_2, \ldots, x_n\) be a sample of \(n\) independent observations of a random variable with distribution function \(F(x)\). We shall denote the empirical distribution function of such a sample by \(F^{(n)}(x)\). The criterion \(\omega_n^2\) is a criterion for testing the simple hypothesis \(F(x)=F_0(x)\), based on the statistic

\[ \omega_n^2(F)=n\int_{-\infty}^{\infty}\bigl[F^{(n)}(x)-F_0(x)\bigr]^2\psi(F_0(x))\,dF_0(x), \tag{1} \]

where \(\psi(u)\ge 0\) \((0\le u\le 1)\) is a certain weight function. We introduce the argument \(F\) to denote the true distribution of the sample. Under the assumption \(F(x)=F_0(x)\), the limiting distribution of the quantity \(\omega_n^2\) as \(n\to\infty\) was studied by N. V. Smirnov \((^1)\) and by Anderson and Darling \((^2)\).

Denote
\[ W^{(n)}(x;F)=\mathbf P\{\omega_n^2(F)<x\} \]
and
\[ P_\alpha^{(n)}(F)=1-W^{(n)}(x_\alpha;F), \]
where \(x_\alpha\) is the root of the equation \(W(x;F_0)=1-\alpha\). At significance level \(\alpha\), \(P_\alpha^{(n)}(F)\) is the power function of the criterion (the probability of rejecting the hypothesis if the alternative \(F(x)\) is true).

Denote

\[ \rho_F^2=\int_{-\infty}^{\infty}\bigl[F(x)-F_0(x)\bigr]^2\psi(F_0(x))\,dF_0(x). \tag{2} \]

Let the function \(\delta(u)\), \(0\le u\le 1\), be such that \(\delta(0)=\delta(1)=0\) and

\[ \int_0^1 \delta^2(u)\psi(u)\,du=1. \tag{3} \]

We shall call the class of functions of the form
\[ F_a(x)=F_0(x)+\frac{a}{\sqrt n}\,\delta(F_0(x)) \]
the class \([\delta(u)]\). By (3), \(a^2=n\rho_{F_a}^2\). For functions \(F_a(x)\in[\delta(u)]\), we introduce the notation \(\omega_n^2(a)\), \(W^{(n)}(x,a)\), and \(P_\alpha^{(n)}(a)\) instead of \(\omega_n^2(F_a)\), \(W^{(n)}(x;F_a)\), and \(P_\alpha^{(n)}(F_a)\).

Denote by \(\lambda_j\) and \(f_j(u)\) \((j=1,2,\ldots)\) the eigenvalues and eigenfunctions of the integral equation

\[ f(u)=\lambda\int_0^1 K(u,v)f(v)\,dv, \]

where
\[ K(u,v)=[\min(u,v)-uv]\sqrt{\psi(u)}\sqrt{\psi(v)},\qquad 0\le u,v\le 1. \]
The kernel \(K(u,v)\) is positive definite, whence \(\lambda_j>0\) \((j=1,2,\ldots)\); we shall assume that \(\lambda_k\ge \lambda_j\) for \(k>j\). In addition, the system of eigenfunctions \(\{f_j(u)\}\) may be chosen orthonormal:

\[ \int_0^1 f_j(u) f_k(u)\,du=\delta_{jk}. \]
By \(D(\lambda)\) we shall denote the Fredholm determinant of the integral equation.

§ 2. Suppose:

I. The functions \(F_0(x)\) and \(\delta(u)\) are continuous.

II. The function \(\psi(u)\) is continuous in any interval \(0<u_1\le u\le u_2<1\), and the integral
\[ \int_0^1 u(1-u)\psi(u)\,du=\int_0^1 K(u,u)\,du \]
exists.

III. The integral
\[ \int_0^1 \delta(u)\psi(u)\,du \]
exists.

IV. For the expansion
\[ \delta(u)\sqrt{\psi(u)}=\sum_{k=1}^{\infty}\delta_k f_k(u), \]
where
\[ \delta_k=\int_0^1 f_k(u)\delta(u)\sqrt{\psi(u)}\,du, \]
the closure condition is satisfied.

Theorem 1. If conditions I–III are satisfied, \(W^{(n)}(x,a)\), for each \(a\), converges weakly as \(n\to\infty\) to \(W(x,a)=\mathbf P\{\omega^2(a)<x\}\), where
\[ \omega^2(a)=\int_0^1 [y(u)+a\delta(u)]^2\psi(u)\,du, \tag{4} \]
\(y(u)\), \(0\le u\le 1\), is a Gaussian random process with \(\mathbf M y(u)=0\), \(\mathbf M y(u)y(v)=\min(u,v)-uv\).

Theorem 2. Under conditions I–IV, \(W(x,a)\) has the characteristic function
\[ \varphi(t,a)=\frac{1}{\sqrt{D(2it)}}\exp\left\{a^2\sum_{k=1}^{\infty}\frac{it\lambda_k\delta_k^2}{\lambda_k-2it}\right\}. \tag{5} \]

Let us note the expansion from which (5) is obtained:
\[ \omega^2(a)=\sum_{k=1}^{\infty}\left(\frac{X_k}{\sqrt{\lambda_k}}+a\delta_k\right)^2, \tag{6} \]
where \(\{X_k\}\) are independent normally \((0,1)\) distributed quantities.

Theorem 3. As \(a\to\infty\),
\[ W(x,a)-\Phi\left(\frac{x-a^2}{2a\sigma}\right)\to 0 \tag{7} \]
uniformly in \(x\). Here
\[ \Phi(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{x} e^{-t^2/2}\,dt \]
and
\[ \sigma^2=\int_0^1\int_0^1 K(u,v)\delta(u)\delta(v)\sqrt{\psi(u)}\sqrt{\psi(v)}\,du\,dv. \tag{8} \]

Proof. The quantity
\[ \xi=\int_0^1 y(u)\delta(u)\psi(u)\,du \]
is normally distributed with parameters \((0,\sigma)\). From (4) and (3) we obtain
\[ W(x,a)=\mathbf P\{\omega^2(a)<x\} =\mathbf P\{\omega^2(0)+2a\xi+a^2<x\} \]
\[ =\mathbf P\left\{\frac{\omega^2(0)}{2a}+\xi<\frac{x-a^2}{2a}\right\}, \tag{9} \]
whence (7) follows.

Chapman [3] was the first to point out the validity of (7), proceeding from somewhat different considerations.

1. It can be shown that, for each \(x\), \(W^{(n)}(x,a)\to W(x,a)\) uniformly in \(a\).

2. From (5) it follows that \(W(x,a)=W(x,-a)\); we shall therefore assume \(a\geqslant 0\).

3. From (6) it follows that, for each \(x>0\), the function \(W(x,a)\) decreases monotonically (in \(a\)), since the distribution function of each of the terms has this property. Hence it follows that \(P_\alpha(a)\) increases monotonically; in particular, \(P_\alpha(a)>P_\alpha(0)=\alpha\), if \(a>0\). Thus, \(\omega^2\) is an asymptotically unbiased test.

4. From (9) it follows that, for all \(x\) and \(a\),

\[ W(x,a)<\Phi\left(\frac{x-a^2}{2a\sigma}\right), \]

which gives a lower bound for the power function:

\[ P_\alpha(a)>1-\Phi\left(\frac{x_\alpha-a^2}{2a\sigma}\right). \]

1. From (8) and (3) we obtain the estimate

\[ \sigma^2=\sum_{k=1}^{\infty}\frac{\delta_k^2}{\lambda_k}<\frac{1}{\lambda_1}. \]

Now from (10) it follows that, for the family of alternatives \(\{F(x)\}\) with

\[ \rho_F\geqslant \frac{\sqrt{x_\alpha}}{\sqrt{n}}, \]

\[ P_\alpha(F)\geqslant 1-\Phi\left(\frac{x_\alpha-n\rho_F^2}{2\rho_F\sqrt{n}}\sqrt{\lambda_1}\right) \]

uniformly for all distribution functions of the family. In particular, if a sequence of alternatives \(\{F_n(x)\}\) is such that \(\rho_{F_n}\sqrt{n}\to\infty\), then \(P_\alpha(F_n)\to 1\).

1. For \(a\leqslant x\)

\[ W(x,a)=\mathrm{P}\{\omega^2(0)+2a\xi<x-a^2\}> \mathrm{P}\left\{\omega^2(0)<x-a^2-2a\frac{1}{\sqrt{a}}\right\}- \]

\[ -\mathrm{P}\left\{\xi>\frac{1}{\sqrt{a}}\right\}\geqslant W(x-a^2-2\sqrt{a},0)- \left[1-\Phi\left(\left(\frac{1}{\sqrt{a}}-\sqrt{x}\right)\sqrt{\lambda_1}\right)\right], \]

or, for \(a\leqslant x_\alpha\),

\[ P_\alpha(a)\leqslant 1-W(x_\alpha-a^2-2\sqrt{a},0)- \left[1-\Phi\left(\left(\frac{1}{\sqrt{a}}-\sqrt{x_\alpha}\right)\sqrt{\lambda_1}\right)\right]. \]

Hence, if a sequence \(\{F_n(x)\}\) is such that \(\rho_{F_n}=o(1/\sqrt{n})\), then \(P_\alpha(F_n)\to\alpha\).

1. From (5) the semi-invariants of \(W(x,a)\) are easily found:

\[ k_n(a)=2^{\,n-1}(n-1)!\left(\sum_{j=1}^{\infty}\frac{1}{\lambda_j^n} +a^2 n\sum_{j=1}^{\infty}\frac{\delta_j^2}{\lambda_j^{\,n-1}}\right) =2^{\,n-1}(n-1)!\left(\int_0^1 K_n(u,u)\,du+\right. \]

\[ \left. +a^2 n\int_0^1\int_0^1 K_{n-1}(u,v)\delta(u)\delta(v)\sqrt{\psi(u)}\sqrt{\psi(v)}\,du\,dv\right), \]

where \(K_n(u,v)\) is the \(n\)-th iteration of the kernel \(K(u,v)\).

1. Put

\[ \nu_n(x_\alpha)=\int_{+0}^{\infty} a^n\,dP_\alpha(a). \]

Using (5), one can find the Laplace transform of \(\nu_n(x)\):

\[ \widetilde{\nu}_n(p)=\int_0^\infty e^{-px}\nu_n(x)\,dx =\Gamma(n/2+1)\left|p\sqrt{D(-2p)}\right| \left(\sum_{k=1}^{\infty}\frac{\lambda_k\delta_k^2 p}{\lambda_k+2p}\right)^{n/2}. \]

§ 3. To compute the asymptotic efficiency of the criterion in the general case, we shall find, for the family of alternatives \([\delta(u)]\), the asymptotically most powerful criterion and its power function \(P_\alpha^*(a)\). Then, if \(P_\alpha^*(a^*)=P_\alpha(a)=1-\beta\), where \(\beta\) is the prescribed error of the second kind, the asymptotic efficiency of the criterion is equal to \(e=a^{*2}/a^2\).

Under condition I, without loss of generality one may take \(F_0(x)=x\), \(0\leq x\leq 1\). Then \(F_a(x)=x+\dfrac{a}{\sqrt n}\delta(x)\).

Theorem 4. Suppose that \(\delta(u)\) is differentiable and that the integral

\[ s^2=\int_0^1[\delta'(u)]^2\,du \]

exists. Then, if \(F(x)=F_a(x)\), the quantity

\[ T^{(n)}=\frac{1}{\sqrt n}\sum_{i=1}^n \delta'(x_i) \]

is asymptotically normally distributed \((as^2,\ s)\), and the criterion based on the statistic \(T^{(n)}\) is the asymptotically most powerful criterion for testing the hypothesis \(F(x)=x\) against the alternative \(F(x)=F_a(x)\).

Proof. Since \(T^{(n)}\) is a sum of independent identically distributed terms, the first assertion follows directly from the central limit theorem. The second assertion follows from the fact that the criterion \(T^{(n)}\) is asymptotically equivalent to the criterion based on the likelihood ratio

\[ \sum_{i=1}^n \ln\left[1+\frac{a}{\sqrt n}\delta'(x_i)\right]. \]

Denote by \(z_\alpha\) the root of the equation \(1-\Phi(z)=\alpha\). The power function of the criterion \(T^{(n)}\) is asymptotically equal to \(\Phi(|a|s-z_\alpha)\). Using (10), we obtain

\[ e\geq \frac{z_\alpha+z_\beta} {s\left(\sigma z_\beta+\sqrt{\sigma^2 z_\beta^2+\chi_\alpha}\right)}. \]

For example, for the criterion with \(\psi\equiv 1\) under the hypothesis \(\Phi(x)\) and the alternatives \(\Phi(x-\mu)\) and \(\Phi\left(\dfrac{x}{1+\vartheta}\right)\), the efficiencies (\(e_\mu\) and \(e_\vartheta\), respectively) will be: for \(\alpha=\beta=0.05\), \(e_\mu\geq 0.53,\ e_\vartheta\geq 0.18\); for \(\alpha=\beta=0.01\), \(e_\mu\geq 0.56,\ e_\vartheta\geq 0.20\); for \(\alpha=\beta=0.001\), \(e_\mu\geq 0.58,\ e_\vartheta\geq 0.21\).

Steklov Mathematical Institute
Academy of Sciences of the USSR

Received
17 XII 1960

REFERENCES

1. N. V. Smirnoff, C. R., 202, 449 (1936); N. V. Smirnov, Matem. sborn., 2 (44), no. 5, 973 (1937); UMN, 4, 4 (32), 196 (1949).
2. T. W. Anderson, D. A. Darling, Ann. Math. Stat., 23, 2, 193 (1952).
3. D. G. Chapman, Ann. Math. Stat., 29, 3, 655 (1958).