Statistical inference for \(L^2\)-distances to uniformity

Abstract

The paper deals with the asymptotic behaviour of estimators, statistical tests and confidence intervals for \(L^2\)-distances to uniformity based on the empirical distribution function, the integrated empirical distribution function and the integrated empirical survival function. Approximations of power functions, confidence intervals for the \(L^2\)-distances and statistical neighbourhood-of-uniformity validation tests are obtained as main applications. The finite sample behaviour of the procedures is illustrated by a simulation study.

This is a preview of subscription content, access via your institution.

References

  1. Abate J, Valkó PP (2004) Multi-precision Laplace transform inversion. Int J Numer Methods Eng 60:979–993

    Article  MATH  Google Scholar 

  2. Achieser NI (1992) Theory of approximation. Dover Publications, New York

    MATH  Google Scholar 

  3. Bahadur RR (1971) Some limit theorems in statistics. SIAM, Philadelphia

    Book  MATH  Google Scholar 

  4. Baringhaus L, Henze N (2017) Cramér-von Mises distance: probabilistic interpretation, confidence intervals, and neighbourhood-of-model validation. J Nonparametr Stat 29:167–188

    MathSciNet  Article  MATH  Google Scholar 

  5. Baringhaus L, Ebner B, Henze N (2017) The limit distribution of weighted \(L^2\)-goodness-of-fit statistics under fixed alternatives, with applications. Ann Inst Stat Math 69:969–995

    Article  MATH  Google Scholar 

  6. Bingham N, Goldie C, Teugels J (1989) Regular variation. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  7. Courant R, Hilbert D (1968) Methoden der Mathematischen Physik I. Springer, Berlin

    Book  MATH  Google Scholar 

  8. Csörgő S, Faraway J (1996) The exact and asymptotic distributions of Cramér–von Mises statistics. J R Stat Soc Ser B 58:221–234

    MATH  Google Scholar 

  9. Durbin J, Knott M (1972) Components of Cramér–von Mises statistics. I. J R Stat Soc Ser B 34:290–307

    MATH  Google Scholar 

  10. Gao F, Hannig J, Lee T-Y, Torcaso F (2003) Laplace transforms via Hadamard factorization. Electron J Probab 8(13):20

    MathSciNet  MATH  Google Scholar 

  11. Henze N, Nikitin Y (2000) A new approach to goodness-of-fit testing based on the integrated empirical process. J Nonparametr Stat 12:391–416

    MathSciNet  Article  MATH  Google Scholar 

  12. Kallenberg WCM, Ledwina T (1987) On local and nonlocal measures of efficiency. Ann Stat 15:1401–1420

    MathSciNet  Article  MATH  Google Scholar 

  13. Kamke E (1983) Differentialgleichungen. Lösungsmethoden und Lösungen I. Teubner, Stuttgart

    MATH  Google Scholar 

  14. Khoshnevisan D, Shi Z (1998) Chung’s law for integrated Brownian motion. Trans Am Math Soc 350:4253–4264

    MathSciNet  Article  MATH  Google Scholar 

  15. Klar B (2001) Goodness-of-fit test for the exponential and the normal distribution based on the integrated empirical distribution function. Ann Inst Stat Math 53:338–353

    Article  MATH  Google Scholar 

  16. Kourouklis S (1989) On the relation between Hodges–Lehmann efficiency and pitman efficiency. Can J Stat 17:311–318

    MathSciNet  Article  MATH  Google Scholar 

  17. Nazarov A, Nikitin Y (2004) Exact \(L_2\)-small ball behavior of integrated Gaussian processes and spectral asymptotics of boundary value problems. Probab Theory Relat Fields 129:469–494

    Article  MATH  Google Scholar 

  18. Nikitin Y (1995) Asymptotic efficiency of nonparametric tests. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  19. Pollard D (1984) Convergence of stochastic processes. Springer, New York

    Book  MATH  Google Scholar 

  20. Serfling R (1980) Approximation theorems of mathematical statistics. Wiley, New York

    Book  MATH  Google Scholar 

  21. Titchmarsh EC (1939) The theory of functions. Oxford University Press, Oxford

    MATH  Google Scholar 

  22. Wellek S (2010) Testing statistical hypotheses of equivalence and noninferiority. CRC Press, Boca Raton

    Book  MATH  Google Scholar 

Download references

Acknowledgements

The authors thank the referees for constructive comments and suggestions.

Author information

Affiliations

Authors

Corresponding author

Correspondence to L. Baringhaus.

Appendix

Appendix

We introduce the entire function

$$\begin{aligned} d(z)=\frac{1}{2}\left( \cosh (z)\sin (z)+\sinh (z)\cos (z)\right) ,~z\in \mathbb {C}, \end{aligned}$$

and note that the real positive zeros of d are just the positive solutions \(\kappa _j,\,j\ge 1,\) of the equation (2.1). We aim to derive the representation of d as an infinite product involving the zeros of d. Obviously, 0 is a simple zero of d. Observing

$$\begin{aligned}\sin (-z)&=-\sin (z),~\sinh (-z)=-\sinh (z),\\ \cos (-z)&=\cos (z),~\cosh (-z)=\cosh (z) \end{aligned}$$

and

$$\begin{aligned} \sin (iz)=i\sinh (z),~\sinh (iz)=i\sin (z),~\cos (iz)=\cosh (z),~\cosh (iz)=\cos (z) \end{aligned}$$

for each \(z\in \mathbb {C},\) we note that if \(0\ne z_0\in \mathbb {C}\) is a zero of d,  also \(-z_0,~iz_0,\) and \(-iz_0\) are zeros of d. Thus, with \(\kappa _k,k\in \mathbb {N},\) as the real positive zeros of d,  we recognise that \(-\kappa _k,k\in \mathbb {N},\) \(i\kappa _k,k\in \mathbb {N},\) and \(-i\kappa _k,k\in \mathbb {N},\) are zeros of d. Adding the zero 0, we assert that this is the whole set of zeros of d. To see that the assertion is true, it suffices to prove the following lemma.

Lemma 4

There is no zero \(z=x+iy\) of d with real positive x and real positive y.

Proof

Assume, there is a zero \(z=x+iy\) of d with real positive x and real positive y. Then,

$$\begin{aligned} \cosh (x+y)\sin (x-y)+\sinh (x-y)\cos (x+y)&=0,\\ \cosh (x-y)\sin (x+y)+\sinh (x+y)\cos (x-y)&=0. \end{aligned}$$

Due to \(\sin u +\sinh u>0\) for real positive u,  it is \(x\ne y.\) We can (and do) assume without loss of generality that \(x>y.\) Putting \(a=x+y,~b=x-y\) we have

$$\begin{aligned} \begin{aligned} \cosh a\sin b+\sinh b\cos a&=0,\\ \cosh b\sin a+\sinh a\cos b&=0. \end{aligned} \end{aligned}$$
(6.1)

From this, we deduce that

$$\begin{aligned} \begin{aligned} \cosh ^2 a \sin ^2 b&= \sinh ^2 b \cos ^2 a,\\ \cosh ^2 b \sin ^2 a&= \sinh ^2 a \cos ^2 b. \end{aligned} \end{aligned}$$
(6.2)

Adding the corresponding terms on the right and the left hand side in (6.2) and using (6.2) we obtain

$$\begin{aligned} \cosh ^2 a=\sinh ^2 b+ \sin ^2 a +\cos ^2 b \end{aligned}$$

and

$$\begin{aligned} \cosh ^2 b=\sinh ^2 a+ \sin ^2 b +\cos ^2 a. \end{aligned}$$

It follows that

$$\begin{aligned} \cosh 2a + \cos 2a =\cosh 2b + \cos 2b. \end{aligned}$$

This is impossible, because the function \(\cosh u + \cos u,\,u>0,\) is strictly increasing. \(\square \)

Theorem 4

(Gao et al. (2003)) The Laplace Transform of the limit distribution is given by

$$\begin{aligned} L(t)&=\left( \frac{2z(t)}{\sinh (z(t))+\sin (z(t))}\right) ^{1/2},~t\ge 0, \end{aligned}$$

where \(z(t)=2^{1/2}(2t)^{1/4}\) for \(t\ge 0.\)

Proof

Note that

$$\begin{aligned} f(z)=\frac{d(z)}{z},~z\in \mathbb {C}, \end{aligned}$$

is an entire function of order 1 and that \(f(0)=1.\) Due to

$$\begin{aligned} \left( j-\frac{1}{2}\right) \pi<\kappa _j<j\pi ,~j\ge 1, \end{aligned}$$

the genus of the canonical product of the primary factors of f is 1. So, by definition, f is of genus 1. By Hadamard’s factorisation theorem, see, e.g., Titchmarsh (1939), Sec. 8.24, it follows that f has the representation

$$\begin{aligned} f(z)=\exp (Q(z))\prod _{k=1}^\infty \left( 1-\frac{z^4}{\kappa _k^4}\right) ,~ z\in \mathbb {C}, \end{aligned}$$

where \(Q(z)=\alpha z, z\in \mathbb {C},\) with \(\alpha \in \mathbb {C}.\) Since \(f(z)=f(-z)\) for \(z\in \mathbb {C}\) it is \(\alpha =0.\) Thus,

$$\begin{aligned} f(z)=\prod _{k=1}^\infty \left( 1-\frac{z^4}{\kappa _k^4}\right) ,~z\in \mathbb {C}, \end{aligned}$$

and the Laplace transform of

$$\begin{aligned} T=\sum _{k=1}^\infty \frac{1}{\kappa _k^4}W_k^2 \end{aligned}$$

is seen to be

$$\begin{aligned} L(t)&=\prod _{k=1}^\infty \left( 1+\frac{2t}{\kappa _k^4}\right) ^{-1/2}\\&=\left( f\left( (-2t)^{1/4}\right) \right) ^{-1/2}\\&= \left( \frac{2z(t)}{\sinh (z(t))+\sin (z(t))}\right) ^{1/2},~t\ge 0. \end{aligned}$$

\(\square \)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Baringhaus, L., Gaigall, D. & Thiele, J.P. Statistical inference for \(L^2\)-distances to uniformity. Comput Stat 33, 1863–1896 (2018). https://doi.org/10.1007/s00180-018-0820-0

Download citation

Keywords

  • Integrated empirical distribution (survival) function
  • Goodness-of-fit tests for uniformity
  • Numerical inversion of Laplace transforms
  • Coverage probability
  • Equivalence test
  • Neighbourhood-of-uniformity validation test