### Citations

374 |
A class of statistics with asymptotically normal distribution
- Hoeffding
- 1948
(Show Context)
Citation Context ...uming that the integral exists. An unbiased estimate of 8 = 8(F) is furnished by the U-statistic ¢(X. , ... ,X. ) 1 1 1 m (2) where the sum extends over all (~) subsets of the Xi' Following Hoeffding =-=[10]-=-, define c=1,2, ... ,m and ~c(xl""'xc) = ~c(xl""'xc) - 8, assuming all expectations exist. Also define 1';0 = 0 , c = 1,2, ... ,m . If for some nonnegative integer d < m, sd 1 ~ 0 but S = 0 for c = O,... |

50 |
On the Asymptotic Distribution of Differentiable Statistical Functions
- Mises
- 1947
(Show Context)
Citation Context ...er suitable normalization [10]. If d ~ 0, the distribution is no longer normal and ma)' be obtained by2 applying the theory of differentiable statistical functionals due to Von Mises and Filippova" (=-=[13]-=-, [7]) . Let F n be the empirical distribution function of Xl' ... 'X n . The distribution of 1 n S(F ) = - I n nm. 1 1 = 1 n .L 1 =1 m </>(X. , ... ,X. ) 1 1 1 m (3) is closely related to the distrib... |

35 |
Numerical inversion of a characteristic function
- Davies
- 1973
(Show Context)
Citation Context ...1-2itA.)-~, can be expressed in closed form. j=l J In subcase (iii), the limit distribution may most conveniently be found by numerical inversion of </let). Various methods are available (e.g. Davies =-=[5]-=-, Bohman [3], Martynov [12]). In subcase (ii), computation of the inner product is necessary. A reason~b1e [2] and consists of expJ'es~ing </let) way to accomplish this is described in N-1 !,: = II (1... |

13 |
Large Sample Theory for U-Statistics and Tests of Fit
- Gregory
- 1977
(Show Context)
Citation Context ...of (4) can be found using integral equation techniques. Filippova "gives an expression for the characteristic function of the asymptotic distribution of (4) in terms of Fredholm determinants. Gregory =-=[8]-=- gives a more concrete representation of the asymptotic distribution of (4) as an infinite series of random variables, in the case m = 2. Below we combine these results to give explicit expressions fo... |

8 |
Some properties of incomplete U-statistics.
- Ray, Blom, et al.
- 1976
(Show Context)
Citation Context ...s on incomplete U-statistics. The calculation of the U-statistic (2) requires the averaging of (~) terms, which may not be practical if m and n are not small. To reduce the volume of computation Blom =-=[1]-=- and Brown and Kildea [4] have proposed the use of incomplete U-statistics of the form U = N l I <I>(X i , ... ,x. ) 1 1 m (11) when the sum in (11) is taken over N specified or randomly selected m-su... |

7 |
Reduced U-statistics and the Hodges-Lehmann Estimator
- BROWN, KILDEA
- 1978
(Show Context)
Citation Context ...ics. The calculation of the U-statistic (2) requires the averaging of (~) terms, which may not be practical if m and n are not small. To reduce the volume of computation Blom [1] and Brown and Kildea =-=[4]-=- have proposed the use of incomplete U-statistics of the form U = N l I <I>(X i , ... ,x. ) 1 1 m (11) when the sum in (11) is taken over N specified or randomly selected m-subsets of the indices. The... |

6 | The distribution of quadratic forms in normal variatea: a small sample theory with applications to spectral analysis. Unpublished paper - GBENANDEB, POLLAK, et al. - 1958 |

6 |
Computation of the distribution functions of quadratic forms of normal random variables. Theory Probab.
- Martynov
- 1975
(Show Context)
Citation Context ...ed in closed form. j=l J In subcase (iii), the limit distribution may most conveniently be found by numerical inversion of </let). Various methods are available (e.g. Davies [5], Bohman [3], Martynov =-=[12]-=-). In subcase (ii), computation of the inner product is necessary. A reason~b1e [2] and consists of expJ'es~ing </let) way to accomplish this is described in N-1 !,: = II (1-2itA J .)-2 </l2(t) where ... |

5 |
Concerning a certain probability problem. Theory of Probability and its Applications 6
- Zolotarev
- 1961
(Show Context)
Citation Context ...o large . m that C N < (2 m~x Itml)-l and use the power series to compute ¢2' In special cases other methods may be used. If the eigenvalues A. are J all positive, the asymptotic results of Zolotarev =-=[14]-=- and Hoeffding [11] furnish good approximations for the tail probabilities. If the eigenvalues are all positive and of multiplicity unity, then Smirnov's formula may be used as an efficient numerical ... |

4 |
Distribution-free tests of independence based on the sample distribution function
- BLUM, REEFER, et al.
- 1961
(Show Context)
Citation Context ...tly be found by numerical inversion of </let). Various methods are available (e.g. Davies [5], Bohman [3], Martynov [12]). In subcase (ii), computation of the inner product is necessary. A reason~b1e =-=[2]-=- and consists of expJ'es~ing </let) way to accomplish this is described in N-1 !,: = II (1-2itA J .)-2 </l2(t) where j=l7 log ¢(t) 00 = -~ I j=N 10g(1-2itA.). J Formally, we have 00 00 00 00 ~ I I (2... |

4 |
From characteristic function to distribution function via Fourier analysis,”
- Bohman
- 1972
(Show Context)
Citation Context ...can be expressed in closed form. j=l J In subcase (iii), the limit distribution may most conveniently be found by numerical inversion of </let). Various methods are available (e.g. Davies [5], Bohman =-=[3]-=-, Martynov [12]). In subcase (ii), computation of the inner product is necessary. A reason~b1e [2] and consists of expJ'es~ing </let) way to accomplish this is described in N-1 !,: = II (1-2itA J .)-2... |

1 |
Von Mises '-tlieorem on the asymptotic behavior of functionals of empirical distribution functions and its statistical applications
- Filippova
- 1962
(Show Context)
Citation Context ...table normalization [10]. If d ~ 0, the distribution is no longer normal and ma)' be obtained by2 applying the theory of differentiable statistical functionals due to Von Mises and Filippova" ([13], =-=[7]-=-) . Let F n be the empirical distribution function of Xl' ... 'X n . The distribution of 1 n S(F ) = - I n nm. 1 1 = 1 n .L 1 =1 m </>(X. , ... ,X. ) 1 1 1 m (3) is closely related to the distribution... |

1 |
On a theorem of V.M
- Hoeffding
- 1964
(Show Context)
Citation Context ...N < (2 m~x Itml)-l and use the power series to compute ¢2' In special cases other methods may be used. If the eigenvalues A. are J all positive, the asymptotic results of Zolotarev [14] and Hoeffding =-=[11]-=- furnish good approximations for the tail probabilities. If the eigenvalues are all positive and of multiplicity unity, then Smirnov's formula may be used as an efficient numerical inversion method; s... |