### Abstract

Let D((0 l] ^{2} ) denote the space of all functions on (0,l] ^{2} with no discontinuities of the second kind. We prove weak invariance principles in the space D((0,l] ^{2} ) for processes of the form ∫h(H _{n+m} (t))dF _{n} (t), m, n ≤ 1, where F _{n} and G _{m} are two independent empirical distribution functions of independent, identically distributed sequences of random variables, H _{n+m} = (n+m+I) ^{-1} (nF _{n} + mG _{m} ), and where h belongs to a certain class of functions on the open unit interval. The appropriate topology in D(((0,l] ^{2} ) is given by uniform convergence on compact sets. This type of processes is central in nonparametric statistics having applications to two-sample linear rank statistics and signed rank statistics.

Original language | English (US) |
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Title of host publication | Probability Theory and Extreme Value Theory |

Publisher | De Gruyter Mouton |

Pages | 310-361 |

Number of pages | 52 |

Volume | 2 |

ISBN (Electronic) | 9783110917826 |

ISBN (Print) | 9789067643856 |

State | Published - Jul 11 2011 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

*Probability Theory and Extreme Value Theory*(Vol. 2, pp. 310-361). De Gruyter Mouton.

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*Probability Theory and Extreme Value Theory.*vol. 2, De Gruyter Mouton, pp. 310-361.

**An invariance principle for processes indexed by two parameters and some statistical applications.** / Denker, Manfred Heinz; Puri, Madan L.

Research output: Chapter in Book/Report/Conference proceeding › Chapter

TY - CHAP

T1 - An invariance principle for processes indexed by two parameters and some statistical applications

AU - Denker, Manfred Heinz

AU - Puri, Madan L.

PY - 2011/7/11

Y1 - 2011/7/11

N2 - Let D((0 l] 2 ) denote the space of all functions on (0,l] 2 with no discontinuities of the second kind. We prove weak invariance principles in the space D((0,l] 2 ) for processes of the form ∫h(H n+m (t))dF n (t), m, n ≤ 1, where F n and G m are two independent empirical distribution functions of independent, identically distributed sequences of random variables, H n+m = (n+m+I) -1 (nF n + mG m ), and where h belongs to a certain class of functions on the open unit interval. The appropriate topology in D(((0,l] 2 ) is given by uniform convergence on compact sets. This type of processes is central in nonparametric statistics having applications to two-sample linear rank statistics and signed rank statistics.

AB - Let D((0 l] 2 ) denote the space of all functions on (0,l] 2 with no discontinuities of the second kind. We prove weak invariance principles in the space D((0,l] 2 ) for processes of the form ∫h(H n+m (t))dF n (t), m, n ≤ 1, where F n and G m are two independent empirical distribution functions of independent, identically distributed sequences of random variables, H n+m = (n+m+I) -1 (nF n + mG m ), and where h belongs to a certain class of functions on the open unit interval. The appropriate topology in D(((0,l] 2 ) is given by uniform convergence on compact sets. This type of processes is central in nonparametric statistics having applications to two-sample linear rank statistics and signed rank statistics.

UR - http://www.scopus.com/inward/record.url?scp=85064856015&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85064856015&partnerID=8YFLogxK

M3 - Chapter

SN - 9789067643856

VL - 2

SP - 310

EP - 361

BT - Probability Theory and Extreme Value Theory

PB - De Gruyter Mouton

ER -