### Abstract

We consider the model with i.i.d. observations Xi, Xn where Xi ~ μ on (X, A). In this model we define a class of statistics Sa,n for a Є [0, 1] as n where Døa denotes the øa-divergence for øa(t) = (t - 1 )2/(a+ (1 - a) t), t >, and are restrictions of the empirical distribution μn and true distribution μ on the events generated by a measurable partition P = Pn = {An1, Anmn} of X. Sa, n is a blend (nonconvex combination) of the Pearson statistic S1,n = n x 2(μP, μP) and Neyman statistic S0.,n = nX2 (μP, μPn) where x2 denotes the chi-square divergence. We describe the goodness of fit tests and minimum distance estimators defined by means of Sa,n with given a Є [0, 1], or by San,n with given sequence an Є [0, 1]. The main results are conditions under which San,n with arbitrary sequence an Є [0, 1] asymptotically coincides with the Pearson S1,n in the sense San,n - S1,n = op (mn) and (San,n - S1,n)/mn → 0 a.s. The results are applied to extension of limit laws proved for S1,n/mn or So,n/mn to all blended statistics San,n/mn.

Original language | English |
---|---|

Pages (from-to) | 239-252 |

Number of pages | 14 |

Journal | Statistics and Risk Modeling |

Volume | 19 |

Issue number | 3 |

DOIs | |

Publication status | Published - Jan 1 2001 |

### Fingerprint

### Keywords

- Asymptotic distributions
- Asymptotic equivalence
- Neyman statistics
- Pearson statistics
- Robust versions

### ASJC Scopus subject areas

- Statistics and Probability
- Modelling and Simulation
- Statistics, Probability and Uncertainty

### Cite this

*Statistics and Risk Modeling*,

*19*(3), 239-252. https://doi.org/10.1524/strm.2001.19.3.239

**A Class of Modified Pearson and Neyman Statistics.** / Györfi, L.; Vajda, I.

Research output: Contribution to journal › Article

*Statistics and Risk Modeling*, vol. 19, no. 3, pp. 239-252. https://doi.org/10.1524/strm.2001.19.3.239

}

TY - JOUR

T1 - A Class of Modified Pearson and Neyman Statistics

AU - Györfi, L.

AU - Vajda, I.

PY - 2001/1/1

Y1 - 2001/1/1

N2 - We consider the model with i.i.d. observations Xi, Xn where Xi ~ μ on (X, A). In this model we define a class of statistics Sa,n for a Є [0, 1] as n where Døa denotes the øa-divergence for øa(t) = (t - 1 )2/(a+ (1 - a) t), t >, and are restrictions of the empirical distribution μn and true distribution μ on the events generated by a measurable partition P = Pn = {An1, Anmn} of X. Sa, n is a blend (nonconvex combination) of the Pearson statistic S1,n = n x 2(μP, μP) and Neyman statistic S0.,n = nX2 (μP, μPn) where x2 denotes the chi-square divergence. We describe the goodness of fit tests and minimum distance estimators defined by means of Sa,n with given a Є [0, 1], or by San,n with given sequence an Є [0, 1]. The main results are conditions under which San,n with arbitrary sequence an Є [0, 1] asymptotically coincides with the Pearson S1,n in the sense San,n - S1,n = op (mn) and (San,n - S1,n)/mn → 0 a.s. The results are applied to extension of limit laws proved for S1,n/mn or So,n/mn to all blended statistics San,n/mn.

AB - We consider the model with i.i.d. observations Xi, Xn where Xi ~ μ on (X, A). In this model we define a class of statistics Sa,n for a Є [0, 1] as n where Døa denotes the øa-divergence for øa(t) = (t - 1 )2/(a+ (1 - a) t), t >, and are restrictions of the empirical distribution μn and true distribution μ on the events generated by a measurable partition P = Pn = {An1, Anmn} of X. Sa, n is a blend (nonconvex combination) of the Pearson statistic S1,n = n x 2(μP, μP) and Neyman statistic S0.,n = nX2 (μP, μPn) where x2 denotes the chi-square divergence. We describe the goodness of fit tests and minimum distance estimators defined by means of Sa,n with given a Є [0, 1], or by San,n with given sequence an Є [0, 1]. The main results are conditions under which San,n with arbitrary sequence an Є [0, 1] asymptotically coincides with the Pearson S1,n in the sense San,n - S1,n = op (mn) and (San,n - S1,n)/mn → 0 a.s. The results are applied to extension of limit laws proved for S1,n/mn or So,n/mn to all blended statistics San,n/mn.

KW - Asymptotic distributions

KW - Asymptotic equivalence

KW - Neyman statistics

KW - Pearson statistics

KW - Robust versions

UR - http://www.scopus.com/inward/record.url?scp=84976446987&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84976446987&partnerID=8YFLogxK

U2 - 10.1524/strm.2001.19.3.239

DO - 10.1524/strm.2001.19.3.239

M3 - Article

VL - 19

SP - 239

EP - 252

JO - Statistics and Risk Modeling

JF - Statistics and Risk Modeling

SN - 2193-1402

IS - 3

ER -