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**On the bias in estimating genetic length and other quantities in simplex constrained models.**
*(English)*
Zbl 1012.62114

Summary: The genetic distance between two loci on a chromosome is defined as the mean number of crossovers between the loci. The parameters of the crossover distribution are constrained by the parameters of the distribution of chiasmata. J. Ott [T. Speed et al. (eds.), Genetic mapping and DNA sequencing. IMA Vol. Math. Appl. 81, 49-63 (1996; Zbl 0860.92025)] derived the maximum likelihood estimator (MLE) of the parameters of the crossover distribution and the MLE of the mean. We demonstrate that the MLE of the mean is pointwise less than or equal to the empirical mean number of crossovers. It follows that the MLE is negatively biased. For small sample sizes the bias can be nonnegligible. We recommend reduced bias estimators. Generalizations to many other problems involving linear constraints on parameters are made. Included in the generalizations are a variety of problems involving simplex constraints as studied recently by C. Liu [J. Am. Stat. Assoc. 95, 109-120 (2000)].

### MSC:

62P10 | Applications of statistics to biology and medical sciences; meta analysis |

92D10 | Genetics and epigenetics |

62F10 | Point estimation |

### Citations:

Zbl 0860.92025### References:

[1] | ANDERSON, T. W. (1971). The Statistical Analysis of Time Series. Wiley, New York. · Zbl 0225.62108 |

[2] | COHEN, A., KEMPERMAN, J. H. B. and SACKROWITZ, H. B. (1994). Unbiased testing in exponential family regression. Ann. Statist. 22 1931-1946. · Zbl 0824.62011 · doi:10.1214/aos/1176325765 |

[3] | LEE, C. C. (1988). Quadratic loss of order restricted estimators for treatment means with a control. Ann. Statist. 16 751-758. · Zbl 0646.62023 · doi:10.1214/aos/1176350833 |

[4] | LIU, C. (2000). Estimation of discrete distributions with a class of simplex constraints. J. Amer. Statist. Assoc. 95 109-120. |

[5] | MATHER, K. (1933). The relation between chiasmata and crossing-over in diploid and triploid Drosophila melanogaster. J. Genetics 27 243-259. |

[6] | MATHER, K. (1938). Crossing-over. Biol. Rev. Cambridge Philos. Soc. 13 252-292. |

[7] | OTT, J. (1996). Estimating crossover frequencies and testing for numerical interference with highly polymorphic markers. In Genetic Mapping and DNA Sequencing (T. Speed and M. S. Waterman, eds.) 49-63. Springer, New York. · Zbl 0860.92025 |

[8] | ROBERTSON, T., WRIGHT, F. T. and DYKSTRA, R. L. (1988). Order Restricted Statistical Inference. Wiley, New York. · Zbl 0645.62028 |

[9] | YU, K. and FEINGOLD, E. (2001). Estimating the frequency distribution of crossovers during meiosis from recombination data. Biometrics 57 427-434. JSTOR: · Zbl 1209.62343 · doi:10.1111/j.0006-341X.2001.00427.x |

[10] | ZANGWILL, W. I. and MOND, B. (1969). Nonlinear Programming: A Unified Approach. Prentice- Hall, Englewood Cliffs, NJ. · Zbl 0195.20804 |

[11] | PISCATAWAY, NEW JERSEY E-MAIL: artcohen@rci.rutgers.edu |

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