Goodness-of-fit test for length-biased data with discussion on prevalence, sensitivity and specificity of the length bias.

*(English)*Zbl 0848.62008Summary: Due to reasons like absence of a proper sampling frame or inaccessibility to population units, data in statistical studies are sometimes contaminated by a phenomenon called length bias (LB). In this article, an asymptotic test statistic is derived to examine the homogeneity of a length biased sample from the Mean Exponential Family (MEF), a new class of distributions introduced by the author [ibid. 23, No. 2, 227-291 (1989; Zbl 0677.62011)]. Expressions for the test statistic are obtained for length biased binomial, Poisson, negative binomial, beta, gamma, normal, Pareto, Laplace, and Raleigh distributions as special cases. The results are illustrated. One among several advantages in our approach is the ability to quantify the specificity and sensitivity of LBs and their intrinsic relations.

##### MSC:

62E15 | Exact distribution theory in statistics |

62F05 | Asymptotic properties of parametric tests |

62E10 | Characterization and structure theory of statistical distributions |

62F03 | Parametric hypothesis testing |

##### Keywords:

Bayes predictive value; characterization; length bias; homogeneity; mean exponential family; binomial; Poisson; negative binomial; beta; gamma; normal; Pareto; Laplace; Raleigh distributions
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\textit{R. Shanmugam}, J. Stat. Plann. Inference 48, No. 3, 277--290 (1995; Zbl 0848.62008)

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