# Asymptotic Estimation Theory of Multipoint Linkage Analysis under Perfect Marker Information

## Ola Hössjer

### Centre for Mathematical Sciences Mathematical Statistics Lund Institute of Technology, Lund University, 2001

##### ISSN 1403-9338
Abstract:
We consider estimation of a disease susceptibility locus $\tau$ at a chromosome. With perfect marker data available, the estimator $\htau_N$ of $\tau$ based on $N$ pedigrees has a surprisingly fast rate of convergence $N^{-1}$ under mild regularity conditions. The limiting distribution is the argmax of a certain compound Poisson process. Our approach is conditional on observed phenotypes, and therefore treats parametric and nonparametric linkage, as well as quantitative trait loci methods within a unified framework. A constant appearing in the asymptotics, the so called asymptotic slope-to-noise ratio, is introduced as a performance measure for a given genetic model, score function and weighting scheme. This enables us to define asymptotically optimal score functions and weighting schemes. Interestingly, traditional $N^{-1/2}$-theory breaks down, in that, for instance, the ML-estimator is not asymptotically optimal. Further, the asymptotic estimation theory automatically takes uncertainty of $\tau$ into account, which is otherwise handled by means of multiple testing and Bonferroni type corrections.
Other potential applications of our approach that we discuss are general sampling criteria for planning of linkage studies, appropriate grid size of marker maps, robustness w.r.t.\ choice of map function (dropping assumption of no chromatid interference) and quantification of information loss due to heterogeneity (with linked or unlinked trait loci).
We also discuss relations to pointwise performance criteria and pay special attention to weak genetic models, so called local specificity models.
Key words:
Argmax of stochastic processes, compound Poisson process, crossovers, linkage analysis, perfect marker information.