Confidence limits and bias correction for estimating angles between directions with applications to paleomagnetism

Details Confidence limits and bias correction for estimating angles between directions with applications to paleomagnetism Confidence limits and bias correction for estimating angles between directions with applications to paleomagnetism

Dec 1995

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1995jgr...10024405d&link_type=abstract

Journal of Geophysical Research, Volume 100, Issue B12, p. 24405-24430

Physics

9

Geomagnetism And Paleomagnetism: Instruments And Techniques

Scientific paper

Many problems in the earth sciences, particularly in the use of paleomagnetic data in plate tectonics, require estimates of the angles between directions and confidence intervals for these angles. To give a set of interrelated simple methods, we approximate all estimated directions by appropriate concentrated Fisher distributions. In preliminary numerical experiments we verify previous results for the distribution of the means of samples drawn from Fisher distributions and study the distributions of six estimators of the Fisher concentration parameter κ. We selected Fisher's original estimator for κ, k=(n-1)/(n-R), for use in our subsequent simulations. We then study the bias in, and confidence intervals around, the estimator θ̂ of the angle between the means of samples drawn from two Fisher distributions (``Fisher means'').
We find an approximate expression for the geometric bias in this angle, B(θ)=-tanθ+[tan2θ+θ2crit]1/2 where θcrit=[1/(R1k1)+1/(R2k2)]1/2, and show that it works well for θ>2θcrit. (When, for example, Rk=250 for both samples, θcrit=5.1°.) We then show that the bias-corrected estimator of θ,θ*=θ̂-B(θ*), is normally distributed with mean equal to the true angle and variance σθ̂2=θcrit2. Thus one can construct a 95% confidence interval for this angle with the formula (θ*-1.96σθ̂,θ*+1.96σθ̂). We show by extensive simulation that coverage probabilities for this confidence interval are slightly conservative for θ>2θcrit and good for θ>3θcrit. We derive related results for angles between various combinations of fixed directions and Fisher means. We find that the estimator α̂ of the angle between two great circles containing a known fixed direction and two different Fisher means is unbiased and normally distributed with variance σα̂2=1/(R1k1sin2θ1)+1/(R2k2sin2θ2) where the θi are the angular distances between the Fisher means and the known fixed direction. Thus the confidence interval on this rotation angle is (α̂-1.96σα̂,α̂+1.96σα̂). We give examples of applying these techniques to paleomagnetic data analysis, especially for determining terrane motions. Our methods provide confidence intervals for poleward displacement. For rotation, simulation suggests that they provide results at least as accurate as previous methods [McWilliams, 1984; Demarest, 1983] over a wide range of relevant parameter values, while being simpler or more flexible. We apply these methods to estimating rates of apparent polar wander and point out an additional bias due to dating errors.

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