LÉVY-BASED ERROR PREDICTION IN CIRCULAR SYSTEMATIC SAMPLING

Kristjana Ýr Jónsdóttir; Eva B. Vedel Jensen

doi:10.5566/ias.v32.p117-125

Authors

Kristjana Ýr Jónsdóttir Aarhus University
Eva B. Vedel Jensen Aarhus University

DOI:

https://doi.org/10.5566/ias.v32.p117-125

Keywords:

Fourier series, Lévy basis, planar particles, stationary stochastic processes, stereology, systematic sampling

Abstract

In the present paper, Lévy-based error prediction in circular systematic sampling is developed. A model-based statistical setting as in Hobolth and Jensen (2002) is used, but the assumption that the measurement function is Gaussian is relaxed. The measurement function is represented as a periodic stationary stochastic process X obtained by a kernel smoothing of a Lévy basis. The process X may have an arbitrary covariance function. The distribution of the error predictor, based on measurements in n systematic directions is derived. Statistical inference is developed for the model parameters in the case where the covariance function follows the celebrated p-order covariance model.

References

Barndorff-Nielsen OE, Schmiegel J (2004). Lévy based tempo-spatial modeling; with applications to turbulence. Uspekhi Mat Nauk 159:63–90.

Cruz-Orive LM (2005). A new stereological principle for test lines in three-dimensional space. J Microsc 219:18–28.

Dufour JM, Roy R (1976). On spectral estimation for a homogeneous random process on the circle. Stoch Proc Appl 4:107–20.

Gual-Arnau X, Cruz-Orive LM (2000). Systematic sampling on the circle and the sphere. Adv Appl Prob SGSA 32:628–47.

Gundersen HJG (1988). The nucleator. J Microsc 151:3–21. 18/23

Hansen LV, Thorarinsdottir TL, Gneiting T (2011). Lévy particles: Modelling and simulating star-shaped random sets. Research report 4, CSGB, Department of Mathematical Sciences, Aarhus University. Submitted.

Hellmund G, Prokešová M, Jensen EBV (2008). Lévy-based Cox point processes. Adv Appl Prob SGSA 40:603–29.

Hobolth A, Jensen EBV (2002). A note on design-based versus model-based variance estimation in stereology. Adv Appl Prob SGSA 34:484–90.

Hobolth A, Pedersen J, Jensen EBV (2003). A continuous parametric shape model. Ann Inst Statist Math 55:227–42.

Holm H, Alouini MS (2004). Sum and difference of two squared correlated Nakagami variates in connection with the McKay distribution. IEEE Trans Commun 52:1367– 76.

Jensen EBV, Gundersen HJG (1993). The rotator. J Microsc 170:35–44.

Jensen JL (1995). Saddlepoint Approximations. Oxford: Clarendon Press.

Jónsdóttir KY, Hoffmann LM, Hobolth A, Jensen EBV (2006). On variance estimation in circular systematic sampling. J Microsc 222:212–6.

Jónsdóttir KY, Rønn-Nielsen A, Mouridsen K, Jensen EBV (2011). Lévy based modelling in brain imaging. Research report 2, CSGB, Department of Mathematical Sciences, Aarhus University. Submitted.

Jónsdóttir KY, Schmiegel J, Jensen EBV (2008). Lévy-based growth models. Bernoulli 14:62–90.

Schneider R (1993). Convex Bodies: The Brunn-Minkowski Theory. Cambridge: Cambridge University Press.