• Bernoulli
  • Volume 22, Number 4 (2016), 2521-2547.

Methods for improving estimators of truncated circular parameters

Kanika and Somesh Kumar

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


In decision theoretic estimation of parameters in Euclidean space $\mathbb{R}^{p}$, the action space is chosen to be the convex closure of the estimand space. In this paper, the concept has been extended to the estimation of circular parameters of distributions having support as a circle, torus or cylinder. As directional distributions are of curved nature, existing methods for distributions with parameters taking values in $\mathbb{R}^{p}$ are not immediately applicable here. A circle is the simplest one-dimensional Riemannian manifold. We employ concepts of convexity, projection, etc., on manifolds to develop sufficient conditions for inadmissibility of estimators for circular parameters. Further invariance under a compact group of transformations is introduced in the estimation problem and a complete class theorem for equivariant estimators is derived. This extends the results of Moors [J. Amer. Statist. Assoc. 76 (1981) 910–915] on $\mathbb{R}^{p}$ to circles. The findings are of special interest to the case when a circular parameter is truncated. The results are implemented to a wide range of directional distributions to obtain improved estimators of circular parameters.

Article information

Bernoulli, Volume 22, Number 4 (2016), 2521-2547.

Received: January 2015
First available in Project Euclid: 3 May 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

admissibility convexity directional data invariance projection truncated estimation problem


Kanika; Kumar, Somesh. Methods for improving estimators of truncated circular parameters. Bernoulli 22 (2016), no. 4, 2521--2547. doi:10.3150/15-BEJ736.

Export citation


  • [1] Batschelet, E. (1981). Circular Statistics in Biology. London: Academic Press.
  • [2] Bhattacharya, C.G. (1984). Two inequalities with an application. Ann. Inst. Statist. Math. 36 129–134.
  • [3] Buss, S.R. and Fillmore, J.P. (2001). Spherical averages and applications to spherical splines and interpolation. ACM Trans. Graph. 20 95–126.
  • [4] Danzer, L., Grünbaum, B. and Klee, V. (1963). Helly’s theorem and its relatives. In Proc. Sympos. Pure Math., Vol. VII 101–180. Providence, RI: Amer. Math. Soc.
  • [5] Ducharme, G.R. and Milasevic, P. (1987). Spatial median and directional data. Biometrika 74 212–215.
  • [6] Eaton, M.L. (1989). Group Invariance Applications in Statistics. Regional Conference Series in Probability and Statistics. Hayward, CA: IMS.
  • [7] Ferguson, T.S. (1967). Mathematical Statistics: A Decision Theoretic Approach. New York: Academic Press.
  • [8] Fisher, N.I. (1993). Statistical Analysis of Circular Data. Cambridge: Cambridge Univ. Press.
  • [9] Gradshteyn, I.S. and Ryzhik, I.M. (1965). Table of Integrals, Series and Products. New York: Academic Press.
  • [10] He, X. and Simpson, D.G. (1992). Robust direction estimation. Ann. Statist. 20 351–369.
  • [11] Holmquist, B. (1991). Estimating and testing the common mean direction of several von Mises–Fisher populations with known concentrations. Statistics 22 369–378.
  • [12] Jammalamadaka, S.R. and SenGupta, A. (2001). Topics in Circular Statistics 5. River Edge, NJ: World Scientific.
  • [13] Jones, M.C. and Pewsey, A. (2005). A family of symmetric distributions on the circle. J. Amer. Statist. Assoc. 100 1422–1428.
  • [14] Kubokawa, T. (2004). Minimaxity in estimation of restricted parameters. J. Japan Statist. Soc. 34 229–253.
  • [15] Kubokawa, T. (2005). Estimation of bounded location and scale parameters. J. Japan Statist. Soc. 35 221–249.
  • [16] Kumar, S. and Sharma, D. (1992). An inadmissibility result for affine equivariant estimators. Statist. Decisions 10 87–97.
  • [17] Marchand, E. and Strawderman, W.E. (2004). Estimation in restricted parameter spaces: A review. In A Festschrift for Herman Rubin. Institute of Mathematical Statistics Lecture Notes – Monograph Series 45 21–44. Beachwood, OH: IMS.
  • [18] Marchand, É. and Strawderman, W.E. (2012). A unified minimax result for restricted parameter spaces. Bernoulli 18 635–643.
  • [19] Mardia, K.V. and Jupp, P.E. (2000). Directional Statistics. Wiley Series in Probability and Statistics. Chichester: Wiley.
  • [20] Mardia, K.V. and Sutton, T.W. (1978). A model for cylindrical variables with applications. J. Roy. Statist. Soc. Ser. B 40 229–233.
  • [21] Moors, J.J.A. (1985). Estimation in truncated parameter space. Ph.D. dissertation, Tilburg Univ., The Netherlands.
  • [22] Moors, J.J.A. (1981). Inadmissibility of linearly invariant estimators in truncated parameter spaces. J. Amer. Statist. Assoc. 76 910–915.
  • [23] Moors, J.J.A. and van Houwelingen, J.C. (1993). Estimation of linear models with inequality restrictions. Stat. Neerl. 47 185–198.
  • [24] Neeman, T. and Chang, T. (2001). Rank score statistics for spherical data. Contemp. Math. 287 241–254.
  • [25] Rueda, C., Fernández, M.A. and Peddada, S.D. (2009). Estimation of parameters subject to order restrictions on a circle with application to estimation of phase angles of cell cycle genes. J. Amer. Statist. Assoc. 104 338–347.
  • [26] Tsai, M.-T. (2009). Asymptotically efficient two-sample rank tests for modal directions on spheres. J. Multivariate Anal. 100 445–458.
  • [27] Udrişte, C. (1994). Convex Functions and Optimization Methods on Riemannian Manifolds. Mathematics and Its Applications 297. Dordrecht: Kluwer Academic.
  • [28] van Eeden, C. (2006). Restricted Parameter Space Estimation Problems. New York: Springer.
  • [29] Watson, G.S. (1983). Statistics on Spheres. New York: Wiley.