Saddlepoint approximations to the distribution of the total distance of the multivariate isotropic and von Mises-Fisher random walks. (English) Zbl 1421.62062
Author’s abstract: This article considers the random walk over \({\mathbb R}^p\) with \(p \geq 2\), where the directions taken by the individual steps follow either the isotropic or the von Mises-Fisher distributions. Saddlepoint approximations to the density and to the upper tail probabilities of the total distance covered by random walk, i.e., of the length of the resultant, are derived. The saddlepoint approximations are one-dimensional and simple to compute, even though the initial problem is \(p\)-dimensional.
The fact that the relevant approximations are one-dimensional follows from a factorization lemma which makes independent the length of the resultant and its direction.
Some numerical examples based on Monte Carlo simulations show the high accuracy of the proposed approximations.
The fact that the relevant approximations are one-dimensional follows from a factorization lemma which makes independent the length of the resultant and its direction.
Some numerical examples based on Monte Carlo simulations show the high accuracy of the proposed approximations.
Reviewer: Fabio Rapallo (Alessandria)
MSC:
| 62H11 | Directional data; spatial statistics |
| 62E17 | Approximations to statistical distributions (nonasymptotic) |
| 60G50 | Sums of independent random variables; random walks |
| 60F10 | Large deviations |
Keywords:
Bessel function; directional distribution; large deviations approximation; relative error; resultant lengthReferences:
| [1] | M. Abramowitz and I. E. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Courier Dover Publ., New York, 1972). · Zbl 0543.33001 |
| [2] | D. E. Amos, “Computation of Modified Bessel Functions and Their Ratios”, Math. Comput. 28, 239-251 (1974). · Zbl 0277.65006 · doi:10.1090/S0025-5718-1974-0333287-7 |
| [3] | A. Banerjee, I. S. Dhillon, J. Ghosh, and S. Sra, “Clustering on the Unit Hypersphere Using von Mises-Fisher Distributions”, J. of Machine Learning Research 6, 1345-1382 (2005). · Zbl 1190.62116 |
| [4] | R. Barakat, “Isotropic Random Flights”, J. Phys. A:Mathematical, Nuclear and General 6, 796-804 (1973). · Zbl 0256.60063 · doi:10.1088/0305-4470/6/6/008 |
| [5] | M.N. Barber and B. W. Ninham, Random and Restricted Walks, Theory and Applications (Gordon and Breach, 1970). · Zbl 0232.60048 |
| [6] | O. E. Barndorff-Nielsen and D. R. Cox, Asymptotic Techniques for Use in Statistics, (Chapman & Hall, 1989). · Zbl 0672.62024 · doi:10.1007/978-1-4899-3424-6 |
| [7] | Y.-H. Chen, D. Wei, G. Newstadt, M. De Graef, J. Simmons, and A. Hero, “Parameter Estimation in Spherical Symmetry Groups”, IEEE Signal Processing Lett. 22, 1152-1155 (2015). · doi:10.1109/LSP.2014.2387206 |
| [8] | H. E. Daniels, “Saddlepoint approximations in statistics”, Ann. Math. Statist. 25, 631-650 (1954). · Zbl 0058.35404 · doi:10.1214/aoms/1177728652 |
| [9] | H. E. Daniels and G. A. Young, “Saddlepoint Approximation for the Studentized Mean, with an Application to the Bootstrap”, Biometrika 78, 169-179 (1991). · doi:10.1093/biomet/78.1.169 |
| [10] | N. G. De Bruijn, Asymptotic Methods in Analysis (Dover, 1982). · Zbl 0082.04202 |
| [11] | A. Feuerverger, “On the Empirical Saddlepoint Approximation”, Biometrika 76, 457-464 (1989). · Zbl 0674.62019 · doi:10.1093/biomet/76.3.457 |
| [12] | C. Field, “Small Sample Asymptotic Expansions forMultivariateM-Estimates”, Ann. Statist. 10, 672-689 (1982). · Zbl 0515.62024 · doi:10.1214/aos/1176345864 |
| [13] | V. A. Field and E. Ronchetti, Small Sample Asymptotics (Inst. Math. Statist, 1990), Vol.13. · Zbl 0742.62016 |
| [14] | C. A. Field and M. A. Tingley, “Small Sample Asymptotics: Applications in Robustness”, in Handbook of Statistics (Elsevier, 1997), Vol. 15, Ch. 18, pp. 513-536. · Zbl 0908.62033 · doi:10.1016/S0169-7161(97)15020-9 |
| [15] | P. J. Flory, Statistical Mechanics of Chain Molecules (Interscience, 1969). |
| [16] | R. Gatto, “Multivariate Saddlepoint Test for the Wrapped Normal Model”, J. Statist. Comput. and Simul. 65, 271-285 (2000). · Zbl 1092.62549 · doi:10.1080/00949650008812002 |
| [17] | Gatto, R., Large Deviations Approximations to Distributions of the Total Distance of Compound Random Walks with von Mises Directions (2017) · Zbl 1403.62092 |
| [18] | R. Gatto and S. R. Jammalamadaka, “A Conditional Saddlepoint Approximation for Testing Problems”, J. Amer. Statist. Assoc. 94, 533-541 (1999). · Zbl 0997.62016 · doi:10.1080/01621459.1999.10474148 |
| [19] | R. Gatto and S. R. Jammalamadaka, “Inference for Wrapped Symmetric α-Stable Circular Models”, Sankhyā, A:Math. Statist. and Probab. 65 (2), 333-355 (2003). · Zbl 1193.62092 |
| [20] | R. Gatto and M. Mayer, “Saddlepoint Approximations for SomeModels of Circular Data”, Statist. Methodology 2, 233-248 (2005). · Zbl 1248.62085 · doi:10.1016/j.stamet.2005.04.002 |
| [21] | K. Hornik and B. Grün, “OnMaximumLikelihood Estimation of the Concentration Parameter of vonMises-Fisher Distributions”, Comput. Statist. 29, 945-957 (2014). · Zbl 1306.65071 · doi:10.1007/s00180-013-0471-0 |
| [22] | P. Jeganathan, R. L. Paige, and A. A. Trindade, “Saddlepoint-Based Bootstrap Inference for the Spatial Dependence Parameter in the Lattice Process”, Spatial Statist. 12, 1-14 (2015). · doi:10.1016/j.spasta.2015.01.001 |
| [23] | J. L. Jensen, Saddlepoint Approximations (Oxford University Press, 1995). · Zbl 1274.62008 |
| [24] | J. T. Kent, K. V. Mardia, and J. S. Rao, “A Characterization of Uniform Distribution on the Circle”, Ann. Statist. 7, 882-889 (1979). · Zbl 0423.62012 · doi:10.1214/aos/1176344737 |
| [25] | J. E. Kolassa, Series Approximation Methods in Statistics, in Lecture Notes in Statistics, 3rd ed. (Springer, 2006), Vol.88. · Zbl 1090.62015 |
| [26] | A. Kume and A. T. A. Wood, “Saddlepoint Approximations for the Bingham and Fisher-Bingham Normalizing Constants”, Biometrika 92, 465-476 (2005). · Zbl 1094.62063 · doi:10.1093/biomet/92.2.465 |
| [27] | R. Lugannani and S. Rice, “Saddle Point Approximation for the Distribution of the Sum of Independent Random Variables”, Advances in Appl. Probab. 12, 475-490 (1980). · Zbl 0425.60042 · doi:10.1017/S0001867800050278 |
| [28] | Y. Ma and E. Ronchetti, “Saddlepoint Test in Measurement Error Models”, J. Amer. Statist. Assoc. 106, 147-156 (2011). · Zbl 1396.62108 · doi:10.1198/jasa.2011.tm10031 |
| [29] | K. V. Mardia and P. E. Jupp, Directional Statistics (Wiley, New York, 2000). · Zbl 0935.62065 |
| [30] | K. V. Mardia, J. T. Kent, and J. M. Bibby, Multivariate Analysis (Academic Press, 1979). · Zbl 0432.62029 |
| [31] | Masoliver, M.; Porrá, J. M.; Weiss, G. H., Some Two and Three-Dimensional PersistentRandom Walks, 469-482 (1993) |
| [32] | E. Orsingher and A. De Gregorio, “Random Flights in Higher Spaces”, J. Theoret. Probab. 20, 769-806 (2007). · Zbl 1138.60014 · doi:10.1007/s10959-007-0093-y |
| [33] | N. Reid, “SaddlepointMethods and Statistical Inference”, Statist. Sci. 3, 213-238 (1988). · Zbl 0955.62541 · doi:10.1214/ss/1177012906 |
| [34] | J. Robinson, “Saddlepoint Approximations for Permutation Tests and Confidence Intervals”, J. Royal Statist. B 44, 91-101 (1982). · Zbl 0487.62016 |
| [35] | J. Robinson, E. Ronchetti, and G. A. Young, “Saddlepoint Approximations and Tests Based on Multivariate M-Estimates”, Ann. Statist. 31, 1154-1169 (2003). · Zbl 1056.62023 · doi:10.1214/aos/1059655909 |
| [36] | E. Ronchetti and A. H. Welsh, “Empirical Saddlepoint Approximations for Multivariate M-Estimators”, J. Royal Statist. Soc., Ser. B (Statist. Methodology) 56, 313-326 (1994). · Zbl 0796.62018 |
| [37] | R. Srinivisan and S. Parthasarathy, Some Statistical Applications in X-ray Crystallography (Pergamon Press, 1976). |
| [38] | W. Stadje, “Exact Probability Distribution for Non-Correlated Random Walk Models”, J. Statist. Phys. 56, 415-435 (1989). · Zbl 0714.60058 · doi:10.1007/BF01044444 |
| [39] | N. M. Temme, “The Uniform Asymptotic Expansion of a Class of Integrals Related to Cumulative Distribution Functions”, SIAM J. Math. Anal. 13, 239-253 (1982). · Zbl 0489.41031 · doi:10.1137/0513017 |
| [40] | S. Wang, “General Saddlepoint Approximations in the Bootstrap”, Statist & Probab. Lett. 13, 61-66 (1992). · doi:10.1016/0167-7152(92)90237-Y |
| [41] | S. Wang, “Saddlepoint Expansions in Finite Population Problems”, Biometrika 80, 583-590 (1993). · Zbl 0785.62012 · doi:10.1093/biomet/80.3.583 |
| [42] | S. Wang, “One-Step Saddlepoint Approximations for Quantiles”, Comput. Statist. and Data Anal. 20, 65-74 (1995). · Zbl 0875.62065 · doi:10.1016/0167-9473(94)00029-I |
| [43] | G. H. Weiss and J. E. Kiefer, “The Pearson RandomWalk with Unequal Step Sizes”, J. Phys. A: Math. and General 16, 489-495 (1983). · Zbl 0508.60059 · doi:10.1088/0305-4470/16/3/009 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
