# Advances in Directional and Linear Statistics: A Festschrift by Barry C. Arnold, Ashis SenGupta (auth.), Martin T. Wells,

By Barry C. Arnold, Ashis SenGupta (auth.), Martin T. Wells, Ashis SenGupta (eds.)

The current quantity involves papers written through scholars, colleagues and collaborators of Sreenivasa Rao Jammalamadaka from a variety of nations, and covers various study issues which he enjoys and contributed immensely to.

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1 i D1 n 1 which, in turn, implies that X n 1 1X Vi n n P ! 3 then follows by Borel–Cantelli theorem. 4 in [4]: that is to verify that conditions (i)–(iii) are satisfied. References 1. Bennett G (1962) Probability inequalities for the sum on independent random variables. J Am Stat Assoc 19:33–45 2. Devroye LP, Wagner TJ (1980) Distribution-free consistency results in nonparametric discrimination and regression function estimate. Ann Stat 8:231–239. 3. Harel M, Puri LM (1996) Conditional U-statistics for dependent random varaibles.

O"t Cs T s "O/0 : ! s D 1; 2; : : :/ for l D 0; 1, if residuals f"Ot g are autocorrelated. 4) described above. X1t ; : : : ; Xrt /0 . i; j /th element of the weighting matrix W. 6) is denoted by S . d. with mean zero and variance 2 IN . The zeros of ! p p X X k k . Âk0 C Âk1 W /z and det IN C det IN C kD1 kD1 are outside the unit circle. Z 0t k Z t k 0 / is finite for k; k 0 D 1; : : : ; p. i; j /th element given by lim is positive definite. Furthermore, we assume ˙" D sum of squares is given by: SD 1 T X 2 2 IN for simplicity.

Still in Sect. 3, we show various graphs of GvMF densities, which feature uni- and multimodality, different modal shapes, girdle and other interesting forms. t. 3) below. Under both criteria, either GvMF distributions or other directional distributions having a GvMF part turn out to be the optima. An important consequence is that these distributions are optimal solutions of a constrained prior selection problem of Bayesian statistics. In our setting, an optimal directional prior distribution is obtained by maximizing the entropy given some prior knowledge on the moments along some directions.