# 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 includes papers written by means of scholars, colleagues and collaborators of Sreenivasa Rao Jammalamadaka from numerous international locations, and covers various study issues which he enjoys and contributed immensely to.

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Extra info for Advances in Directional and Linear Statistics: A Festschrift for Sreenivasa Rao Jammalamadaka

Example text

Z/ ! h; x/ ! x/; n ! h; x/ ! h; x/ ! 1; x/ ! z/d z in probability as n ! 1 for -almost all x. It remains to prove the uniform integrability. 33) 24 E. Elharfaoui et al. 1 is proved. 3 below. j / j. This norm is equivalent to the Euclidian norm and easy to work with here. 3 below see also [1]. 3. 35) i 1 Then n 1 n X Vi ! 0 with probability 1, as n ! 1: i D1 Proof. For > 0, 1X Vi n n P ! j / ˇ Vi ˇ max ˇ ˇ 1Äj Äd ˇ n ! j / ˇ Vi ˇ ˇ ˇn ˇ ! j / ˇ Vi ˇ ˇ ˇ ˇn i D1 ! 1 Ä r rE n ˇr ! 2 of [3], one has that ˇ n ˇr !

U and where 0 log 0 D 0 by assumption. t. P or mean information per observation of P for discriminating of Q from P . s. P jQ/ is not a metric: it violates the symmetry and the triangle rules. x/; Sp 1 where 0 log 0 D 0 is assumed. P / measures the uncertainty inherent in P or in f . P / measures the expected amount of information gained on obtaining a direction from P , based on the principle that the rarer an event, the more informative its occurrence. P /. x/ D ˛r ; where i1 Ä : : : Ä ik 2 f1; 2; : : :g, mr 2 S p 1 , ˛r 2 R and r D 1; : : : ; k.

This norm is equivalent to the Euclidian norm and easy to work with here. 3 below see also [1]. 3. 35) i 1 Then n 1 n X Vi ! 0 with probability 1, as n ! 1: i D1 Proof. For > 0, 1X Vi n n P ! j / ˇ Vi ˇ max ˇ ˇ 1Äj Äd ˇ n ! j / ˇ Vi ˇ ˇ ˇn ˇ ! j / ˇ Vi ˇ ˇ ˇ ˇn i D1 ! 1 Ä r rE n ˇr ! 2 of [3], one has that ˇ n ˇr ! 38) i D1 From the above two inequalities, one deduces that ˇr ! j / ˇ Vi ˇ ˇ ˇ ˇn ! <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.