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 contains papers written by way of scholars, colleagues and collaborators of Sreenivasa Rao Jammalamadaka from a variety of nations, and covers various learn subject matters which he enjoys and contributed immensely to.

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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. J Multivar Anal 57:84–100 4. Harel M, Puri LM (2004) Universally consistent conditional U-statistics for absolutely regular processes and its applications for hidden markov models.

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. 1. Suppose « D fBm;n ; k0 ; k1 ; Âl0 ; Âl1 g, with m D 1; : : : ; N I n D 1; : : : ; rI k D 1; : : : ; pI l D 1; : : : ; q. The least square estimates «O are asymptotically normally distributed with mean « and with covariance matrix given by: 34 X.

S. Rao where 8 p P ˆ ˆ . < Xnt C @"i t D P p ˆ @Bm;n ˆ . t k/ if i ¤ m kD1 @"i t "i t are independent with mean zero and @B are finite and independent of "t . 0 T @Bm;n in probability as T ! t k/ "i t have zero mean and uncorrelated over time t. 0 T T @ k0 in probability as T ! t k/ : uD1 Since W is a known matrix, the asymptotic properties of Therefore, the proof will be omitted. t @Âl0 1 @S . d. t l/ "i t / D 0 for l D 1; : : : ; q. s l/ "i s / D 0 for t ¤ s. t l/ : 38 X. S. Rao we can prove: 1 @S !

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