By Jordan M. Stoyanov
Counterexamples (in the mathematical feel) are strong instruments of mathematical thought. This e-book covers counterexamples from chance concept and stochastic approaches. This new accelerated version contains many examples and the most recent study effects. the writer is considered one of many premiere specialists within the box. comprises numbers examples.
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Extra resources for Counterexamples in probability
66) is second-order AMU and efficient, and that E . _ _ (p - 7jJ)(1 - 2p7jJ + 7jJ2) _ 2p + ( -1) ePqML - P n(1 _ p7jJ)(l _ 7jJ2) non. 67) are . 68) PqML = +;;: PqML' E ePqML = P - -;;:- + 0 n .. (1 2). respectively. ~~f XtXt+! ~=1 Xf ' neglecting the terms of order Op(pn), which do not disturb our asymptotic theory. 33 0- 2 Case 6. , and p are known parameters. It is not so hard to show B(1/;) Let ;PqML = (1/; - f) p)(l + 1/;2 - 21/;p - p2 + 31/;2p2 - 21/;3 p) (1- 1/;2)2(1 - 1/;p)(l - p2)' . 69) be the quasi-maximum likelihood estimator of 1/;.
If we modify the maximum likelihood estimator of (J to be third-order AMU, then it is third-order asymptotically efficient in D. 4. Normalizing transformations of some statistics of Gaussian ARMA processes In the area of multivariate analysis several authors have considered transformations of statistics which are based upon functions of the elements of sample covariance matrix, and derived the Edgeworth expansions of the transformed statistics. Konishi(1978) gave a transformation of the sample correlation coefficient which extinguishes a part of the second-order terms of the Edgeworth expansion.
2 = (1qML + 3n (1qML - + I)}. e = (12), and that 7jJ and p are known (12. 2 n (1- p2)(I_7jJ2(qML is second-order AMU and efficient. 55) we obtain (12 Case 5. , and 7jJ are known parameters. Then it is not difficult to show e = p), and that Let PqML be the quasi-maximum likelihood estimator of p. 66) is second-order AMU and efficient, and that E . _ _ (p - 7jJ)(1 - 2p7jJ + 7jJ2) _ 2p + ( -1) ePqML - P n(1 _ p7jJ)(l _ 7jJ2) non. 67) are . 68) PqML = +;;: PqML' E ePqML = P - -;;:- + 0 n .. (1 2). respectively.
Counterexamples in probability by Jordan M. Stoyanov