By William D. Siuru

Colour AND BLACK AND WHITE PHOTOS;

HISTORY,

THE males in the back of IT

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**Example text**

E. not depending on x). Introducing eqns. 112) whence, if the estimator is unbiased, eqn. 97b results. Integration of eqn. t. 113) where ^l(-) is a function of 0 only and A2(') is a function of x only. 114) x where q(x) = 0, B(O) = -A(O)E(O]O) and h(x) = № \ Therefore, if an estimate O which reaches the CRLB does exist, the likelihoodfunction is of the exponential type. Moreover, it is readily shown that if (a) the hypotheses (i) and (H), under which the CRLB was derived, hold and (b) the likelihood function is of the exponential type, the estimator $ = q(x) ofeqn.

It corresponds to the computation of the Hermitian form q = zTQz* and its comparison with the threshold T. 53) where Ag(z), i = 1, 2, . . 57) where U1, i=l, 2, . . A^, are the eigenvectors of Q. The pertaining block diagram is shown in Fig. 9 [39]. An equivalent way to define the optimum test is based upon the concept of simultaneous reduction to principal axes of the two Hermitian forms in the square brackets of expr. 5). e. 64) where X(i), *= 1, 2, . N, are the eigenvalues of the pencil. Therefore, the optimum test (eqn.

95) An important theorem, generally known as the Cramer-Rao lower bound (CRLB) or Frechet inequality states that there is a lower bound to the conditional variance of any scalar estimator. Therefore, if we are able to find an unbiased estimator whose variance equals that predicted by the said theorem, we can be sure that the estimator is the best in the class: it is also known as an efficient estimator. 1. 97 a) with a probability of 1. If the estimator is unbiased, eqns. 976) The CRLB (eqns. 97) can be derived as follows: (i) The PDF of the observation, p(x\6), should be greater than zero and differentiable in a given part Dx of the x domain; this part shall not depend on 6.