New PDF release: Wiley Survey of Instrumentation and Measurement

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Example text

As discussed in reference 10, Rayleigh considers observations described by cˆ W−1 = (XH W −1 X)−1 XH W −1 w α{sinc2 [2π (x − β1 )] + sinc2 [2π (x − β2 )]} (29) For observations with a distribution different from normal, this estimator is no longer maximum likelihood but is still best linear unbiased. If, in addition, the period is unknown, the estimation problem is recognized as a separable nonlinear least squares problem. The model is linear in the 2K + 1 Fourier coefficients and nonlinear in the period d.

Then the Cram´er Rao inequality states that cov[t(w), t(w)] ≥ M−1 (11) In this expression, cov[t(w), t(w)] is the covariance matrix of the estimator t(w). That is, the ( p,q)-th element of this matrix is defined as the covariance of the p-th element t p(w) and the q-th element tq(w). Therefore, the diagonal elements are the variances of t1(w), . . , t K (w), respectively. In Eq. (11) expresses that the difference of the matrix cov[t(w), t(w)] and the matrix M−1 is positive semidefinite. A property of positive semidefinite matrices is that their diagonal elements cannot be negative.

Maximum likelihood estimation of peak height, width and location from Poisson distributed observations. Suppose that observations w1, . . , w N are available made on the spectral peak of Example 6, and that these observations are independent and Poisson distributed. ) (14) n with respect to t. This shows that with normally distributed observations, maximum likelihood estimation is equivalent to a weighted least squares measurement with W −1 as weighting matrix. Least squares estimation is also often used if the distribution of the observations is not known or is known to be not normal.

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