To a close approximation [xj|Uj] is normal with mean u and variance ¿2/J. Therefore, (VJxj/¿)2 has a non-central chi-squared distribution with non-centrality parameter A = U2 /(¿2/J). We recognize as the moment-generating function (MGF) of a non-central chi-square distribution3. It transpires that
This expression yields the likelihood (i.e., using Eq. (7.1.2)) of y and n as a function of Uj (instead of xj) and the parameters a2 and £2. It is worth pondering the form of this 'effective detection function' and its implications.
(1) When distance is measured with error, we can use the mean observed distance in the half-normal distance function. This supports the heuristic notion in the related trapping grid problem of using the mean location of capture as an individual covariate (Boulanger and McLellan, 2001).
(2) Note that the effective scale parameter is a2(1 + ^2/a2) which tends to a2 as
^ 0, as it should. Moreover, the 'intercept' of the implied distance function (1 + ^)1/2, is less than 1, but tends to 1 as ¿2 ^ 0.
(3) This intuitive form also supports the general phenomenon (in regression problems) that measurement error attenuates (flattens) the regression curve toward the mean. In this case, it flattens the detection function.
(4) That the intercept of the implied distance function is non-zero suggests that the phenomenon of measurement error might appear to be 'imperfect detection on the line'. We see that adjustment for the latter phenomenon will not necessarily resolve the bias inherent in the scale parameter that results from measurement error. That is, if we were to use two observers to account for imperfect detection on the line - we might not be addressing the correct problem - measurement error.
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