Let X be a random variable with a density function f(X | [xi]), where [xi] = ([theta], [delta]), [theta] is the parameter of interest, and [delta] is a

nuisance parameter.

In contrast, under the cause specific hazard rate framework [17] introduced a semiparametric Bayesian method assuming that each cumulative baseline cause-specific hazard rate function has a gamma prior distribution, and a marginal likelihood function based on data and the prior parameter values was proposed for the estimation of regression parameters by considering cumulative baseline cause-specific hazard rate functions as a

nuisance parameter.

(2010) proposed a LM (hereafter [ILT.sup.*]) panel unit root test that is invariant to the

nuisance parameters. Following Lee and Strazicich (2009), the dependency of the test statistic on the

nuisance parameter can be removed with the following transformation:

This type of model arises when knowledge of all measurement variables is subject to significant error, and the model assumes a true value for each variable, which is to be estimated at least as a

nuisance parameter. It is true that least squares and similar methods have produced and are producing useful results when the errors in the independent variables are small, but sometimes they are large enough to have a substantial effect on the conclusions.

"Optimal Tests When a

Nuisance Parameter Is Present Only under the Alternative." Econometrica, 62, 1994, 1383-414.

Finally, when the focus is on parameter estimation despite model uncertainty, Bayesian analysis allows the model indicator itself to become a random variable treated as a

nuisance parameter. The posterior of parameters of interest can then be inferred averaged across models, taking into account the posterior probabilities of all models under consideration.

This

nuisance parameter represents some trait that might influence a systematist's choice of taxa and is expected to be similar in close relatives.

Note that because the precise CI (13) depends on the

nuisance parameter [mu], this paper shows how to address the

nuisance parameter [mu] based on a generalized pivotal quantity, and an exact CI for [sigma] is proposed based on the generalized pivotal quantity.

To remove these unwanted parameters (

nuisance parameters), we can multiply the likelihood given in (4) by the (assigned) probability distribution for each of the error standard deviations embodied in (6) and integrate the result with respect to the unwanted parameters (error standard deviations) to give an integrated likelihood function with the following form [15]:

According to (1), the unknown parameters, to be estimated from the data vectors [{y[n]}.sup.N.sub.n=1], are [zeta] = [[[[zeta].sub.1], ..., [[zeta].sub.K]].sup.T], directional parameters [nabla] = [[[[phi].sub.1], [[phi].sub.K]].sup.T] and [phi] = [[[[phi].sub.1]; [[phi].sub.K]].sup.T] and noise covariance level [[sigma].sup.2], where the directional parameters [phi] and [psi] are the ones of interest, while [zeta] and [[sigma].sup.2] are the unknown

nuisance parameters.

The fourth model was based on the model MO but included a localized and eroded WM regressor to form a local estimate of

nuisance parameters while avoiding gray matter signals in the regions of interest as follows:

We are interested primarily in estimating the association between A(t) and survival, [beta], while treating as

nuisance parameters the underlying hazard functions,