This study is concerned with the simultaneous detection and least squares estimation of vector random processes. The problem is formulated in the following context: A random process, out of a countably infinite collection of (not necessarily Gaussian) vector random processes with known distributions, is observed with additive white Gaussian noise. The <i>a priori</i> probability, that a specific random process will be observed, is specified for each one in the collection. The least squares estimate of the random process that is being observed is to be found in terms of the hypothesis conditioned estimates. It is shown that the best estimate is the linear combination of the hypothesis conditioned estimates weighted by the <i>a posteriori</i> probabilities of the hypotheses conditioned on the observations. A Radon-Nikodym derivative representation is derived for the <i>a posteriori</i> probability by using the specific structure of the product probability measure for this problem. It is shown that this Radon-Nikodym derivative can be expressed in terms of the Radon-Nikodym derivatives of measures induced by the random processes in the collection with respect to Wiener measure. By using the recent results on likelihood functions, an expression for the <i>a posteriori</i> probability is found in terms of the conditioned estimates. In this connection, an extended version of the partition theorem of parameter adaptive estimation is proved. The unique stochastic differential equation, that each <i>a posteriori</i> probability satisfies with its associated <i>a posteriori</i> probability as the initial condition, is derived for the case of finitely many hypotheses along with an expression for the conditional error covariance in terms of the hypothesis conditioned error covariances. The results are applied to the parameter adaptive estimation problem in linear continuous and discrete stochastic dynamic systems. In the continuous case, the solution is also obtained through an alternate approach using nonlinear filtering theory. An application of the theory to the design of a digital flight control system which is tolerant of sensor failures is presented with real-time hybrid computer simulation results. A review of random processes and statistical decision theory is also included. / Ph. D.