  The purpose of this paper is twofold. First, we propose a novel algorithm for
estimating parameters in one-dimensional Gaussian mixture models (GMMs). The
algorithm takes advantage of the Hankel structure inherent in the Fourier data
obtained from independent and identically distributed (i.i.d) samples of the
mixture. For GMMs with a unified variance, a singular value ratio functional
using the Fourier data is introduced and used to resolve the variance and
component number simultaneously. The consistency of the estimator is derived.
Compared to classic algorithms such as the method of moments and the maximum
likelihood method, the proposed algorithm does not require prior knowledge of
the number of Gaussian components or good initial guesses. Numerical
experiments demonstrate its superior performance in estimation accuracy and
computational cost. Second, we reveal that there exists a fundamental limit to
the problem of estimating the number of Gaussian components or model order in
the mixture model if the number of i.i.d samples is finite. For the case of a
single variance, we show that the model order can be successfully estimated
only if the minimum separation distance between the component means exceeds a
certain threshold value and can fail if below. We derive a lower bound for this
threshold value, referred to as the computational resolution limit, in terms of
the number of i.i.d samples, the variance, and the number of Gaussian
components. Numerical experiments confirm this phase transition phenomenon in
estimating the model order. Moreover, we demonstrate that our algorithm
achieves better scores in likelihood, AIC, and BIC when compared to the EM
algorithm.
