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In this paper we introduce a new technique for estimating the parameters of the Keplerian model commonly used in radial velocity data analysis for extrasolar planet detection. The un known parameters in the Keplerian model, namely eccentricity e, orbital frequency f, periastron passage time T, longitude of periastron ω, and radial velocity amplitude K are estimated by a new approach named SPICE (a semi-parametric iterative covariance-based estimation technique). SPICE enjoys global convergence, does not require selection of any hyperparameters, and is computationally efficient (indeed computing the SPICE estimates boils down to solving a numerically efficient linear program (LP)). The parameter estimates obtained from SPICE are then refined by means of a relaxation-based maximum likelihood algorithm (RELAX) and the significance of the resultant estimates is determined by a generalized likelihood ratio test (GLRT). A real-life radial velocity data set of the star HD 9446 is analyzed and the results obtained are compared with those reported in the literature.