Asymptotically robust algorithms for detection and recognition of signals

Improving the accuracy of estimates of the pulse sequence period using the methodology of complete sufficient statistics

“…Taking into account that the parameter (classical in the theory of maximum likelihood estimates) is non-energetic, in estimating the lower bound of the accuracy of pulse repetition period measurement, we take as a basis the expression for the elements of the Fisher matrix: where is the uncertainty function of the vector . Since the inverse Fisher matrix is the correlation matrix of the MLE of all simultaneously estimated parameters 29 , and the dispersions of the MLE of these parameters are on the main diagonal of this matrix, and taking into account that the focus is only on one non-energy parameter – the pulse arrival time ( ), it can be written as: where is the MLE of the pulse arrival time. Assuming that the uncertainty function is stationary with respect to the parameter it gives the following expression: where E is a signal energy.…”

“…The solution of this problem using the methodology of CSS is based on the theorem of the uniqueness of an effective estimate 29 , which is a consequence of the Lehmann–Scheffe theorem 13 .…”

“…A practical way to find sufficient statistics is to use the factorization theorem 29 , 31 , 32 , according to which the conditional probability density distribution must be represented as: …”

“…Here the parameter C( θ ) is obtained as C( θ ) = (− ϑ 3 /π) n + 1 exp{C 1 ( θ ) + C 2 ( θ )} , where C 1 ( θ ) and C 2 ( θ ) are represented by formulas ( 20 ) and ( 21 ). Thus, the conditional probability density takes the form: where and and according to the factorization theorem 29 , 31 , 32 , the statistics expressed by formula ( 23 ) are sufficient statistics.…”

“…For unknowns, , T, the parameter takes values from the region (0, ∞) × (0, ∞) × (− ∞, 0). We use the completeness theorem 27 , according to which if the probability distribution density belongs to the class of exponential distributions and the set of parameter values contains an n -dimensional interval, then the statistics Y is complete 29 . Then we can apply the theorem on the uniqueness of the effective estimate.…”