Sequential estimation procedures can be defined in two stages (i) a stopping rule and (ii) the estimation procedure once the stopping rule is determined. We consider the class of estimation procedures related with a sequence of sequential sampling procedures where the size of sample tends stochastically to infinity and is asymptotically constant in the sense that its coefficient of variation tends to zero. It is shown that, under suitable regularity conditions, the discretized likelihood estimation procedure with a properly determined stopping rule is second order asymptotically efficient in the sense that the estimation procedure attains the Bhattacharyya type bound up to the second order. We also consider the sum of mean squared error and the expected size of sample multiplied by a constant and show that the bias-adjusted maximum likelihood estimation procedure combined with an appropriate stopping rule is second order asymptotically efficient in terms of the risk. Further the result is extended to the sequential estimation of a linearly combined parameter.