Let F = {F θ : θ Θ} denote the class of natural exponential family of distributions having power variance function, (NEF-PVF). We consider the problem of sequentially estimating the mean μ of F θ F, based on i.i.d. observations from F θ . We propose an appropriate sequential estimation procedure under a combined loss of estimation error and sampling cost. We provide expansion for the regret R a and study its asymptotic properties. We show that R a = cv 2 (μ) + o(1) as a → ∞, where c > 0 is a known constant and v(μ) denotes the coefficient of variation of F θ .