The Study For The Mills' Ratio Of The Skew-Generalized Normal Distribution

In probability theory,the Mills' ratio of one distribution has association with its tail probability.Therefore,the Mills' ratio is widely used in finance,insurance,actuary.We aim to study the the Mills' ratio of the SGN(skew-generalized normal)distribution,that is I(x),and establish some lower and upper bounds for I(x).I(x)is not a completely monotonic function.However,we find I(x)has some propositions which is similar to complete monotonicity.Thus we introduce a new concept named multiple order completely monotonic function,as well as multiple order Bernstein function and their propositions.Then using variable substitution,we find the recursions among these multiple order derivatives,and our lower and upper bounds are established.Finally,for demonstrating the efficiency of our estimates,some numerical computations are provided.