Sharp mean-variance bounds for Jensen-type inequalities. (English) Zbl 0705.60017

Pittenger, A. O.

Stat. Probab. Lett. 10, No. 2, 91-94 (1990).

Suppose \(\phi\) is a convex function and Y is a random variable such that \(\mu =E(Y)\) and \(E(\phi(Y))\) are finite. Then \(\phi(\mu)\leq E[\phi(Y)]\) by Jensen’s inequality. Suppose that \(\sigma^ 2=Var Y\) is finite. Let \(x_ 0\) belong to the domain of \(\phi\). If \(\mu=x_ 0\), define \(a_ 0=x_ 0\), and \(p_ 0=1\); otherwise let \[ p_ 0=(\mu-x_ 0)^ 2/[\sigma^ 2+(\mu -x_ 0)^ 2],\;a_ 0=\mu +\sigma^ 2/(\mu-x_ 0). \] If \(Y\leq x_ 0\) a.s. and if \((\phi (x)-\phi (x_ 0))/(x-x_ 0)\) is concave on \([-\infty,x_ 0]\cap dom(\phi)\), then it is shown that \[ (*)\;p_ 0\phi (a_ 0)+(1-p_ 0)\phi (x_ 0)\leq E[\phi(Y)]. \] If, in addition, \(\phi\) is convex, then the left side of (*) is greater than or equal to \(\phi(\mu)\). Conditions for equality in (*) are discussed. Several special cases have been derived.

Reviewer: B.L.S.Prakasa Rao

MSC:

60E15

Inequalities; stochastic orderings

Keywords:

convex function; Jensen’s inequality

References:

[1]	Agnew, R.A., Inequalities with application in economic risk analysis, J. appl. probab., 9, 441-444, (1972) · Zbl 0239.90011
[2]	Cartwright, D.I.; Field, M.J., A refinement of the arithmetic mean — geometric Mean inequality, Proc. amer. math. soc., 71, 36-38, (1978) · Zbl 0392.26010
[3]	Chao, M.T.; Strawderman, W.E., Negative moments of positive random variables, J. amer. statist. assoc., 67, 429-431, (1972) · Zbl 0238.60008
[4]	Eckberg, A.E., Sharp bounds on laplace—stieltjes transforms, with applications to various queueing problems, Math. oper. res., 2, 135-142, (1977) · Zbl 0405.60017
[5]	Karlin, S.; Studden, W.J., Tchebycheff systems with applications in analysis and statistics, (1966), Wiley New York · Zbl 0153.38902
[6]	Skibinsky, M., The range of the (n + 1)th moment for distributions on [0, 1], J. appl. probab., 4, 543-552, (1967) · Zbl 0189.18803