Some Probability Inequalities of Multivariate Normal and Multivariate t

Abstract Applying the result of expectations of non-negative random variables, inequalities for the one-sided and two-sided rectangular probabilities of higher-dimensional multivariate normal and multivariate t distributions are obtained. It is shown that their k-dimensional rectangular probabilities is lower bounded by the (k/m)th power of their m-dimensional rectangular probabilities for every k≥ m ≥ 1, and this bound is quite good when k/m is close to one. Applications to multiple decision problems are considered.

[1]  Z. Šidák.  ON MULTIVARIATE NORMAL PROBABILITIES OF RECTANGLES: THEIR DEPENDENCE ON CORRELATIONS' , 1968 .

[2]  J. Gurland.  Inequalities of Expectations of Random Variables Derived by Monotonicity or Convexity , 1968 .

[3]  J. Gurland.  The Teacher's Corner: Inequalities of Expectations of Random Variables Derived by Monotonicity or Convexity , 1968 .

[4]  C. Antle,et al.  Straight Line Confidence Regions for Linear Models , 1967 .

[5]  Z. Šidák.  Rectangular Confidence Regions for the Means of Multivariate Normal Distributions , 1967 .

[6]  D. C. Bowden,et al.  Linear Segment Confidence Bands for Simple Linear Models , 1967 .

[7]  S. Gupta.  Probability Integrals of Multivariate Normal and Multivariate $t^1$ , 1963 .

[8]  D. Slepian.  The one-sided barrier problem for Gaussian noise , 1962 .

[9]  C. Dunnett.  A Multiple Comparison Procedure for Comparing Several Treatments with a Control , 1955 .

[10]  C. Dunnett,et al.  Approximations to the probability integral and certain percentage points of a multivariate analogue of Student's t-distribution* , 1955 .

[11]  K. Pillai,et al.  ON THE DISTRIBUTION OF THE RATIO OF THE iTH OBSERVATION' IN AN ORDERED SAMPLE FROM A NORMAL POPULATION TO AN INDEPENDENT ESTIMATE OF THE STANDARD DEVIATION , 1954 .

[12]  C. Dunnett,et al.  A BIVARIATE GENERALIZATION OF STUDENT'S t-DISTRIBUTION, WITH TABLES FOR CERTAIN SPECIAL CASES , 1954 .

[13]  R. Bechhofer.  A Single-Sample Multiple Decision Procedure for Ranking Means of Normal Populations with known Variances , 1954 .