Estimating the Ratio of Multivariate Means

Document Type


Presentation Date


Conference Name

2012 Joint Statistical Meetings

Conference Location

San Diego, CA


Consider two correlated populations, X and Y, each with two correlated measurements; (X1, X2) and (Y 1, Y 2), respectively. Further suppose µx =( µ1, µ2), µy =( α µ1, α µ2), α ≠ 0, where α is the common ratio of bivariate means, µy i/ µ xi , i=1,2. The ratio of two means plays a significant role in areas such as Risk Analysis. The values such as the mean interest rate paid per mean amount of credit used, the mean amount of profit earned per mean amount invested in business, the mean amount of operating expense per mean revenue, or the mean amount of depreciation expense per mean amount of revenue are a few examples of meaningful ratios. The estimation of the unknown value of α has been investigated previously by several authors. The existing methods are based on two criteria: one by combining two estimates of α, each based on a test statistic Wi , where W i , is a linear function of X i, and Yi ; the otherby combining two estimates of α, each based on a test statistic Vi , where V i , is a linear function of the ratios, ri ,where rI = (Y i/X i), i =1, 2.

In this paper we provide an alternative way of estimating α by combining the ratios r i under some specific conditions. Under these conditions the authors have discussed the approximate distribution of each ratio ri as well as the approximate distribution of the statistic based on the combined ratios. The proposed estimation methodology can be extended to address the case of three or more correlated measurements. Further, the properties of the confidence intervals of α, based on the proposed estimates have been investigated and simulation study is used to compare the proposed confidence intervals of α with the intervals computed from existing methods. It is assumed that the variance covariance matrices ∑x and ∑y are unknown and unequal.


ratio of means, multivariate


Statistics and Probability

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