#### Title

Estimating the Ratio of Multivariate Means

#### Document Type

Presentation

#### Presentation Date

7-30-2012

#### Conference Name

2012 Joint Statistical Meetings

#### Conference Location

San Diego, CA

#### Abstract

Consider two correlated populations, X and Y, each with two correlated measurements; (X_{1}, X_{2}) and (Y _{1, }Y _{2}), respectively. Further suppose µ_{x} =( µ_{1, }µ_{2}), µ_{y }=( α µ_{1, }α µ_{2}), α ≠ 0, where α is the common ratio of bivariate means, µ_{y 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 W_{i }, where W_{ i }, is a linear function of X _{i, }and Y_{i }; the otherby combining two estimates of α, each based on a test statistic V_{i }, where V_{ i }, is a linear function of the ratios, r_{i },where r_{I =} (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 r_{i }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.

#### Keywords

ratio of means, multivariate

#### Disciplines

Statistics and Probability

#### Opus Citation

Chand K. Chauhan Dr and Yvonne Zubovic (2012).
*Estimating the Ratio of Multivariate Means*. Presented at 2012 Joint Statistical Meetings, San Diego, CA.

https://opus.ipfw.edu/math_facpres/107