Читать книгу Applied Univariate, Bivariate, and Multivariate Statistics - Daniel J. Denis - Страница 43

We often wish to analyze data simultaneously on several response variables. For this, we require vector and matrix notation to express our responses. The matrix operations presented here are surveyed more comprehensively in the Appendix and in any book on elementary matrix algebra.

Figure 2.8 Because the sum of deviations about the arithmetic mean is always zero, it can be conceptualized as a balance point on a scale.

Consider the following vector:

where y₁ is observation 1 up to observation y_n.

We can write the sample mean vector for several variables y₁ through y_p as

where is the mean of the p^th variable.

The expectation of individual observations within each vector is equal to the population mean μ, of which the expectation of the sample vector y is equal to the population vector, μ. This is simply an extension of scalar algebra to that of matrices:

Likewise, the expectations of individual sample means , , … are equal to their population counterparts, μ₁, μ₂, … μ_p. The expectation of the sample mean vector is equal to the population mean vector, μ:

We note also that is an unbiased estimator of μ since .

Recall that we said that the mean is the first moment of a distribution. We discuss the second moment of a distribution, that of the variance, shortly. Before we do so, a brief discussion of estimation is required.