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Sampling from a MVN Distribution. BMTRY 726 1/17/2014. Sample Mean Vector. We can estimate a sample mean for X 1, X 2, …, X n. Sample Mean Vector. Now we can estimate the mean of our sample But what about the properties of ? It is an unbiased estimate of the mean

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sample mean vector
Sample Mean Vector
  • We can estimate a sample mean for X1,X2, …, Xn
sample mean vector1
Sample Mean Vector
  • Now we can estimate the mean of our sample
  • But what about the properties of ?
    • It is an unbiased estimate of the mean
    • It is a sufficient statistic
    • Also, the sampling distribution is:
sample covariance
Sample Covariance
  • And the sample covariance for X1,X2, …, Xn
  • Sample variance
  • Sample Covariance
sample mean vector2
Sample Mean Vector
  • So we can also estimate the variance of our sample
  • And like , S also has some nice properties
    • It is an unbiased estimate of the variance
    • It is also a sufficient statistic
    • It is also independent of
  • But what about the sampling distribution of S?
wishart distribution
Wishart Distribution

Given , the distribution of is called a Wishart distribution with n degrees of freedom.

has a Wishart distribution with n -1 degrees of freedom

The density function is

where A and S are positive definite

wishart cont d
Wishart cont’d
  • The Wishart distribution is the multivariate analog of the central chi-squared distribution.
    • If are independent then
    • If then CAC’ is distributed
    • The distribution of the (i, i) element of A is
large sample behavior
Large Sample Behavior
  • Let X1,X2, …, Xnbe a random sample from a population with mean and variance (not necessarily normally distributed)

Then and Sare consistentestimators for m and S. This means

large sample behavior1
Large Sample Behavior
  • If we have a random sample X1,X2, …, Xna population with mean and variance, we can apply the multivariate central limit theorem as well
  • The multivariate CLT says
checking normality assumptions
Checking Normality Assumptions
  • Check univariate normality for each component of X
    • Normal probability plots (i.e. Q-Q plots)
    • Tests:
      • Shapiro-Wilk
      • Correlation
      • EDF
  • Check bivariate (and higher)
    • Bivariate scatter plots
    • Chi-square probability plots
univariate methods
Univariate Methods
  • If X1, X2,…, Xn are a random sample from a p-dimensional normal population, then the data for the ith trait are a random sample from a univariate normal distribution (from result 4.2)
  • -Q-Q plot
    • Order the data
    • Compute the quantiles according to
    • Plot the pairs of observations
correlation tests
Correlation Tests
  • Shapiro-Wilk test
  • Alternative is a modified version of Shapiro-Wilk test
  • Uses correlation coefficient from the Q-Q plot
  • Reject normality if rQ is too small (values in Table 4.2)
empirical distribution tests
Empirical Distribution Tests
  • Anderson-Darling and Kolmogrov-Smirnov statistics measure how much the empirical distribution function (EDF)

differs from the hypothesized distribution

  • For a univariate normal distribution
  • Large values for either statistic indicate observed data were not sampled from the hypothesized distribution
multivariate methods
Multivariate Methods
  • You can generate bivariate plots of all pairs of traits and look for unusual observations
  • A chi-square plot checks for normality in p> 2 dimensions
    • For each observation compute
    • Order these values from smallest to largest
    • Calculate quantiles for the chi-squared distribution with p d.f.
multivariate methods1
Multivariate Methods
  • Plot the pairs

Do the points deviate too much from a straight line?

things to do with non mvn data
Things to Do with non-MVN Data

Apply normal based procedures anyway

Hope for the best….

Resampling procedures

Try to identify an more appropriate multivariate distribution

Nonparametric methods

Transformations

Check for outliers

transformations
Transformations
  • The idea of transformations is to re-express the data to make it more normal looking
  • Choosing a suitable transformation can be guided by
    • Theoretical considerations
      • Count data can often be made to look more normal by using a square root transformation
    • The data themselves
      • If the choice is not particularly clear consider power transformations
power transformations
Power Transformations
  • Commonly use but note, defined only for positive variables
  • Defined by a parameter l as follows:
  • So what do we use?
    • Right skewed data consider l< 1 (fractions, 0, negative numbers…)
    • Left skewed data consider l> 1
power transformations1
Power Transformations
  • Box-Cox are a popular modification of power transformations where
  • Box-Cox transformations determine the best l by maximizing:
transformations1
Transformations
  • Note, in the multivariate setting, this would be considered for every trait
  • However… normality of each individual trait does not guarantee joint normality
  • We could iteratively try to search for the best transformations for joint and marginal normality
    • May not really improve our results substantially
    • And often univariate transformations are good enough in practice
  • Be very cautious about rejecting normality
next time
Next Time
  • Examples of normality checks in SAS and R
  • Begin our discussion of statistical inference for MV vectors
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