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Sampling from a MVN DistributionPowerPoint Presentation

Sampling from a MVN Distribution

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Sample Mean Vector

- We can estimate a sample mean for X1,X2, …, Xn

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

- And the sample covariance for X1,X2, …, Xn
- Sample variance
- Sample Covariance

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

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

- 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

- 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 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

- 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

- 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

- 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

- 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

- 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 Methods

- Plot the pairs
Do the points deviate too much from a straight line?

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

- 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

- Theoretical considerations

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 Transformations

- Box-Cox are a popular modification of power transformations where
- Box-Cox transformations determine the best l by maximizing:

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

- Examples of normality checks in SAS and R
- Begin our discussion of statistical inference for MV vectors

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