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Hi everyone,

I teach our multivariate stats courses. When we talk about normality I always emphasize that univariate normality is necessary but not sufficient for MV normality. Do any of you have, or know of, an example in which all variables are normal but MV normality does not hold? I would be happy with a real dataset, a contrived made-up dataset, or a description of the situation so that I could then generate a simulated dataset. Or if you have a reference that provides such an example, please pass it along. Thanks!!

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Robert Pearson
Assistant Professor
University of Northern Colorado
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Nagaraj Neerchal
Professor and Chair
UMBC
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This falls under the "contrived example" category.

Let X ~ N(0,1). Define Y=X if  abs(X) > M; and Y= - X if abs(X) <= M for a fixed positive number M.
Now, the components of the vector (Y,X) are normally distributed, but it fails to satisfy the definition of
MVN.

Building on that, define a multivariate function on [X,Y] such that
f(x,y) is   MVN(x0,y0) if x>x0 and y>y0)
        and MVN(x0,y0) if x<x0 and y<y0
this will create two distinct "half MV normals" (actually quarter normals)
that will marginally be normal.

In general, you should be able to create a large number of bimodal bivariate distributions,
that marginally appear to be normal.

Ray

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Raymond Hoffmann
Professor
Medical College of Wisconsin
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Original Message:
Sent: 10-31-2012 11:49
From: Nagaraj Neerchal
Subject: Multivariate normality



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Nagaraj Neerchal
Professor and Chair
UMBC
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This falls under the "contrived example" category.

Let X ~ N(0,1). Define Y=X if  abs(X) > M; and Y= - X if abs(X) <= M for a fixed positive number M.
Now, the components of the vector (Y,X) are normally distributed, but it fails to satisfy the definition of
MVN.

In fact, you can combine any two univariate normal distributions with any copula except the normal copula, and the marginals will be univariate normal, but the bivariate distribution cannot be bivariate normal, since the copula of a bivariate normal distribution is the normal copula.  (This works for higher dimensions than 2 also.)

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Christopher Monsour
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Original Message:
Sent: 10-31-2012 12:06
From: Raymond Hoffmann
Subject: Multivariate normality

Building on that, define a multivariate function on [X,Y] such that
f(x,y) is   MVN(x0,y0) if x>x0 and y>y0)
        and MVN(x0,y0) if x<x0 and y<y0
this will create two distinct "half MV normals" (actually quarter normals)
that will marginally be normal.

In general, you should be able to create a large number of bimodal bivariate distributions,
that marginally appear to be normal.

Ray

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Raymond Hoffmann
Professor
Medical College of Wisconsin
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I just wanted to thank everyone for all of the useful suggestions and references I received. I was very impressed by the amount and quality of feedback I received. Thanks!

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Robert Pearson
Assistant Professor
University of Northern Colorado
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