```Applied Multivariate Analysis
Prof. Amit Mitra
Prof. Sharmishtha Mitra
Department of Mathematics and Statistics
Indian Institute of Technology, Kanpur
Lecture No. # 02
Basic concepts on Multivariate Distribution – II
Let us start the day with some definitions. Suppose we have x, a p dimensional random
vector such that, we have expectation of x as the mean vector mu and the covariance
matrix of x to be denoted by sigma. We define the characteristic function.
(Refer Slide Time: 00:20)
Characteristic function of x is defined as phi x t to be expectation of e to the power i t
prime x, where we have i to be square root of minus 1 and t, a vector belonging to R to
the power p right. Now, this is the joint characteristic function of the components of x.
So, it is the joint characteristic function for the random vector x. Now given the
can actually get to the marginal characteristic functions of the respective components,
that make up this particular random vector x; similar to what we had seen in the last
lecture for the movement generating function.
Now, this characteristic function ofcourse completely determines a multivariate
distribution. And the knowledge of this characteristic function about the multivariate
random vector completely determines also the characteristic function and hence, the
distribution of the respective components of x. Sometimes a quantity which is used to
condense the information about, which is present in this covariance matrix sigma; other
two following quantities just put it as definitions. So, suppose the covariance matrix of x
is sigma; it is a p by p matrix. So, there are unknown elements which are present in this p
by p matrix; it is a symmetric matrix.
So, the total number of unknown quantities in this particular matrix is p in to p plus 1 by
2. Now, in order to summarize this the information, that is present in the covariance
matrix. The two following quantities are defined. The first is the total variance or total
variation in x. So, the total variation in x is given by trace of the sigma matrix. So, it
condenses the information that we get through the covariance matrix in terms of single
quantity, which is given through the trace of this particular matrix. The second quantity
which is of interest, what is referred to as a generalized variance?
Generalized variance of this random vector x and that is given by determinant of sigma.
So, once again it summarizes or rather condenses the information, that is present in this
covariance matrix sigma in terms of two single quantities either trace of sigma or
determinant of sigma. The first one is called the total variation in x. As we will see later
on, then this total variation in x is the quantity of interest and which actually is looked
upon as preserving type of approach, when we look at principle components analysis.
Now, it should be noted that, the two quantities that we have defined just now.
The trace of sigma, which is the total variation in x and the generalized variance of x are
not sufficient to replace this p by p matrix sigma. So, just put it as a note that, this trace
of sigma and determinant of sigma cannot actually replace this sigma. It is obvious
actually, if you look at this determinant and then, the corresponding condensation in
terms of trace of sigma and determinant of sigma. Say for example, if you have a sigma
matrix, suppose sigma is a 2 by 2 matrix which has elements say 2 3 on the diagonal; 1 1
on the off diagonal.
Suppose this is the first sigma matrix covariance matrix; we can have another sigma
matrix, which is of 2 by 2 matrix; which has entries as 2 3, the same entries on the
diagonal; minus 1 and plus 1 on the two off diagonal. So, these are two different
covariance matrices. However, if we look at the total variation that is, it is given through
trace of sigma. Trace of sigma 1 will be equal to trace of sigma 2 and that is equal to the
some of the two diagonal elements that is 5. We can look at the determinant of the two
covariance matrixes sigma 1 and sigma 2. So, these two quantities would be 2 into 3
minus 1. So, that is equal to 5 as well right.
So, the total variation that is there in this sigma 1 and sigma 2 are both equal to 5. The
generalized variance of the underlined random vector x, which is given through this
sigma 1 and sigma 2; both of them are also equal to 5. So, they have both of these two
covariance matrixes have the same measures as far as total variation and the generalized
variance of the associated random vector is concerned. However, the two covariance
matrixes themselves are quite or clearly different right. So, these two quantities are not
enough actually to replace the covariance matrix as such. However, the condensation that
one gets is of interest at times.
(Refer Slide Time: 06:23)
Now, let us move in to an important concept, which is random sampling from
multivariate distributions or rather random sampling from multivariate distributions.
Now, random sampling from multivariate distribution; what is what is basically the need
of this random sampling? Suppose we have a multivariate multivariable population
Suppose we have a multivariable population with unknown mean vector as mu, which is
p by 1 and unknown covariance matrix as sigma. Since for all practical purposes, when
we considering a multivariate population these two quantities; mu, the mean vector and
the covariance matrix sigma are unknown.
So, the numbers of unknown quantities, the parameters which are actually present in that
particular population are the following. So, this mean vector will lead us to p unknown
quantities, because it is a p dimensional vector. So, p unknown quantities and when we
look at the sigma matrix, which is p by p matrix. This is a symmetric matrix; it is a
covariance matrix. Hence, it is a symmetric matrix and hence, the total number of
unknown quantities which are present in this sigma matrix are p into p plus 1 by 2. So,
the total number of unknown quantities which are present in this mu vector and the
covariance matrix are p plus p into p plus 1 by 2 right.
So, in order to have inference about these unknown quantities and to have some
estimators corresponding to the mean vector mu and the covariance matrix sigma, what
is done is to go for random sampling from that particular population. Now, let us denote
by the following quantities. Let x 1 vector, x 2 vector, x n vector be a random sample
from the multivariate population. So, if we have that, then on the basis of this n random
samples, now each of these x i's; x i vectors are p by 1; because that is basically random
sample drawn from the p variate, a multivariate population. And hence in order to have
the inference about the mean vector mu and the covariance matrix sigma, we will be
using this particular set of random sample.
Now, let us denote by capital x, the random sample matrix. The random sample matrix,
which is a p by n matrix basically comprises of the following vectors. So, where this x 1
vector, which is the first random sample to up to x n vector, which is the nth random
sample. So, we have this p by n random matrix, which is actually having this entire
random sample of dimension n. Now, this can alternatively be written in terms of the
following. So, let us write this as y 1 prime, y 2 prime and y p prime, where these
quantities are having the following interpretation.
(Refer Slide Time: 10:26)
So, we have this x j vector is basically the j th observation vector. We have n such
observations. So, j equal to 1 to up to n, where this x j vector is given by the following,
that this is say x 1 j, x p j. So, each of these are p dimensional, because that is j th
observation vector corresponding to this p dimensional vector random vector. Similarly,
this y i vector that we had defined here, that this is y 1 prime, y 2 prime, y n prime. This
y i prime is what is holding the observations corresponding to the i th variable. So, these
are basically y 1 prime. So, corresponding to the i th row of that matrix.
So, this would be x i 1, x i 2, x i n; where this i now is from 1 to up to p. So, this
basically is the structure of data matrix; where this y i prime is basically the i th row of
the x matrix and x j is the j th column of the x matrix right. In terms of these vectors, that
we have defined through the random sampling; we will introduce the two important
quantities, which is the random mean vector or sample mean vector and the sample
variance covariance matrix, which would be used actually in order to have inference or
rather the inference about the unknown mean vector mu and the unknown covariance
matrix, that is sigma right.
So, we will have this being defined, which is the x bar matrix or x bar vector rather. This
is the sample mean vector. well In order to have the same notation as this one, what we
will denote this by capital x i’s. So, let us be consistent with this definition. So, you will
have a capital x i’s to denote the corresponding random variables and we will use small x
i's to denote the observations. So, x capital x bar vector is the sample mean vector and it
is given by the following. So, this is a p dimensional vector, which holds x 1 bar, x 2 bar
and the p th variables mean x p bar.
So, this would be given by 1 upon n, then the summation of all the observations
corresponding to the first variable. So, that is x 1 j say, this j is equal to 1 to up to n and
the last element is 1 upon n summation j equal to 1 to n summation x p j right. So, these
are the corresponding sample means of the respective variables in the x vector
component. Now, in terms of this observation vector, this can be written in the following
way. So, you can take 1 upon n outside and then, what we have here is? This is basically
y 1 prime multiplied by I which is of dimension n; I n.
So, it is a column vector of dimension n with 1 as all the entries and the last one is, what
is corresponding to the all the variables for the p th variable; all the observation
corresponding to the p th variable, where this will just complete this particular stuff here.
So that, this in terms of the random matrix, what we have is 1 upon n x times I n; where
this I n is an n by 1 column vector, which has 1 as all its entries. So, this x bar which is
the sample mean vector; it is a random vector once again; because it is comprising of the
random variables that we have taken. So, the entries of that particular x vector are given
by these, which in terms of the observation vectors that we had defined earlier, the p
observation vectors.
So, these basically are these observation vector, y i primes and that random sample mean
vector in terms of the random matrix, that the data matrix that we have defined. It is 1
upon n x times I n. Now, let us move on to looking at now this x bar vector, which is the
sample mean vector is what is going to be used for inference that is based; that is
inference about the population mean vector, which is unknown; which is the mu
quantity. Now, in order to look at the other unknown quantity which is present; which is
sigma the variance covariance matrix.
(Refer Slide Time: 15:50)
We introduce the sample variance covariance matrix. So, the sample variance covariance
matrix can have divisor either n similar to the univariate set up can have a divisor as n
can have a divisor n minus 1. Suppose we define that in terms of a divisor n, then this is
how the sample variance covariance matrix is going to look like; so, this sample variance
covariance matrix which is going to be based once again on the random samples that we
have drawn. It is a p by p random matrix, which has the following entries. So, the 1 1th
element of that is going to be given by this x 1 j minus x 1 bar.
Now, x 1 bar is what we have already defined through this random vector here, the
sample mean vector. So, that is what is going to be used here. Then, the 1 2 th element is
going to be given by this x 1 j minus x 1 bar; this into x 2 j minus x 2 bar and the last
entry here do not have enough space actually to write it. So, this would be given by this x
1 j minus x 1 bar that multiplied by x p j minus x p bar right. So, this is the matrix here.
Then, this is going to be a symmetric matrix. So, this we only need to write the upper try
upper triangle of this particular matrix.
The 2 2 th element would be the one, which is corresponding to the second variable; this
is x 2 j minus x 2 bar whole square. The last entry in this row would be x 2 j minus x 2
bar this into x p j minus x p bar j equal to 1 to n and the last diagonal entry the p p th
element of this matrix would be j equal to 1 to up to n x p j minus x p bar whole square.
Now, if you look carefully at this sample variance covariance matrix that we have listed
out here, it basically is holding say for example, this particular element; this you can
associate with the variance covariance variance for the first component.
So, we can denote that by s 1 1 say, then 1 upon n of this quantity is what is sample
covariance between the first and the second variable. The last entry in the first row here
is the sample covariance between the first and the p th variable. So, this would be s 2 1; s
2 1 is just this element s 1 2 itself; because this is the variance covariance matrix. This is
s 2 2, s 2 p and this would be s p p element, where the s i j is the i j th entry of the
particular sample variance covariance matrix right. Now, if we define the sample
variance covariance matrix through divisor n minus 1, we will see later on that; one of
them would be unbiased estimator of the population covariance matrix sigma.
And the other is going to be associated with the maximum likelihood estimator, when we
talk about random sampling from a multivariate normal distribution. So, one could have
also define this, in terms of this your yes n minus 1 quantity. So, we can say that n times
s n would be in such a situation will be given by n minus 1 s n minus 1 right; where the s
n minus 1 matrix would be this matrix what we have with a divisor here as n minus 1
right. Now, let us try to write this particular sample variance covariance matrix, what we
have defined in terms of the data matrix x that we had introduced.
(Refer Slide Time: 20:43)
So, we will have the n minus 1 s n minus 1 is equal to n times s n. Now, that if we look at
this particular matrix here, now this element here can be written in terms of summation
following. I will just write it out here that this entry here; we can write it as x 1 j square
this minus n times x 1 bar square. So, all these entries similarly can be written like this.
Say for example, the 1 2 th element here can be written as summation x 1 j x 2 j minus n
times x 1 bar in to x 2 bar. So, keeping that in mind, we can write this particular matrix
in two parts. The first part will hold the sum of squares and the cross products.
So, we can right that as x 1 j square; this j equal to 1 to n. The second entry, we will just
have the first quantity which I said x 1 j x 2 j; the last entry here is summation from j
equal to 1 to n x 1 j into x p j. This is going to be x 2 j square j equal to 1 to n and the last
entry here similarly would be j equal to 1 to n x 2 j x p j and the last entry p p th element
of this matrix is just the sum of squares corresponding to the p th component. So, this is
the first entry, the first block actually and then, this minus n times the sum of squares
entries which will be getting like here; n times x 1 bar will come here; n times x 1 bar
into x 2 bar will come here and similarly, the other entries will follow.
So, this can be written in terms of x 1 bar square. The second entry will be x 1 bar x 2
bar. The last entry in this first row would be x 1 bar x p bar x 2 bar square x 2 bar into x
p bar. Then, the last diagonal entry would be x p bar square right. So, once we have
written it in this particular form, it is easy to realize that the first matrix that we have
written here in terms of the data matrix; that we had introduced that x. It is just x x
transpose right. The x matrix we had defined here which was well we can actually we
had written this as this matrix here. Let me see yeah this particular matrix here, this
basically is the x matrix.
So, if we write the corresponding entries here x 1 vector, the first component; so, it will
have the p components out there. Then, this x n will have once again the p components
of that n th observation vector. And then, this matrix p by p would actually lead us to the
product x x transpose, which would now be having the diagonal entries as the sum of the
squares of the respective components and the off diagonal entries will hold the products.
Now, this particular matrix block here can similarly be written in terms of the x bar
vector. So, this is x bar vector that multiplied by this x bar vectors transpose.
What is x bar vector? x bar vector is what we have defined here as the respective x bar
components for the p entries (( )) p elements. Now, we from this particular form, it is
sometimes useful to reduce it to this particular form. Now, this is not yet in terms of
entirely the data matrix. Now, as we had seen that, this x bar vector can be written in
terms of the data matrix as x bar vector equal to 1 upon n x I n. So, we can use that same
thing out here. So, it is 1 upon n x times I n and then, the transpose of that particular
vector. So, this can be written as x x transpose minus 1 upon n; this one will get canceled
out.
So, will have one one upon n remaining and then, this is x I n, then the transpose of this
quantity. So, what will be having is I n prime and x prime. So, we can rewrite this in a
more compact form as x I n minus 1 upon n I n I n prime; this multiplied by x transpose.
So, this particular form what we have is what is now expressing this the sample variance
covariance matrix with either the divisor n or the divisor n minus 1 in terms of the
observed data matrix. Now, here I n of course is an n by n identity matrix. So, this is
what we get. Now, an alternate form of this sample variance covariance matrix is also
sometimes useful.
(Refer Slide Time: 26:36)
I will just write that, say n minus 1 s n minus 1 n times s n that what we have derived as
x x transpose into x bar transpose. So, since this x vector, x matrix rather; x matrix is of
this particular form x x transpose can be given by the following that, this is summation j
equal to 1 to n x j x j transpose minus n times x bar x bar transpose. So, this can be
written as following x j minus x bar vector into x j minus x bar transpose j equal to 1 to n
right. So, this form also we will today itself use this form, in order to derive an unbiased
estimator for the population covariance matrix, that is sigma. Now, similar to the
population correlation matrix, one can also define the sample correlation matrix.
Say, the sample correlation matrix say denoted by R would be say D half inverse times s,
where s is either with the divisor s n or n minus 1; this into D half to the power minus 1;
where this D matrix is the diagonal matrix which is holding the sample variances of the
respective components p in number right. So, through this diagonal matrix, if we look at
pre and post multiplication using D half inverse; So, what will be getting is the sample
correlation matrix; where the i j th element will actually be the sample correlation
random variable for the x i and x j component of this right. Now, let us look at a bit of
geometric interpretation for this random sampling.
interpretations. Suppose we look at these as observation vectors, let x 1 x 2 x n be n
observation vectors. Now, from these observation vectors we had seen that, one can also
write this in terms of y i’s. So, this y 1 vector, y 2 vector, y p vector; this would be
observations corresponding to p variables. Now, if we look at the projection of any y i on
one projection of say any y i vector on I n is going to be given by y i prime 1 divided by
1 prime 1; this multiplied by this 1 vector and what is this equal to?
This is the projection this is the projection vector, which is the projection of y i on I n.
So, there are p such vectors, i equal to 1 to up to p. Corresponding to each of these p
vectors, which now are holding all the observations corresponding to these variables. So,
this is what is going to give us the sum of all the entries, which we have corresponding to
the i th variable and 1 prime 1; this is I n I just drop this I n from all these places. So,
later on actually without loss of any generality will just drop this particular I n index will
say that, it is a vector which is holding 1’s and its (( )) particular dimensions conforming
to the other vector.
So, this is just the sum of all these observations and this is going to give us n. So, this is
what we will be having is x i bar times I n. So, this basically is the vector, which is of
dimension n and has entries x i bar at all the places. So, that is basically the projection.
So, this is the mean mean of the i th variable and this is the n dimensional vector and that
is what has got the interpretation that, it is just a projection of the y i vector. The i th
vector correspond the vector corresponding to the i th variable are holding all the n
observations on this I n vector and that is what is this.
(Refer Slide Time: 32:11)
Now, let us also look at the following deviation vectors. Let us denote by say d i, the
deviation vector. Now, the deviation vector is the deviation, which we are going to
define as y i; this minus x i bar times I right. So, this is what is going to hold the
following quantities that, we have x i 1 minus x i bar; this is x i n, the nth observation
minus x i bar. So, this deviation vector is what is going to give us in each of the
components, the deviation of the respective observations corresponding to that i th
variable.
This is the first observation in the i th variable that minus the mean corresponding to all
the n observation of that i th variable. So, these are this is the deviation of the first
observation from its mean from the mean corresponding to that variable and likewise, we
have all these n entries like this. If we look at the square of the norm of this deviation
vector; if we just look at this d i prime d i, what is that going to give us? That is going to
give us the sum of squares of these deviation quantities. So, this is going to be given by x
i j minus x i bar square this j equal to 1 to n. So, what is this quantity? This quantity is n
minus 1 times s i i.
So, this is actually if we have the deviation vectors, d i is being denoted by this deviation
as we have discussed that, these are the deviations basically from the respective mean
components. Then, d i prime d i is nothing but the sum of squares of this l these entries in
the deviation vector, which is associated with n minus 1 times s i i; where s i i is the
sample observed variance components corresponding to the i th components. So, we
have this deviation vectors p in numbers and hence, this also will be p in numbers. So, it
is basically the norms square of this deviation vector are associated with the sample
variances of the respective components with the multiplier n minus 1 as the divisor.
Now, similarly also what we can actually see that this d i prime d k, which is the dot
product of the deviation for the i th variable and the dot product for the k th variable.
This would be the cross product j equal to 1 to n x i j minus x i bar. So, this is x i j minus
x i bar into x k j minus x k bar right. And this is what this quantity, which is the cross
product between the deviation vector for the i th variable and k th variable. It is nothing
but the covariance n minus 1 times s i k. So, where s i k is 1 upon n minus 1 of this
product, which is the covariance between the i th and the k th variable.
Let us also now look at the angle between these two deviation vectors. Let us denote by
theta i k. Let theta i k be the angle between d i, the deviation vector for the i th variable
and d k, the deviation vector corresponding to the k th variable. Then, what we have is
this cosine of this theta i k is going to be given by this, which is d i prime d k; this
divided by d i prime d i, this for the i th vector d i; this into d k prime d k whole raise to
the power half right. So, if we have these two vectors d i and d k, then the cosine of the
angle between the two vectors two deviation vectors is given by this and what is this
equal to?
As we have seen that, this d i prime d k is nothing but our n minus 1 times s i k, the
covariance that divided by now this d i prime d i, as we had seen out here is n minus 1
times s i i. So, this comes down to n minus 1; this as s i i. So, that is coming from this
and this d k prime d k is n minus 1 times s k k. So, this is this s k k whole raise to the
power half. So, what we sees that, this n minus 1 factor cancels out and what we have as
the cosine of the angle between these two deviation vectors is s i k divided by under root
of s i i into s k k.
(Refer Slide Time: 38:06)
That is, the cosine of this angle theta i k is s i k that divided by s i i in to s k k whole raise
to the power half and what is that equal to? That is, just the correlation the sample
correlation between the i th and the k th variable. So, this gives us a nice geometric
interpretation about these deviation vectors, that are associated with this random
sampling; that the cosine of the angle between the two is nothing but the correlation
between two random variables i and k. Now, given this particular expression here we can
say that, if this theta i k the angle between the two deviation vector is 0. So, they are
basically in the same direction.
So, if this theta i k the angle between the two deviation vector is 0, then what we have
this r i k is cosine of that angle 0 and hence, this is equal to 1. That justifies actually our
intuition that, if two deviation vectors are in the same direction, then the correlation
between the two random variables would be perfect linear correlation. So, that the
correlation is equal to 1. Now if on the other hand, the two are orthogonal if theta i k
equal to pi by two. So, we have the two deviation vectors orthogonal to one another.
Then, what is the value of the correlation coefficient between the two as we would
expect the two are orthogonal?
So, it is moving in orthogonal directions and hence, the correlation would just be equal to
0. In the other extreme, if they move in opposite direction not orthogonal, move perfectly
in opposite direction; that is, if the angle between the two deviation vectors is pi. So, the
two vectors are moving in completely different, but perfectly opposite direction. Then,
what we have this r i k is what we expect that, they are exactly in the oppose perfect
negative correlation. So, this gives us a nice feeling about verifying the intuition that,
what happens to the deviation vectors and the angle that they are making and the
corresponding values of the measure of association between the two any two variables.
So, this cosine theta i k ofcourse, it is for every pair i and k taken from the set of possible
p variables right. Now you remember that, we had defined at the start that, two quantities
which are associated with the covariance matrix sigma, the total variation in x given
through the trace of sigma matrix and then, the generalized variance of x given by the
determinant of this sigma matrix. Similar to that, one can also define the sample
quantities. So, the sample quantities would be the sample total variance. One can define
the sample total variation as trace of s matrix s either with a divisor n or with a n minus
1.
And the sample generalized variance as the determinant of s matrix and as we had argued
that these ofcourse, gives us compression of the sample variance covariance matrix.
However, these are not sufficient enough to replace s. Because we can construct in a
similar way to what we had seen for the sigma matrix that, it is possible to have two
different sample covariance matrix giving us the same total sample variation and the
sample generalized radiance right. Now, we move on to one important result regarding
this estimation procedure.
(Refer Slide Time: 42:35)
Now, as we had seen that, we are we have the two following quantities, which is the x
bar vector; which we are going to associate with the population mean vector, which is
mu and we have the sample variance covariance matrix say with a divisor n minus 1.
Now, that is what is going to be used in order to have inference about the population
variance covariance matrix that is sigma. The way of the following result, let me first
state the result. So, let our x 1, x 2, x n be a random sample be a random sample from a
from a multivariate population with mean vector unknown as mu and covariance matrix
as sigma.
Then, the two quantities that we have defined as x bar and s n minus 1 has the following
properties. Number one: expectation of this x bar vector is going to be equal to the mean
vector mu and covariance matrix of this x bar vector is going to be sigma by n and
number two: for the covariance matrix sample variance covariance matrix s n minus 1,
expectation of s n minus 1 only is equal to sigma. In other words, we are trying to say
that this x bar vector, the sample mean vector is an unbiased estimator of the population
mean vector, which is mu.
The sample variance covariance matrix with a divisor n minus 1 is an unbiased estimator
of the population variance covariance matrix, that is sigma right. And the covariance
matrix of this x bar vector is going to be given by sigma by n. This gives us the feeling of
generalization of the univariate result what we have to the multivariate set up. We had
ofcourse; similar result when we had univariate set up. Let us quickly look into the proof
of this particular result. Now, in order to prove this one, one is quite straight forward
that, expectation of this x bar is expectation of the respective components.
So, its expectation of 1 upon n x bar; now x bar is going to be given by this x i bar. So,
its all the all these observations i equal to 1 to n. Now, when we look at expectation of
this; its expectation operators comes inside and what we have is i equal to 1 to n.
Expectation of these x i components and expect x 1, x 2, x n are random sample from the
same multivariate population. Each having identical mean as mu and hence, expectation
of each of these exercise will be equal to mu. So, what will have this as n times mu; this
divided by n and hence, that is equal to mu. So, we have this first component of this
result.
We can also now look at the second result, which gives us the covariance of this x bar
vector. Which by definition of the covariance matrix of any random vector would be
given by x bar minus its expectation vector, which is mu as what we have derived; that
multiplied by the transpose of that same quantity. So, what we have this is the following
that, one can write this as 1 upon n summation i equal to 1 to n. Then, we will have this x
i minus mu and then, the transpose of the same quantity, which is this i equal to 1 to n x i
minus mu transpose. Now, what is this quantity equal to? This quantity would be equal
to the following.
(Refer Slide Time: 47:10)
Let us just split this or rather write it term by term. So, the covariance matrix of this x bar
vector is going to be given by expectation. Now, what is the first element? The first
element is 1 upon n, as we had seen it is summation of those components. So, that it will
hold these quantities. So, this is x 1 minus mu plus the last term is nth random sample
vector. Now, this multiplied by the transpose of this. Transpose of this would be given
by 1 upon n and then, the transpose of the first entry, which is x 1 minus mu transpose
plus x n minus mu transpose right.
Now, when we look at taking expectation term by term of each of these entries with
these; now, if we look at expectation of this term multiplied by this term, the first term
here; what will be getting is the covariance matrix of this x, which is sigma apart from
this multiplier. Now, when we look at the cross product say this with the next element
here; now remember that this x 1 vector, x 2, x n; they are random sample and hence,
they are independent. So that, we will have the expectation of the cross product of this
with any of these here, except the first entry would be 0; because we have x i x any x i be
independent of x j, if i is not equal to j.
And hence, the covariance matrix between the two would be equal to 0. So that, what
will be having is the following; after we take expectation is 1 upon n square will be there
and then, we will have this. When this is multiplied with each of these entries and then,
expectation being taken only the first element would give us sigma and all the rest of the
elements will be 0. When we look at the second entry here, once again the same entry
here will give us the sigma matrix and the rest of the entries will be zeros. So, we will
have these n sigma components. So, these are n in numbers.
Corresponding to this type of product 1 being 1 with 1, 2 with 2 and n with n, all the
cross product entries will be zeros; because of independence of these components x 1
bar, x 1, x 2 and x n. So, this thus gets us down to 1 upon n square n times sigma matrix
and that is nothing but, this sigma by n matrix right. So, we have proved the first part of
this particular result. Let us now move on to proving the second part of the result, which
establishes actually the unbiasness elements of this covariance matrix. So, what we had
seen? this is the second part of the result. So, what we now have is, this n minus 1 s n
minus 1; that is what, we had seen earlier is x j minus x bar into x j minus x bar
transpose, this j equal to 1 to up to n right.
Now, we had also seen that, this particular term is written equivalently in the form that,
this is j equal to 1 to n x j x j prime minus n times x bar x bar prime right. Now, we look
at proving the results. So, if we now look at expectation of this n minus 1 s n minus 1.
This is going to be given by expectation of this particular entire quantity. So, we take
expectation term by term. We will have this as expectation of x j x j prime, this minus n
times expectation of x bar x bar transpose. Let us give an equation number 1 to this
because will be requiring this latter on. So, we need to find out those two quantities.
What those expectations are? Expectation of x j x j prime and expectation of x bar x bar
prime.
(Refer Slide Time: 51:52)
So, what we realize is that expectation this covariance matrix of x j. x j is the random
sample drawn from that multivariate population. So, the covariance matrix of x j is
nothing but sigma. So, that is equal to expectation of this x j minus mu into x j minus mu
transpose. And hence, this is equal to as we had seen in the last lecture, this is nothing
but x j x j prime; this minus mu mu prime. And hence, this would imply that expectation
of x j x j prime, which is a quantity that we would be requiring in order to evaluate that
expression 1 is given by this right.
Furthermore, if we recall the result that we had proved in a first part, covariance matrix
of x bar is sigma by n and that is equal to expectation of x bar minus mu, its mean into x
bar minus mu transpose right. So, by the same approach what one can show is that, this
is expectation of x bar x bar prime; this minus mu mu prime right. So, what we will be
having is this also, that expectation of x bar x bar prime is going to be equal to sigma by
n; this minus mu mu prime. So, this is one that we have going to use and this is one we
are going to use.
So, we will use this 1 and 2 or rather using 2 and 3 in 1. 1 is what; this particular
expression. So, what will be having is this that expectation of n minus 1 s n minus 1; that
is summation j equal to 1 to n, then expectation of this particular quantity. So, that what
will be having here is sigma mu mu transpose; this minus we have this as n times
expectation of this quantity. So, that sigma this is sigma by n; this minus mu mu
transpose right. So, if one simplifies, this will cancel out.
So, will have one sigma from here; you will have n minus 1 sigma from here and then,
this mu mu transpose term. Just a minute this is plus sign out here. Because if you take
this expectation of x bar x bar prime to this side, then we will have sigma by n plus mu
mu prime. So, this is the plus sign out here not a minus sign; then what will be having
here is just n minus 1 times sigma. This would imply that, expectation of s n minus 1 is
just going to be equal to sigma. So, this will imply that, s n minus 1 is an unbiased
estimator of the population variance covariance matrix that is sigma.
(Refer Slide Time: 55:27)
Now, if on the other hand, we take s n is not going to be an unbiased estimator of sigma.
Why? Simply, because this n times s n minus s n is n minus 1 times s n minus 1 and what
we have proved is that, expectation of s n minus 1 that multiplied by n minus 1. So, that
would also be equal to expectation of n times s n; that is equal to sigma. This would
imply that, this was actually equal to n minus 1 times this sigma. This is n minus 1 times
sigma. So, this would imply that, expectation of s n is going to be equal to n minus 1 by
n times sigma and which is not equal to sigma. So, this proves that, though s n minus 1
with the sample covariance matrix with a divisor n minus 1 is an unbiased estimator of
sigma. However, this s n is not an unbiased estimator of sigma. However, as n goes to
infinity, this will be an unbiased system estimator and hence, this s n will be an unbiased
estimator in the limit.
```