# Multivariate Analysis (ISS (Statistical Services) Statistics Paper II (Old Subjective Pattern)): Questions 1 - 6 of 8

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## Question number: 1

» Multivariate Analysis » Estimation » Covariance Matrix

Appeared in Year: 2014

### Describe in Detail

Let ,

Find ρ such that X _{1 } +X _{2 } +X _{3 } and X _{1 } -X _{2 } -X _{3 } are independent.

### Explanation

Let

Given that Y and Z are independent, So

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## Question number: 2

» Multivariate Analysis » Distribution of Hotelling T2 Statistic » Use for Testing

Appeared in Year: 2015

### Describe in Detail

Show that T ^{2} statistic is invariant under changes in the unit of measurements for a p×1 random vector X of the form __Y__ =C __X __ + __d __ where C is a p×p nonsingular matrix, __d__ is a p×1 vector.

### Explanation

T ^{2} statistic can be computed from __X__. Here we have to compute it form __Y__ and both will be same. Here

Let , then and

So, T ^{2} -statistic comes into existence for testing hypothesis of the form

Since is known vector,

If we have a setup

and

T ^{2} computed from __X__ is

This is the T ^{2} statistic which is computed fro…

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## Question number: 3

» Multivariate Analysis » Multivariate Normal Distribution » Mutliple Correlation Coefficient

Appeared in Year: 2015

### Describe in Detail

Let X= (X _{1}, X _{2}, X _{3}) ’ be distributed as N _{3} (µ, ∑) where µ’= (2, -3, 1) and

(i) Find the distribution of 3X _{1} -2X _{2} +X _{3}.

(ii) Find a 2 × 1 vector a such that X _{2} and are independent.

### Explanation

(i) the distribution of 3X _{1} -2X _{2} +X _{3} is

The mean is

The variance-covariance matrix is

So, the Y has a normal distribution N (13, 9).

(ii) X _{2} and are independent if

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## Question number: 4

» Multivariate Analysis » Multivariate Normal Distribution » Partial Correlation Coefficient

Appeared in Year: 2015

### Describe in Detail

Let __X__ = (X _{1}, X _{2}, X _{3}) ’ be distributed as N _{3} (__µ__, ∑) where __ µ__ = (10, -7, 2) ’ and

Find the partial correlation between X _{1} and X _{2} given X _{3}.

### Explanation

The partial correlation between X _{1} and X _{2} given X _{3} is defined as

where these terms are defined by this equation

Given that

So, the value of equation is

The partial correlation is

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## Question number: 5

» Multivariate Analysis » Estimation » Covariance Matrix

Appeared in Year: 2015

### Describe in Detail

Find the maximum likelihood estimator of the 2×1 mean vector µ and the 2×2 covariance matrix ∑ based on the random sample.

from a bivariate normal distribution.

### Explanation

We known that the maximum likelihood estimator of multivariate normal distribution is

Assume

So,

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## Question number: 6

» Multivariate Analysis » Multivariate Normal Distribution » Mutliple Correlation Coefficient

Appeared in Year: 2015

### Describe in Detail

Let __X__ _{1}, __X__ _{2}, …, __X__ _{n} be a random sample from an population with , a positive definite matrix. Derive 100 (1-α) % simultaneous confidence interval for for all .

### Explanation

Let __X__ _{1}, __X__ _{2}, …, __X__ _{n} be a random sample from an and assume the linear combination of the random sample is

From, the theorem of linear combinations of multivariate normal distribution is that every linear combination of X follows an univariate normal distribution.

The sample mean and variance of the observed values are

where and …

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