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

Essay Question▾

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

… (162 more words) …

Question number: 2

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

Appeared in Year: 2015

Essay Question▾

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

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Question number: 3

» Multivariate Analysis » Multivariate Normal Distribution » Mutliple Correlation Coefficient

Appeared in Year: 2015

Essay Question▾

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

… (269 more words) …

Question number: 4

» Multivariate Analysis » Multivariate Normal Distribution » Partial Correlation Coefficient

Appeared in Year: 2015

Essay Question▾

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

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Question number: 5

» Multivariate Analysis » Estimation » Covariance Matrix

Appeared in Year: 2015

Essay Question▾

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

Essay Question▾

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 sa

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