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

Get 1 year subscription: Access detailed explanations (illustrated with images and videos) to 39 questions. Access all new questions we will add tracking exam-pattern and syllabus changes. View Sample Explanation or View Features.

Rs. 200.00 or

Question number: 1

» Multivariate Analysis » Estimation » Covariance Matrix

Appeared in Year: 2014

Essay Question▾

Describe in Detail

Let Equation , Equation

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

Explanation

Let

Equation

Equation

Given that Y and Z are independent, So

Equation

Equation

Equation

Equation

Equation

Equation

Equation

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

Equation

Let Equation, then Equation and Equation

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

Equation

Since Equation is known vector,… (35 more words) …

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

Equation

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

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

Explanation

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

Equation

The mean is

Equation

Equation

The variance-covariance matrix is

Equation

Equation

Equation

Equation

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

(ii) X 2 and Equation are independent if

Equation

Equation

Equation

Equation

Equation

Equation

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

Equation

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

Equation

where these terms are defined by this equation

Equation

Given that

Equation

So, the value of equation is

Equation

Equation

Equation

The partial correlation is

Equation

Equation

Equation

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.

Equation

from a bivariate normal distribution.

Explanation

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

Equation

Assume

Equation

So,

Equation

Equation

Equation

Equation

Equation

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 Equation population with Equation , a positive definite matrix. Derive 100 (1-α) % simultaneous confidence interval for Equation for all Equation .

Explanation

Let X 1, X 2, …, X n be a random sample from an Equation and assume the linear combination of the random sample is

Equation

From, the theorem of linear combinations of multivariate normal distribution is that every linear combination of X follows an univariate normal distribution.… (91 more words) …

f Page
Sign In