Multivariate Analysis (ISS 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▾

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Let X~N3(0,Σ) , Σ=(1ρ0ρ1ρ0ρ1)

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

Explanation

Let

Y=X1+X2+X3=(111)(X1X2X3)=AX

Z=X1X2X3=(111)(X1X… (101 more words) …

Question number: 2

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

Appeared in Year: 2015

Essay Question▾

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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

X_CX_+d_

Let X_~Nm(μ_,Σ), then μ_C… (333 more words) …

Question number: 3

» Multivariate Analysis » Multivariate Normal Distribution » Mutliple Correlation Coefficient

Appeared in Year: 2015

Essay Question▾

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

Σ=(111132122)

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

(ii) Find a 2 × 1 vector a such that X 2 and X2a(X1X2) are independent.

Explanation

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

Y=3X12X2+X3=(321)(X1X2X3)=AX_

The mean is

E(Y)=… (302 more words) …

Question number: 4

» Multivariate Analysis » Multivariate Normal Distribution » Partial Correlation Coefficient

Appeared in Year: 2015

Essay Question▾

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Let X = (X 1, X 2, X 3) ’ be distributed as N 3 (µ, ∑) where µ = (10, -7, 2) ’ and

Σ=(213153335)

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

rX1X2.X3=sX1X2.X3sX1X1.X3sX2X2.X3

where these… (175 more words) …

Question number: 5

» Multivariate Analysis » Estimation » Covariance Matrix

Appeared in Year: 2015

Essay Question▾

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

X=[34546477]

from a bivariate normal distribution.

Explanation

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

μ=X_andΣ=i=1n(X_iX_)(X_iX_)n

Assume

X_… (206 more words) …

Question number: 6

» Multivariate Analysis » Multivariate Normal Distribution » Mutliple Correlation Coefficient

Appeared in Year: 2015

Essay Question▾

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Let X 1, X 2, …, X n be a random sample from an NP(μ_,Σ) population with Σ , a positive definite matrix. Derive 100 (1-α) % simultaneous confidence interval for l_μ_ for all l_ϵIRP{0_} .

Explanation

Let X 1, X 2, …, X n be a random sample from an NP(μ_,Σ) and assume the linear combination of the random sample is

Y_=l1X_1+l2X_2+… (264 more words) …

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