ISS (Statistical Services) Statistics Paper II (Old Subjective Pattern): Questions 33 - 37 of 39

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Question number: 33

» Hypotheses Testing » Unbiased Test

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Appeared in Year: 2015

Essay Question▾

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Explain the notion of unbiasedness with regards to a test of a hypothesis. Examine the validity of the statement. A most powerful test is invariably unbiased.

Explanation

A test T of the null hypothesis is said to be an unbiased test if the probability of rejection H 0 when it is false is at least as much as the probability of rejecting H 0 when it is true.

where

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Question number: 34

» Estimation » Optimal Properties » Minimum Variance Bound Estimators

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Appeared in Year: 2015

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Let X 1, X 2, …, X n be a random sample from the probability distribution with density

= 0; otherwise

where 0 < θ < ∞. Show that is a minimum variance bound estimator and has variance .

Explanation

By using Cramer-Rao lower bound we find the minimum variance

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Question number: 35

» Estimation » Optimal Properties » Complete Statistics

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Appeared in Year: 2015

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Let X 1, X 2, … X n be a random sample from the binomial distribution with probability mass function

Examine whether the statistic is complete for this distribution.

Explanation

We know that is sufficient statistic for this distribution. Then,

The definition of completene

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Question number: 36

» Multivariate Analysis » Multivariate Normal Distribution » Mutliple Correlation Coefficient

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Appeared in Year: 2015

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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 distrib

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Question number: 37

» Multivariate Analysis » Principal Components » Correlations

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Appeared in Year: 2015

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Let

Determine

(i) The principal components y 1, y 2 and y 3.

(ii) The proportion of variance explained each one of them.

(iii) Correlation between the first principal components y 1 and the third original random variable.

Explanation

First find the eigen value and eigen vector pairs

To solve this equation, we get the eigen values is

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