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

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Question number: 33

» Hypotheses Testing » Unbiased Test

Appeared in Year: 2015

Essay Question▾

Describe in Detail

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 Equation 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.

Equation

where Equation. The power of a… (92 more words) …

Question number: 34

» Estimation » Optimal Properties » Minimum Variance Bound Estimators

Appeared in Year: 2015

Essay Question▾

Describe in Detail

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

Equation

= 0; otherwise

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

Explanation

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

Equation

Equation

Equation

Equation

The Fisher information is

Equation

Equation

Here E (X) =θ. The unbiased estimator of exponential distribution is Equation .

Equation

Equation

The Lower bound is

Equation

Equation

Question number: 35

» Estimation » Optimal Properties » Complete Statistics

Appeared in Year: 2015

Essay Question▾

Describe in Detail

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

Equation

Examine whether the statistic Equation is complete for this distribution.

Explanation

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

Equation

Equation

The definition of completeness is

Equation

Equation

Equation

Assume Equation is free from Equation and Equation

This equation is true iff

Equation

This is also true if and only if this is a polynomial of n th… (25 more words) …

Question number: 36

» 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) …

Question number: 37

» Multivariate Analysis » Principal Components » Correlations

Appeared in Year: 2015

Essay Question▾

Describe in Detail

Let

Equation

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

Equation

Equation

Equation

To solve this equation, we get the eigen values is

Equation

Equation

The corresponding eigen vector for each eigen value is by using normalize the eigen vector by the equation

Equation

Equation

For λ 1 =6,

Equation

Equation<span class="more">… (131 more words) …</span>

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