# ISS (Statistical Services) Statistics Paper II (Old Subjective Pattern): Questions 26 - 30 of 39

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## Question number: 26

» Statistical Quality Control » Sequential Sampling Plans

Appeared in Year: 2015

### Describe in Detail

To test the hypothesis against for the distribution

Develop the sequential probability ratio test.

### Explanation

Let x _{1}, x _{2}, …, x _{n} are independently and identically distributed with distribution is

Let type I and type II errors are defined as

P (rejecting the lot when θ=θ _{0}) =α

P (accpeting the lot when θ=θ _{1}) =β

Under H _{0} and H _{1} the pmf is

Thus the likelihood ratio is given as

The decision about the acceptance, rejection or continuanc…

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## Question number: 27

» Multivariate Analysis » Multivariate Normal Distribution » Partial Correlation Coefficient

Appeared in Year: 2015

### 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 partial correlation is

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## Question number: 28

» Multivariate Analysis » Estimation » Covariance Matrix

Appeared in Year: 2015

### 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: 29

» Estimation » Estimation Methods » Methods of Moments

Appeared in Year: 2015

### Describe in Detail

The observations

3.9, 2.4, 1.8, 3.5, 2.4, 2.7, 2.5, 2.1, 3.0, 3.6, 3.6, 1.8, 2.0, 4.0, 1.5

are a random sample from a rectangular population with pdf

Estimate the parameters by the method of moments.

### Explanation

Let X _{1}, X _{2}, …, X _{n} be a random sample from a rectangular population. We known that

The estimating equations are

The above equation for solving the parameter, we get

By method of moments, we see the observations

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## Question number: 30

» Estimation » Optimal Properties » Cramer-Raoinequality

Appeared in Year: 2015

### Describe in Detail

Stating the regularity conditions, give the Cramer-Rao lower bound for the variance of an unbiased estimator of a parameter. Give an example, each, of a situation where the regularity conditions (i) does not hold (ii) holds

### Explanation

Suppose that X _{1}, …, X _{n} is a sample from a distribution with joint pdf f _{n} (__x__, θ) and T (__X__) is an estimator. Also assume that f _{n} () satisfies the conditions that allow

(i) Interchange of differentiation and integration operations i. e.

(ii) and

(iii)

Then, satisfies the inequality

The example where the regularity condition doe…

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