# Estimation (ISS (Statistical Services) Statistics Paper II (Old Subjective Pattern)): Questions 9 - 13 of 14

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## Question number: 9

» Estimation » Optimal Properties » Sufficient Estimator

Appeared in Year: 2015

### Describe in Detail

Obtain the sufficient statistics for the following distribution.

(i)

(ii)

### Explanation

By using factorization theorem, the condition is that

where h (x) is free from θ and g (. ) depends on __X__ only through T.

(i)

The joint pdf of random sample is

Let T= . By factorization theorem

h (x) =1,

So, h

## Question number: 10

» 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

## Question number: 11

» 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.

## Question number: 12

» Estimation » Optimal Properties » Rao-Blackwell Theorem

Appeared in Year: 2015

### Describe in Detail

Explain how the Rao-Blackwell theorem helps one to find a uniformly minimum variance unbiased estimator (UMVUE) of an unknown parameter. What is the relevance of the Lehman-Scheffe theorem in this scenario? If X _{1}, X _{2}, …, X _{n} are Bin (1, p) variates, find the UMVUE of p.

### Explanation

Let U be an unbiased estimator of θ and T be a sufficient statistic for θ, then E (U|T) is free from θ and it is an estimation. Using the identity , we have

True for all θ. Then is an unbiased for θ.

Next, we find

## Question number: 13

» Estimation » Optimal Properties » Minimum Variance Bound Estimators

Appeared in Year: 2015

### Describe in Detail

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

The Fisher information is

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

The Lower bound is