# Estimation-Optimal Properties (ISS (Statistical Services) Statistics Paper II (Old Subjective Pattern)): Questions 7 - 8 of 8

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## Question number: 7

» 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

## Question number: 8

» Estimation » Optimal Properties » Complete Statistics

Appeared in Year: 2015

### Describe in Detail

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 completeness is

Assume is free from and

This equation is true iff

This is also true if and only if this is a polynomial of n ^{th}