Estimation-Optimal Properties (ISS 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

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: 8

» 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) …

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