Estimation (ISS 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

Essay Question▾

Describe in Detail

Obtain the sufficient statistics for the following distribution.

(i) Equation

(ii) Equation

Explanation

By using factorization theorem, the condition is that

Equation

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

(i)

Equation

The joint pdf of random sample is

Equation

Equation

Let T= Equation. By factorization theorem

h (x) =1, Equation

So, h… (70 more words) …

Question number: 10

» Estimation » Estimation Methods » Methods of Moments

Appeared in Year: 2015

Essay Question▾

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

Equation

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

Equation

Equation

The estimating equations are

Equation

Equation

The above equation for solving the parameter, we get

Equation

Equation

By method of moments, we see the observations

Equation

Equation<span class="more">… (6 more words) …</span>

Question number: 11

» Estimation » Optimal Properties » Cramer-Raoinequality

Appeared in Year: 2015

Essay Question▾

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.… (126 more words) …

Question number: 12

» Estimation » Optimal Properties » Rao-Blackwell Theorem

Appeared in Year: 2015

Essay Question▾

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 Equation, we have

Equation

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

Next, we find… (125 more words) …

Question number: 13

» 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

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