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
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… (70 more words) …
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.… (126 more words) …
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… (125 more words) …
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