Estimation (ISS Statistics Paper II (Old Subjective Pattern)): Questions 8 - 12 of 14

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Question number: 8

» Estimation » Optimal Properties » Confidence Interval Estimation

Appeared in Year: 2015

Essay Question▾

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Let y 1, y 2, …, y n be a random sample from N (µ, σ 2) where µ and σ 2 are both unknown. Obtain a confidence interval of µ with confidence coefficient (1-α)

Explanation

When population mean and population standard deviation in not know. If Y is the samplemean and replace σ by its estimate s and t α/2 be the critical value of the student t-test such that have of the area on the left hand side and other half on the… (113 more words) …

Question number: 9

» Estimation » Optimal Properties » Sufficient Estimator

Appeared in Year: 2015

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Obtain the sufficient statistics for the following distribution.

(i) f(x,θ)=1θexθ;0<x<,θ>0

(ii) f(x,θ)=(1θ)xθ;x=0,1,2,.,0<θ<1

Explanation

By using factorization theorem, the condition is that

fn(x1,x2,,xn;θ)=h(x)g(t;θ)

where h (x) is free from θ and g (. ) depends on X only… (268 more words) …

Question number: 10

» Estimation » Estimation Methods » Methods of Moments

Appeared in Year: 2015

Essay Question▾

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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

f(x;a,b)={1ba,axb0,otherwise

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

μ1=a+b2=m1

μ2=(ba)212+(a+b2… (165 more words) …

Question number: 11

» Estimation » Optimal Properties » Cramer-Raoinequality

Appeared in Year: 2015

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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.… (312 more words) …

Question number: 12

» Estimation » Optimal Properties » Rao-Blackwell Theorem

Appeared in Year: 2015

Essay Question▾

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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 Eθ[Eθ(XY)]=EθX, we have

Eθ… (548 more words) …

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