Estimation-Optimal Properties (ISS Statistics Paper II (Old Subjective Pattern)): Questions 1 - 6 of 8

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Question number: 1

» Estimation » Optimal Properties » Complete Statistics

Appeared in Year: 2014

Essay Question▾

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Define completeness. Verify whether Bin (1, p) is complete.

Explanation

Completeness: It is a property of a statistic in relation to a model for a set of observed data. In essence, it is a condition which ensures that the parameters of the probability distribution representing the model can all be estimated on the basis of the statistic: it ensures… (541 more words) …

Question number: 2

» Estimation » Optimal Properties » Sufficient Estimator

Appeared in Year: 2014

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Show that Z=16(X1+2X2+3X3) is not a sufficient estimator of the Bernoulli parameter θ.

Explanation

Let X i is a ith random variable follows Bernoulli distribution with parameter θ. Then, the random variable is defined as

Xi=(1,withprobabilityθ0,withp… (195 more words) …

Question number: 3

» 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: 4

» Estimation » Optimal Properties » Sufficient Estimator

Appeared in Year: 2015

Essay Question▾

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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: 5

» Estimation » Optimal Properties » Cramer-Raoinequality

Appeared in Year: 2015

Essay Question▾

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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: 6

» 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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