Estimation (ISS Statistics Paper II (Old Subjective Pattern)): Questions 1 - 7 of 14

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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 » Estimation Methods » Methods of Moments

Appeared in Year: 2014

Essay Question▾

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For the Pareto distribution with pdf

f(x;λ)=λxλ+1x1,λ>0

Show that method of moments fails if 0 < λ < 1. State the method of moments estimator when λ > 1. Is it consistent? Justify your answer.

Explanation

Let X 1 , X 2 , …, X n be a simple random sample of Pareto random variables with density

f(x;λ)=λxλ+1x1,λ>0

The mean and variance are respectively

μ=λλ… (197 more words) …

Question number: 3

» Estimation » Estimation Methods » Methods of Moments

Appeared in Year: 2014

Essay Question▾

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X 1, X 2, …, X n be a random sample from U (0, θ). Obtain the moment estimator of θ. Also find its variance.

Explanation

Let X 1, X 2, …, X n be a random sample from U (0, θ). We known that

μ1=θ2=m1

The estimating equation is

m1=1ni=1nXi=θ… (137 more words) …

Question number: 4

» Estimation » Estimation Methods » Maximum Likelihood

Appeared in Year: 2014

Essay Question▾

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X 1 , X 2 , …, X n are i. i. d. random variables from N (θ, 1) where θ is an integer. Obtain MLE of θ.

Explanation

X 1 , X 2 , …, X n are i. i. d. random variables from N (θ, 1). The density function of X is

fX(x)=12πexp(12(xθ)2)

The likelihood… (183 more words) …

Question number: 5

» Estimation » Optimal Properties » Sufficient Estimator

Appeared in Year: 2014

Essay Question▾

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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: 6

» Estimation » Estimation Methods » Maximum Likelihood

Appeared in Year: 2014

Essay Question▾

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x1 , x 2 , …, x n be a random sample from the following distribution

f(x,α)=(e(xα);xα0;otherwise)

Explanation

Let x 1 , x 2 , …, x n be a random sample from f (x, α) and let L (α| x) denote the likelihood function. Then

L(α,x)=i=1nf(xi,α)=i… (144 more words) …

Question number: 7

» Estimation » Estimation Methods » Maximum Likelihood

Appeared in Year: 2014

Essay Question▾

Describe in Detail

Let X be exponentially distributed with parameter θ. Obtain MLE of θ based on a sample of size n, from the above distribution.

Explanation

Let X be exponentially distributed with parameter θ.

f(x;θ)=θeθx;x>0

The likelihood function is

L(θ,x)=i=1nf(xi;θ)=i… (82 more words) …

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