Estimation (ISS (Statistical Services) Statistics Paper II (Old Subjective Pattern)): Questions 1 - 8 of 14

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Question number: 1

» Estimation » Optimal Properties » Complete Statistics

Appeared in Year: 2014

Essay Question▾

Describe in Detail

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

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Question number: 2

» Estimation » Estimation Methods » Methods of Moments

Appeared in Year: 2014

Essay Question▾

Describe in Detail

For the Pareto distribution with pdf

Equation

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

Equation

The mean and variance are respectively

Equation

In this we have only one parameter λ. Thus, we will only need to determine the first moment

Equation

To find

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Question number: 3

» Estimation » Estimation Methods » Methods of Moments

Appeared in Year: 2014

Essay Question▾

Describe in Detail

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

Equation

The estimating equation is

Equation

The above equation is solving for the parameter, we get the estimator by using method of moments

Equation

The variance of this estimator

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Question number: 4

» Estimation » Estimation Methods » Maximum Likelihood

Appeared in Year: 2014

Essay Question▾

Describe in Detail

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

Equation

The likelihood function is

Equation

Equation

The log likelihood function is

Equation

Differentiate this with respect to θ and equating to zero,

Equation

Equation

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Question number: 5

» Estimation » Optimal Properties » Sufficient Estimator

Appeared in Year: 2014

Essay Question▾

Describe in Detail

Show that Equation 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

Equation

For i = 1, 2, …, n

Now

Equation

So, the pmf of Z is

Equation e conditional distribution of (X 1 , X 2 , …, X

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Question number: 6

» Estimation » Estimation Methods » Maximum Likelihood

Appeared in Year: 2014

Essay Question▾

Describe in Detail

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

Equation

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

Equation

Equation

The log-likelihood function is

Equation

We do not differentiable log L with respect to α because this is free from

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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.

Equation

The likelihood function is

Equation

The log-likelihood function is

Equation

Differentiable with respect to θ, equating to zero

Equation

Equation

Question number: 8

» Estimation » Optimal Properties » Confidence Interval Estimation

Appeared in Year: 2015

Essay Question▾

Describe in Detail

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 Equation 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 right

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