Estimation-Estimation Methods (ISS (Statistical Services) Statistics Paper II (Old Subjective Pattern)): Questions 1 - 6 of 6

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Question number: 1

» 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

… (109 more words) …

Question number: 2

» 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

… (19 more words) …

Question number: 3

» 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

… (17 more words) …

Question number: 4

» 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

… (65 more words) …

Question number: 5

» 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: 6

» Estimation » Estimation Methods » Methods of Moments

Appeared in Year: 2015

Essay Question▾

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

Equation

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

Equation

Equation

The estimating equations are

Equation

Equation

The above equation for solving the parameter, we get

Equation

Equation

By method of moments, we see the observations

Equation

Equation

… (6 more words) …

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