# 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

### Describe in Detail

For the Pareto distribution with pdf

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

The mean and variance are respectively

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

To find

## Question number: 2

» Estimation » Estimation Methods » Methods of Moments

Appeared in Year: 2014

### 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

The estimating equation is

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

The variance of this estimator

## Question number: 3

» Estimation » Estimation Methods » Maximum Likelihood

Appeared in Year: 2014

### 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

The likelihood function is

The log likelihood function is

Differentiate this with respect to θ and equating to zero,

## Question number: 4

» Estimation » Estimation Methods » Maximum Likelihood

Appeared in Year: 2014

### Describe in Detail

x_{1 }, x _{2 }, …, x _{n } be a random sample from the following distribution

### 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

The log-likelihood function is

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

## Question number: 5

» Estimation » Estimation Methods » Maximum Likelihood

Appeared in Year: 2014

### 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.

The likelihood function is

The log-likelihood function is

Differentiable with respect to θ, equating to zero

## Question number: 6

» Estimation » Estimation Methods » Methods of Moments

Appeared in Year: 2015

### 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

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

The estimating equations are

The above equation for solving the parameter, we get

By method of moments, we see the observations