# 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

### 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 that the distributions corresponding to different values of the parame

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## Question number: 2

» 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 the method of moments estimator for λ, we solve λ as a fu

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## Question number: 3

» 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 is

Here the sample are independ

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## Question number: 4

» 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,

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## Question number: 5

» Estimation » Optimal Properties » Sufficient Estimator

Appeared in Year: 2014

### Describe in Detail

Show that 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

For i = 1,2, …, n

Now

So, the pmf of Z is

e conditional distribution of (X _{1 }, X _{2 }, …, X _{n }) given Z = k is given by

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## Question number: 6

» 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 parameter. So, according to maximum likelihood principle log

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## Question number: 7

» 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

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## Question number: 8

» Estimation » Optimal Properties » Confidence Interval Estimation

Appeared in Year: 2015

### 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 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 side that is

For first inequality we get

From secon

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