# ISS (Statistical Services) Statistics Paper II (Old Subjective Pattern): Questions 1 - 7 of 39

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## Question number: 1

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Appeared in Year: 2015

Essay Question▾

### Describe in Detail

Distinguish between the single sampling plan and double sampling plan. Discuss how the O. C curves can be used for comparing two sampling plans.

### Explanation

A single sampling plan in which a decision about the acceptance or rejection of a lot is based on one sample that has been inspected where double sampling plan when a decision about the acceptance or rejection of a lot has not been reached after single sample inspection from a submitted lot, a decision will always be reached when the second sample

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## Question number: 2

» Hypotheses Testing » Likelihood Ratio Test » ASN Function

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Appeared in Year: 2015

Essay Question▾

### Describe in Detail

Derive the likelihood ratio test for comparing the means of k independent homoscedastic normal populations.

### Explanation

Given that there are k independent homoscedastic normal populations that is the variance is same i. e. ; i = 1,2, …, k. We have to test

In the X population the sample is = { (x i1, x i2, …, x ini)

The parameter space is

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## Question number: 3

» Estimation » Optimal Properties » Complete Statistics

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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 that the distributions corresponding to different values of the paramet

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## Question number: 4

» Estimation » Estimation Methods » Methods of Moments

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Appeared in Year: 2014

Essay Question▾

### 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

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## Question number: 5

» Estimation » Estimation Methods » Methods of Moments

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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

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

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## Question number: 6

» Linear Models » Theory of Linear Estimation » Gauss-Markoff Setup

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Appeared in Year: 2014

Essay Question▾

### Describe in Detail

Define estimability of a linear parametric function in a Gauss Markov model. State and prove a necessary and sufficient condition for estimability.

### Explanation

Estimability : The linear parametric function c’β is an estimable function if there exists a vector

a R n such that

If X is of full column rank then all linear combinations of β are estimable, since is unique, that is

Suppose we are dealing with the model of estimability. Then a necessary and

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## Question number: 7

» Estimation » Estimation Methods » Maximum Likelihood

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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

The likelihood function is

The log likelihood function is

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