ISS (Statistical Services) Statistics Paper II (Old Subjective Pattern): Questions 30 - 32 of 39

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Question number: 30

» Estimation » Optimal Properties » Cramer-Raoinequality

Appeared in Year: 2015

Essay Question▾

Describe in Detail

Stating the regularity conditions, give the Cramer-Rao lower bound for the variance of an unbiased estimator of a parameter. Give an example, each, of a situation where the regularity conditions (i) does not hold (ii) holds

Explanation

Suppose that X 1, …, X n is a sample from a distribution with joint pdf f n (x, θ) and T (X) is an estimator. Also assume that f n () satisfies the conditions that allow

(i) Interchange of differentiation and integration operations i.

… (126 more words) …

Question number: 31

» Estimation » Optimal Properties » Rao-Blackwell Theorem

Appeared in Year: 2015

Essay Question▾

Describe in Detail

Explain how the Rao-Blackwell theorem helps one to find a uniformly minimum variance unbiased estimator (UMVUE) of an unknown parameter. What is the relevance of the Lehman-Scheffe theorem in this scenario? If X 1, X 2, …, X n are Bin (1, p) variates, find the UMVUE of p.

Explanation

Let U be an unbiased estimator of θ and T be a sufficient statistic for θ, then E (U|T) is free from θ and it is an estimation. Using the identity Equation , we have

Equation

True for all θ. Then Equation is an unbiased for θ.

Next, we find

… (125 more words) …

Question number: 32

» Statistical Quality Control » Control Charts » Variable

Appeared in Year: 2015

Essay Question▾

Describe in Detail

Sample of sizes n = 5 are taken from a manufacturing process every hour. A quality characteristic is measured, and Equation and R are computed for each sample. After 25 samples have been analyzed, we have Equation and Equation . Assume that the quality characteristic is normally distributed.

(i) Find the control limit for the Equation and R charts.

(ii) Assume that both chart exhibit control, if specifications are 26.40±0.50, estimate the fraction nonconforming. Express your answers in terms of CDF of N (0, 1) random variable.

[For n = 5, A 2 =0.577, A = 1.342, A 3 =1.427, D 1 =0, D 2 =4.918, D 3 =0, D 4 =2.115 and d 2 =2.326]

Explanation

Given that Equation and Equation

(i) For Equation chart, the control limits are

Equation

Equation

Equation

Equation

Equation

Equation

Equation

For R chart, the control limits are

Equation

Equation

Equation

Equation

Equation

Equation

Equation

(ii) UCL = 26.40 + 0.50 = 26.90

LCL = 26.40 - 0.50 = 25.90

Equation

… (15 more words) …

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