ISS Statistics Paper II (Old Subjective Pattern): Questions 29 - 32 of 39

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Question number: 29

» 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

f(x;a,b)={1ba,axb0,otherwise

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

μ1=a+b2=m1

μ2=(ba)212+(a+b2… (165 more words) …

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.… (312 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 Eθ[Eθ(XY)]=EθX, we have

Eθ… (548 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 X and R are computed for each sample. After 25 samples have been analyzed, we have i=125xj=662.50 and i=125Rj=9.00 . Assume that the quality characteristic is normally distributed.

(i) Find the control limit for the X 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 X=26.5 and R=0.36

(i) For X chart, the control limits are

U.C.LX=X+A3R

=26.5+1.427×0.36

=27.0137

C.LX=X… (152 more words) …

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