Exam Details
| Subject | statistics | |
| Paper | paper 2 | |
| Exam / Course | indian economic service and indian statistical service examination (ies/iss) | |
| Department | ||
| Organization | union public service commission | |
| Position | ||
| Exam Date | 2015 | |
| City, State | central government, |
Question Paper
Let X be a random variable having probability density function
0 otherwise.
Find an unbiased estimator of B^2 based on a sample of size 1.
Consider the matrix
1 1 1 1 1 1 1
1 1 1 1 0 0 0 0
0 0 0 0 1 1 1 1
1 2 3 4 1 2 3 4
Find a basis of the column space and obtain a g-inverse of X'X.
Consider the Gauss-Markov model
2B1 B2
B1 -B2
B1 aB2,
with usual assumptions. Determine a so that the best linear unbiased estimators (BLUEs) of B1 and B2 are uncorrelated.
Let Y1, Y2, ... Yn be a random sample from 0"2) where M are both unknown. Obtain a confidence interval for M with confidence coefficient
Show that, if T is a sufficient statistic for any solution of the likelihood equation will be the function of this statistic. State true or false. A minimal sufficient statistic is always sufficient.
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
does not hold
holds.
Show, in the context of Gauss Markov model XB, that the projection of an unbiased estimator of an estimable linear parametric function is also an unbiased estimator of A'B.
Consider an unbalanced one way fixed effects model
yij M +ai i 1...k, j 1...ni;
where E(eij) 0 for all j. Obtain the constraint needed for estimability of the parameters. Discuss the analysis of variance in a situation where the model is applied.
Consider the linear model y 0,disp V known and positive definite and 2 is unknown. Show that the estimator is the BLUE for iff is uncorrelated with all unbiased estimators of zero.
Let X1, X2 ... Xn be a random sample from the probability distribution with density
fx 0 x 00
,otherwise
where 0 B 00, Show that X 1/n E Xi 1 to is a minimum variance bound estimator and has variance 0'2.
Write Tukey's non-additive model for a set of two-way classified data with one observation per cell. Suggest an estimator of the nonadditivity parameter and find the distribution under additivity. Derive a test for additivity for such a model.
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, 1·5 are a random sample from a rectangular population with pdf
o otherwise
Estimate the parameters by the method of moments.
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? Xl, X2 ..., are Bin variates, find the UMVUE of p.
For a completely balanced two-way random effects model, find unbiased estimators of different variance components. Explain how to obtain the variance of an unbiased estimator of any one variance component corresponding to a main effect. Find an approximate confidence Interval to any arbitrary linear function of variance components.
Consider the following cross classified model without replication.
Yij M ai Bj eij, E(eij) 2. j 3.
Where Y (Yll Y12 Y13 Y21 Y22
a1, a2,B1,B2,B3) and
1 1 0 1 0 0
1 1 0 0 1 0
1 1 1 0 0 1
0 1 0 0 1 0
1 0 1 0 1 0
1 0 1 0 0 1
Write down the normal equations and find all solutions. Show that a1 -a2 and B1 -2B2 B3 are estimable and give their least squares estimators.
Obtain the sufficient statistics for the following distributions:
1/e x 00; O.
x 1,2...;
0<8<1.
Let Xl, X2 ... Xn be a random sample from the binomial distribution with probability mass function.
f O^x X 1;0 B<1
0 otherWise
Examine whether the statistic T =Ln is complete for this distribution.
Let X2, X3) be distributed as where and
1 1 1
1 3 2
1 2 2
Find the distribution of 3X1-2X2+X3
Find a 2x1 vector a such that X2 and X2) are independent
Find the maximum likelihood estimators of the 2x1 mean vector M and the 2x2 covariance matrix E based on the random sample
4 5 4
6 4 7 7
from a bivariate normal population
Explain the notion of unbiasedness with regard to a test of a hypothesis. Examine the validity of the statement. A Most Powerful test is invariably unbiased.
To test the hypothesis Ho B eo against e1 for the distribution
X
0 B 1
0 otherwise,
develop the sequential probability ratio test.
Distinguish between the single sampling plan and double sampling plan. Discuss how the O.C curves can be used for comparing two sampling plans.
Let X2, be distributed as where M and
2 3
5
3 5
Find the partial correlation between Xl and X2 given X3.
Show that T^2 statistic is invariant under changes in the units of measurements for a p x 1 random vector X of the form Cx d where C is a p x p nonsingular matrix, is a p x 1 vector.
Distinguish between control chart for variables and control chart for attributes. Give an example for each. When will you say that a process is in control? Suppose that all the points in a control chart falls above the central line. What will be your conclusion
Derive the likelihood ratio test for comparing the means of k independent homoscedastic normal populations.
Let X1,X2,..., Xn be a random sample from an Np(jj, population with a positive definite matrix. Derive simultaneous confidence intervals for l'J.L for alliE
Samples of size 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 LXj 662·50 2S and 9·00. Assume that the quality1=1 characteristic is normally distributed.
Find the control limits for the X and R charts.
Assume that both charts exhibit controL if specifications are 26·40 ± estimate the fraction nonconforming. Express your answer in terms of CDF of random variable.
[For n A2 0·577, A 1.342, A3 1.427, D1 D2 4.918, D3 D4 2·115 and d2 2.326]
Find a Most Powerful test for. testing the simple hypothesis Ho: 02 against the simple alternative H1 based on n random observations from where j.1. is known. Show that this MP test is UMP (uniformly most powerful).
Let Ai be distributed as Wishart 2 and A1,A2 be independent. Show that Al A2 is distributed as Wml+ m2(A1+ A2|E).
If A is distributed as Wm(A then CA is distributed as where C is a nonsingular matrix of order m.
Explain the terms average outgoing quality and average total inspection (ATI).
Let N 10,000, n 89 and C 2. Determine Pa the probability of acceptance and use it to determine AOQ and ATI.
For a sequential probability ratio test of strength fJ) and stopping bounds A and B.(B show that
A and B
Let
3 1 1
1 3 1
1 1 5
Determine
the principal components y1,y2 and y3.
the proportion of variance explained by each one of them.
correlation between the first principal component yl and the third original random variable.
What is meant by acceptance sampling by attributes? Outlining the criteria for the goodness of sampling inspection plan, give briefly the steps in the Dodge inspection plan.
0 otherwise.
Find an unbiased estimator of B^2 based on a sample of size 1.
Consider the matrix
1 1 1 1 1 1 1
1 1 1 1 0 0 0 0
0 0 0 0 1 1 1 1
1 2 3 4 1 2 3 4
Find a basis of the column space and obtain a g-inverse of X'X.
Consider the Gauss-Markov model
2B1 B2
B1 -B2
B1 aB2,
with usual assumptions. Determine a so that the best linear unbiased estimators (BLUEs) of B1 and B2 are uncorrelated.
Let Y1, Y2, ... Yn be a random sample from 0"2) where M are both unknown. Obtain a confidence interval for M with confidence coefficient
Show that, if T is a sufficient statistic for any solution of the likelihood equation will be the function of this statistic. State true or false. A minimal sufficient statistic is always sufficient.
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
does not hold
holds.
Show, in the context of Gauss Markov model XB, that the projection of an unbiased estimator of an estimable linear parametric function is also an unbiased estimator of A'B.
Consider an unbalanced one way fixed effects model
yij M +ai i 1...k, j 1...ni;
where E(eij) 0 for all j. Obtain the constraint needed for estimability of the parameters. Discuss the analysis of variance in a situation where the model is applied.
Consider the linear model y 0,disp V known and positive definite and 2 is unknown. Show that the estimator is the BLUE for iff is uncorrelated with all unbiased estimators of zero.
Let X1, X2 ... Xn be a random sample from the probability distribution with density
fx 0 x 00
,otherwise
where 0 B 00, Show that X 1/n E Xi 1 to is a minimum variance bound estimator and has variance 0'2.
Write Tukey's non-additive model for a set of two-way classified data with one observation per cell. Suggest an estimator of the nonadditivity parameter and find the distribution under additivity. Derive a test for additivity for such a model.
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, 1·5 are a random sample from a rectangular population with pdf
o otherwise
Estimate the parameters by the method of moments.
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? Xl, X2 ..., are Bin variates, find the UMVUE of p.
For a completely balanced two-way random effects model, find unbiased estimators of different variance components. Explain how to obtain the variance of an unbiased estimator of any one variance component corresponding to a main effect. Find an approximate confidence Interval to any arbitrary linear function of variance components.
Consider the following cross classified model without replication.
Yij M ai Bj eij, E(eij) 2. j 3.
Where Y (Yll Y12 Y13 Y21 Y22
a1, a2,B1,B2,B3) and
1 1 0 1 0 0
1 1 0 0 1 0
1 1 1 0 0 1
0 1 0 0 1 0
1 0 1 0 1 0
1 0 1 0 0 1
Write down the normal equations and find all solutions. Show that a1 -a2 and B1 -2B2 B3 are estimable and give their least squares estimators.
Obtain the sufficient statistics for the following distributions:
1/e x 00; O.
x 1,2...;
0<8<1.
Let Xl, X2 ... Xn be a random sample from the binomial distribution with probability mass function.
f O^x X 1;0 B<1
0 otherWise
Examine whether the statistic T =Ln is complete for this distribution.
Let X2, X3) be distributed as where and
1 1 1
1 3 2
1 2 2
Find the distribution of 3X1-2X2+X3
Find a 2x1 vector a such that X2 and X2) are independent
Find the maximum likelihood estimators of the 2x1 mean vector M and the 2x2 covariance matrix E based on the random sample
4 5 4
6 4 7 7
from a bivariate normal population
Explain the notion of unbiasedness with regard to a test of a hypothesis. Examine the validity of the statement. A Most Powerful test is invariably unbiased.
To test the hypothesis Ho B eo against e1 for the distribution
X
0 B 1
0 otherwise,
develop the sequential probability ratio test.
Distinguish between the single sampling plan and double sampling plan. Discuss how the O.C curves can be used for comparing two sampling plans.
Let X2, be distributed as where M and
2 3
5
3 5
Find the partial correlation between Xl and X2 given X3.
Show that T^2 statistic is invariant under changes in the units of measurements for a p x 1 random vector X of the form Cx d where C is a p x p nonsingular matrix, is a p x 1 vector.
Distinguish between control chart for variables and control chart for attributes. Give an example for each. When will you say that a process is in control? Suppose that all the points in a control chart falls above the central line. What will be your conclusion
Derive the likelihood ratio test for comparing the means of k independent homoscedastic normal populations.
Let X1,X2,..., Xn be a random sample from an Np(jj, population with a positive definite matrix. Derive simultaneous confidence intervals for l'J.L for alliE
Samples of size 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 LXj 662·50 2S and 9·00. Assume that the quality1=1 characteristic is normally distributed.
Find the control limits for the X and R charts.
Assume that both charts exhibit controL if specifications are 26·40 ± estimate the fraction nonconforming. Express your answer in terms of CDF of random variable.
[For n A2 0·577, A 1.342, A3 1.427, D1 D2 4.918, D3 D4 2·115 and d2 2.326]
Find a Most Powerful test for. testing the simple hypothesis Ho: 02 against the simple alternative H1 based on n random observations from where j.1. is known. Show that this MP test is UMP (uniformly most powerful).
Let Ai be distributed as Wishart 2 and A1,A2 be independent. Show that Al A2 is distributed as Wml+ m2(A1+ A2|E).
If A is distributed as Wm(A then CA is distributed as where C is a nonsingular matrix of order m.
Explain the terms average outgoing quality and average total inspection (ATI).
Let N 10,000, n 89 and C 2. Determine Pa the probability of acceptance and use it to determine AOQ and ATI.
For a sequential probability ratio test of strength fJ) and stopping bounds A and B.(B show that
A and B
Let
3 1 1
1 3 1
1 1 5
Determine
the principal components y1,y2 and y3.
the proportion of variance explained by each one of them.
correlation between the first principal component yl and the third original random variable.
What is meant by acceptance sampling by attributes? Outlining the criteria for the goodness of sampling inspection plan, give briefly the steps in the Dodge inspection plan.