Exam Details
| Subject | statistics | |
| Paper | paper 3 | |
| 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
Show that for SRSWOR, the sample proportion is an unbiased estimator of the population proportion of members possessing a certain character. Hence obtain the variance of the estimator.
Explain the principles of following strata and clusters. What are the differences between stratified sampling and cluster sampling
Explain why the variance of the estimate of population mean based on a single systematic sample is not unbiasedly estimable.
If two independent systematic samples of size four each from a population of 36 units have y-values of a characteristic y as
24,26,22,28 and 27, 25, 23, 29
respectively, calculate an unbiased estimate of population mean of y and an unbiased estimate of its variance.
Explain briefly the main principles of design of experiments and their analogy in sample surveys with the corresponding concepts.
Discuss how you would proceed with the analysis if data on one plot is missing in a Latin Square Design.
Suppose an experimenter wants to conduct a 3^3 experiment in blocks of size 9 plots, Write down the treatments to 3 blocks with AB2 as the confounding contrast.
Consider a population U u2 The values of the study variable are i 1(1)4.
A sample of size 2 is drawn from the population with, SRSVOR SRSWR. Calculate V1 Var (y|SRSWOR) and V2 Var(y|SRSWR) where y denotes the sample mean.
A linear systematic sample of 2 is drawn when the units of the population are arranged as follows:
Arrangement 1 u1, u4, u2 ,u3
Arrangement 2 u1, u2, u4, u3
Calculate Var (y|arrangement V3 and Var (y|arrangement v4.
Show that v4 v1 v 2 v 3.
For any sampling design, describe the ratio method of estimating the population total Y of a study variate when auxiliary information on a related variable x is available with a known population total X. Denote this estimator by YR and derive its approximate bias.
Consider two ratio estimators r1 and r2 for a parameter e based on a sample of size n with biases and b2= nb1 Using this information, obtain an unbiased estimator of e and its variance expression.
Consider a population U 5}. For a sampling design, probabilities of samples drawn are 0.2, 0.1, 0.3, 0.2, Pr 0.2. Calculate the first-order inclusion probabilities. Obtain estimates Eliyi/ni of the parametric function L liYi where li's are i=l known real numbers, not all zero.
For a two-stage sampling design, let n villages be selected with probabilities proportional to a given size and with replacement from a population of N villages. From th.e ith selected village consisting of M. 1 households mi households are selected by simple random sampling without replacement. Write down an unbiased estimator N M j i=l j=l of y " LJ LJ Y.. where Y .. IJ IJ is the value of a study variate for the hh of the ith village. Also write down an unbiased estimator of Var
Describe how uniformity trials in experimental designs and pilot studies in sample surveys are conducted, mentioning their uses.
Find the C-matrix for the following design and obtain independent estimable treatment contrasts:
B1 B2 B3 B4
Explain what is meant by a split-plot design. Suppose that factor A has p levels Which are arranged in a Randomized Block Design having r replicates: Let factor B have q levels which are applied to plots of a block after subdividing each plot into q sub plots. Write down the model, clearly explaining the notations and assumptions and present a blank ANOVA table with sources of variation and degrees of freedom.
State the conditions to be satisfied by a Partially Balanced Incomplete Block (PBIB) design. Prove the follo'OJing restrictions of a PBIBD
E nj to
E nj Aj to
What are the advantages of stratified sampling? Consider the aBc cation of sample size n to strata. given by ni oc Nioi,where Ni is the stratum size, cr, is the within stratum. standard deviation of the 1 i th stratum, ...., and o is real. Write down an expression for the Var(Yst) where Yst stands for the unbased estimator of the population mean Y of the study variate y based on Simple Random Sample without replacement in each stratum.
Consider a population U ...un}. Observations of study variate and auxiliary variable(x) are yi, x(ui xi,i ...., N.
Let p. Xi/X where X EXi 1 to N
A sample of size n is drawn with PPSWR. Show that for population total T Eyi ,the estimator Ypps 1/n E yi/pi is an unbiased estimator. Obtain the variance of the estimator of the population total.
Verify whether the following arrangement is a symmetrical BIB design. Consider each set a block:
12,
Is it a connected BIBD
32 plots are arranged in the form of a 4 x 8 rectangle. Give a design for 25 factorial experiment with factors D and E in these plots confounding the effects ABC, CDE, ABDE with the rows and AB, CD, ABeD, BDE, ADE, BCE, ACE with the columns.
List the main components of a time series. Explain the method of link relative for measurement of seasonal fluctuations of a time series.
Discuss the considerations in the choice of
base period and
the formula (method), for constructing price index numbers.
Describe exact and near multicollinearity in a regression model. Discuss, along with justifications the effect of these situations on
sampling variance and
prediction.
Describe the problem of heteroscedicity in linear regression models. Outline any One method for overcoming this problem.
Explain the use of Pareto curve in the study of income distribution.
Describe in detail Indirect Least Squares method for estimating structural parameters. the estimator unbiased and/or consistent?
State the problem of autocorrelation in a general linear model (GLM). Why does this problem arise? Demonstrate its effect on ordinary least squares estimator(OLSE).
What do you mean by forecasting in economic models? Define best linear un biased predictor and obtain the same in a simple linear regression model satisfying all the basic assumptions.
Explain variate-difference method for trend analysis. How is the appropriate order of differencing determined
In the construction of price index numbers, explain the terms 'time reversal test' and 'factor reversal test'. Show that both these tests are satisfied by the Fisher index.
Explain the identification problem in a system of simultaneous equations. State, without proof, the rank and order conditions for identifiability of an equation.
Examine for identifiability of the following model:
Demand Mt Bo B1Yt B2R1 B3Pt U1t
Supply Mt ao +a1Yt +U2t
where M money, Y income,
R rate of interest, P price.
Assume that R and P are predetermined.
Describe the two-stage least squares (2SLS) procedure for structural estimation in c. simultaneous equations model. Show that it coincides with indirect least squares method where the equation is exactly identified.
Discuss how the Engel curve lS constructed on the basis of family budget data. Explain and interpret the Engel law in this context How are variations in household size and composition handled?
Write down the auto correlation function of order K. for an model Xt= E t,where Et is a white noise process. Show that this model can be expressed as a moving average process of infinite order. Check the model for stationarity.
The following information concerns changes in price and consumption (quantity of certain major components of the consumption-basket of the labour class
Year 2000 Year 2014
Sl. No. Item Unit Price Consumption Price Consumption
1 Rice Quintal 500 16 640 20
2 Wheat Quintal 240 12 320 10
3 Cloth Metre 16 50 20 35
Compute price index using
Fisher's method;
Marshall-Edgeworth method.
Also interpret the results.
Discuss the main aspects of the sampling design used by the NSSO for collection of data on household consumer expenditure. If you were to improve upon this design, explain what modifications you would Suggest, with a clear justification.
Define periodogram. State the connection between the periodogram and the autocovariance function of a time series. Show that defined by
exp(-iwt)
is of period 2n, where is the auto covariance function.
Define partial auto correlation function (PACF). Determine this function for the model
xt 0.7 0.3Xt-2 Et
where is a sequence of uncorrelated random variables with mean zero and variance O2E.
Explain the principles of following strata and clusters. What are the differences between stratified sampling and cluster sampling
Explain why the variance of the estimate of population mean based on a single systematic sample is not unbiasedly estimable.
If two independent systematic samples of size four each from a population of 36 units have y-values of a characteristic y as
24,26,22,28 and 27, 25, 23, 29
respectively, calculate an unbiased estimate of population mean of y and an unbiased estimate of its variance.
Explain briefly the main principles of design of experiments and their analogy in sample surveys with the corresponding concepts.
Discuss how you would proceed with the analysis if data on one plot is missing in a Latin Square Design.
Suppose an experimenter wants to conduct a 3^3 experiment in blocks of size 9 plots, Write down the treatments to 3 blocks with AB2 as the confounding contrast.
Consider a population U u2 The values of the study variable are i 1(1)4.
A sample of size 2 is drawn from the population with, SRSVOR SRSWR. Calculate V1 Var (y|SRSWOR) and V2 Var(y|SRSWR) where y denotes the sample mean.
A linear systematic sample of 2 is drawn when the units of the population are arranged as follows:
Arrangement 1 u1, u4, u2 ,u3
Arrangement 2 u1, u2, u4, u3
Calculate Var (y|arrangement V3 and Var (y|arrangement v4.
Show that v4 v1 v 2 v 3.
For any sampling design, describe the ratio method of estimating the population total Y of a study variate when auxiliary information on a related variable x is available with a known population total X. Denote this estimator by YR and derive its approximate bias.
Consider two ratio estimators r1 and r2 for a parameter e based on a sample of size n with biases and b2= nb1 Using this information, obtain an unbiased estimator of e and its variance expression.
Consider a population U 5}. For a sampling design, probabilities of samples drawn are 0.2, 0.1, 0.3, 0.2, Pr 0.2. Calculate the first-order inclusion probabilities. Obtain estimates Eliyi/ni of the parametric function L liYi where li's are i=l known real numbers, not all zero.
For a two-stage sampling design, let n villages be selected with probabilities proportional to a given size and with replacement from a population of N villages. From th.e ith selected village consisting of M. 1 households mi households are selected by simple random sampling without replacement. Write down an unbiased estimator N M j i=l j=l of y " LJ LJ Y.. where Y .. IJ IJ is the value of a study variate for the hh of the ith village. Also write down an unbiased estimator of Var
Describe how uniformity trials in experimental designs and pilot studies in sample surveys are conducted, mentioning their uses.
Find the C-matrix for the following design and obtain independent estimable treatment contrasts:
B1 B2 B3 B4
Explain what is meant by a split-plot design. Suppose that factor A has p levels Which are arranged in a Randomized Block Design having r replicates: Let factor B have q levels which are applied to plots of a block after subdividing each plot into q sub plots. Write down the model, clearly explaining the notations and assumptions and present a blank ANOVA table with sources of variation and degrees of freedom.
State the conditions to be satisfied by a Partially Balanced Incomplete Block (PBIB) design. Prove the follo'OJing restrictions of a PBIBD
E nj to
E nj Aj to
What are the advantages of stratified sampling? Consider the aBc cation of sample size n to strata. given by ni oc Nioi,where Ni is the stratum size, cr, is the within stratum. standard deviation of the 1 i th stratum, ...., and o is real. Write down an expression for the Var(Yst) where Yst stands for the unbased estimator of the population mean Y of the study variate y based on Simple Random Sample without replacement in each stratum.
Consider a population U ...un}. Observations of study variate and auxiliary variable(x) are yi, x(ui xi,i ...., N.
Let p. Xi/X where X EXi 1 to N
A sample of size n is drawn with PPSWR. Show that for population total T Eyi ,the estimator Ypps 1/n E yi/pi is an unbiased estimator. Obtain the variance of the estimator of the population total.
Verify whether the following arrangement is a symmetrical BIB design. Consider each set a block:
12,
Is it a connected BIBD
32 plots are arranged in the form of a 4 x 8 rectangle. Give a design for 25 factorial experiment with factors D and E in these plots confounding the effects ABC, CDE, ABDE with the rows and AB, CD, ABeD, BDE, ADE, BCE, ACE with the columns.
List the main components of a time series. Explain the method of link relative for measurement of seasonal fluctuations of a time series.
Discuss the considerations in the choice of
base period and
the formula (method), for constructing price index numbers.
Describe exact and near multicollinearity in a regression model. Discuss, along with justifications the effect of these situations on
sampling variance and
prediction.
Describe the problem of heteroscedicity in linear regression models. Outline any One method for overcoming this problem.
Explain the use of Pareto curve in the study of income distribution.
Describe in detail Indirect Least Squares method for estimating structural parameters. the estimator unbiased and/or consistent?
State the problem of autocorrelation in a general linear model (GLM). Why does this problem arise? Demonstrate its effect on ordinary least squares estimator(OLSE).
What do you mean by forecasting in economic models? Define best linear un biased predictor and obtain the same in a simple linear regression model satisfying all the basic assumptions.
Explain variate-difference method for trend analysis. How is the appropriate order of differencing determined
In the construction of price index numbers, explain the terms 'time reversal test' and 'factor reversal test'. Show that both these tests are satisfied by the Fisher index.
Explain the identification problem in a system of simultaneous equations. State, without proof, the rank and order conditions for identifiability of an equation.
Examine for identifiability of the following model:
Demand Mt Bo B1Yt B2R1 B3Pt U1t
Supply Mt ao +a1Yt +U2t
where M money, Y income,
R rate of interest, P price.
Assume that R and P are predetermined.
Describe the two-stage least squares (2SLS) procedure for structural estimation in c. simultaneous equations model. Show that it coincides with indirect least squares method where the equation is exactly identified.
Discuss how the Engel curve lS constructed on the basis of family budget data. Explain and interpret the Engel law in this context How are variations in household size and composition handled?
Write down the auto correlation function of order K. for an model Xt= E t,where Et is a white noise process. Show that this model can be expressed as a moving average process of infinite order. Check the model for stationarity.
The following information concerns changes in price and consumption (quantity of certain major components of the consumption-basket of the labour class
Year 2000 Year 2014
Sl. No. Item Unit Price Consumption Price Consumption
1 Rice Quintal 500 16 640 20
2 Wheat Quintal 240 12 320 10
3 Cloth Metre 16 50 20 35
Compute price index using
Fisher's method;
Marshall-Edgeworth method.
Also interpret the results.
Discuss the main aspects of the sampling design used by the NSSO for collection of data on household consumer expenditure. If you were to improve upon this design, explain what modifications you would Suggest, with a clear justification.
Define periodogram. State the connection between the periodogram and the autocovariance function of a time series. Show that defined by
exp(-iwt)
is of period 2n, where is the auto covariance function.
Define partial auto correlation function (PACF). Determine this function for the model
xt 0.7 0.3Xt-2 Et
where is a sequence of uncorrelated random variables with mean zero and variance O2E.