UPSC IFS Statistics Syllabus

Paper - I

Probability: 

1. Sample space and events
2. probability measure and probability space
3. random variable as a measurable function
4. distribution function of a random variable
5. discrete and continuous-type random variable
6. probability mass function, probability density function
7. vector-valued random variable
8. marginal and conditional distributions
9. stochastic independence of events and of random variables
10. expectation and moments of a random  variable
11. conditional expectation
12. convergence of a sequence of random variable in distribution in probability
13. pth mean and almost every where
14. criteria and inter-relations
15. Borel-Cantelli lemma
16. Chebyshev’s and Khinchine’s weak laws of large numbers
17. strong law of large numbers and Kolmogorov’s theorems
18. Glivenko-Cantelli theorem
19. probability generating function
20. characteristic function
21. inversion theorem
22. Laplace transform
23. related uniqueness and continuity theorems
24. determination of distribution by its moments
25. Linderberg and Levy forms of central limit theorem
26. standard discrete and continuous probability distributions
27. their interrelations and limiting cases
28. simple properties of finite Markov chains

Statistical Inference: 

1. Consistency
2. Unbiasedness
3. Efficiency
4. Sufficiency
5. minimal sufficiency
6. completeness
7. ancillary statistic
8. factorization theorem
9. exponential family of distribution and its properties
10. uniformly minimum variance unbiased (UMVU) estimation
11. Rao-Blackwell and Lehmann- Scheffe theorems
12. Cramer-Rao inequality for single and several-parameter family of distributions
13. minimum variance bound estimator and its properties
14. modifications and extensions of Cramer-Rao inequality
15. Chapman-Robbins inequality, Bhattacharya’s bounds
16. estimation by methods of moments, maximum likelihood
17. least squares
18. minimum chisquare
19. modified minimum chi-square properties of maximum likelihood
20. other estimators,
21. idea of asymptotic efficiency
22. idea of prior and posterior distributions
23. Bayes
24. estimators
25. Non-randomised and randomised tests
26. critical function
27. MP tests
28. Neyman- Pearson lemma
29. UMP tests, monotone likelihood ratio
30. generalised Neyman- Pearson lemma
31. similar and unbiased tests
32. UMPU tests for single and severalparameter families of distributions
33. likelihood rotates and its large sample properties
34. chi-square goodness of fit test and its asymptotic distribution.
35. Confidence bounds and its relation with tests
36. uniformly most accurate (UMA) and UMA unbiased confidence bounds.
37. Kolmogorov’s test for goodness of fit and its consistency
38. sign test and its optimality
39. Wilcoxon signed-ranks test and its consistency
40. Kolmogorov-Smirnov twosample test
41. run test
42. Wilcoxon-Mann- Whitney test and median test
43. their consistency and asymptotic normality
44. Wald’s SPRT and its properties
45. OC and ASN functions
46. Wald’s fundamental identity
47. sequential estimation

Linear Inference and Multivariate Analysis:

1. Linear statistical models
2. theory of least squares and analysis of variance
3. Gauss- Markoff theory
4. normal equations
5. least squares estimates and their precision
6. test of significance and interval estimates based on least squares theory in one- way,two-way and three-way classified data
7. regression analysis
8. linear regression
9. curvilinear regression and orthogonal polynomials
10. multiple regression
11. multiple and partial correlations
12. regression diagnostics and sensitivity analysis
13. calibration problems
14. estimation of variance and covariance components
15. MINQUE theory
16. multivariate normal distribution
17. Mahalanobis
18. D2 and Hotelling’s T2 statistics and their applications and properties
19. discriminant analysis
20. canonical correlations
21. one-way MANOVA
22. principal component analysis
23. elements of factor analysis

Sampling Theory and Design of Experiments: 

1. An outline of fixed-population and superpopulation approaches,
2. distinctive features of finite population sampling
3. probability sampling designs
4. simple random sampling with and without replacement
5. stratified random sampling
6. systematic sampling and its efficacy for structural populations
7. cluster sampling
8. two-stage and multi-stage sampling
9. ratio and regression
10. methods of estimation involving one or more auxiliary variables
11. two-phase sampling
12. probability proportional to size sampling with and without replacement the Hansen-Hurwitz and the Horvitz-Thompson estimator
13. nonnegative variance estimation with reference to the Horvitz-Thompson estimators non-sampling errors
14. Warner’s randomised response technique for sensitive characteristics.
15. Fixed effects model (two-way classification) random and mixed effects models (two-way classification with equal number of observation per cell), CRD, RBD,LSD and their analysis, incomplete block designs, concepts of orthogonality and balance, BIBD, missing plot technique, factorial designs: 2n, 32 and 33, confounding in factorial experiments, splitplot and simple lattice designs.

Paper – II

I. Industrial Statistics: 

1. Process and product control
2. general theory of control charts
3. different types of control charts for variables and attributes, X, R, s, p, np and c
4. charts
5. cumulative sum chart
6. V-mask
7. single, double, multiple and sequential sampling plans for attributes
8. OC, ASN, AQQ and ATI curves
9. concepts of producer’s and consumer’s risks
10. AQL
11. LTPD and AOQL
12. sampling plans for variables
13. use of Dodge-Roming and Military Standard tables
14. Concepts of reliability
15. maintainability and availability
16. reliability of series and parallel systems and other simple configurations
17. renewal density and renewal function
18. survival models (exponential, Weibull, lognormal, Rayleigh, and bath-tub)
19. different types of redundancy and use of redundancy in reliability improvement
20. Problems in lifetesting censored and truncated experiments for exponential models.

II. Optimization Techniques:

1. Different types of models in Operational Research
2. their construction and general methods of solution
3. simulation and Monte-Carlo methods
4. the structure and formulation of linear programming (LP) problem
5. simple LP model and its graphical solution
6. the simplex procedure
7. the two-phase method and the Mtechnique with artificial variables
8. the duality theory of LP and its economic interpretation, sensitivity analysis, transportation and assignment problems
9. rectangular games
10. two-person zero- sum games
11. method of solution (graphical and algebraic).
12. Replacement of failing or deteriorating items
13. group and individual replacement policies
14. concept of scientific inventory management
15. analytical structure of inventory problems
16. simple models with deterministic and stochastic demand with and without lead time
17. storage models with particular reference to dam type.
18. Homogeneous discrete-time Markov chains
19. transition probability matrix
20. classification of states and ergodic theorems
21. homogeneous continuoustime Markov chains
22. Poisson process
23. elements of queuing theory
24. M/M/1, M/M/K, G/M/1 and M/G/1 queues
25. Solution of statistical problems on computers using well-known statistical software packages like SPSS.

III. Quantitative Economics and Official Statistics:

1. Determination of trend, seasonal and cyclical components,
2. Box-Jenkins method
3. tests for stationery of series
4. ARIMA models and determination of orders of autoregressive and moving average components, forecasting.
5. Commonly used index numbers
6. Laspeyre’s, Paashe’s and Fisher’s ideal Index numbers
7. chain-base index numbers
8. uses and limitations of index number
9. index number of wholesale prices
10. consumer price index number
11. index numbers of agricultural and industrial production
12. test for index numbers like proportionality test
13. timereversal test
14. factor-reversal test
15. circular test and dimensional invariance test
16. General linear model
17. ordinary least squares and generalised least squares methods of estimation
18. problem of multicollinearity
19. consequences and solutions of multi-collinearity
20. autocorrelation and its consequences
21. heteroscedasticity of disturbances and its testing
22. test for independence of disturbances
23. Zellner’s seemingly unrelated regression equation model and its estimation
24. concept ofstructure and model for simultaneousequations
25. problem of identification-rank and order conditions of identifiability
26. twostageleast squares method of estimation
27. Present official statistical system in India relating to population
28. Agriculture
29. industrial production
30. trade and prices
31. methods of collection of official statistics
32. their reliability and limitation and the principal publications containing such statistics various official agencies responsible for data collection and their main functions.

IV. Demography and Psychometry: 

1. Demographic data from census, registration
2. NSS and other surveys, and their limitation and uses
3. Definition
4. construction and uses of vital rates and ratios
5. measures of fertility
6. reproduction rates, morbidity rate
7. standardized death rate
8. complete and abridged life tables
9. construction of life tables from vital statistics and census returns
10. uses of life tables
11. logistic and other population growth curve
12. fitting a logistic curve
13. population projection
14. stable population theory
15. uses of stable population
16. quasi-stable population techniques in estimation of demographic parameters
17. morbidity and its measurement
18. standard classification by cause of death
19. health surveys and use of hospital statistics.
20. Method of standardisation of scales and tests
21. Z-scores, standard scores
22. Tscores, percentile scores
23. intelligence quotient and its measurement and uses
24. validity of test scores and its determination
25. use of factor analysis and path analysis in psychometry