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REVISED SYLLABUS
M.A./M.Sc. STATISTICS
SCHEME OF EXAMINATION
PART I : Examination 2007 and onwards Duration Max. Marks
(hrs.)
Paper I : Analysis 3 75
Paper II : Linear Algebra 3 75
Paper III : Probability Theory 3 75
Paper IV : Statistical Methodology 3 75
Paper V : Stochastic Processes 3 75
Paper VI : Statistical Inference-I 3 75
Paper VII : Survey Sampling 3 75
Paper VIII : Design of Experiments 3 75
Paper IX : Practical-I (Data Structure and Statistical Computing)
comprising the following two sections:
(A) relevant theory for conducting the practicals 2 20
(B) practical applications 2 55*
Paper X : Practical-II: Based on Papers II, IV, VI to VIII 4 75**
Internal Continual Assessment 250
Total 1000
PART II : Examination 2008 and onwards
Paper XI : Statistical Inference-II 3 75
Paper XII : Multivariate Analysis 3 75
Paper XIII : Demography, Statistical Quality 3 75
Control and Reliability
Paper XIV : Econometrics and 3 75
Time Series Analysis
Paper XV : Generalized Linear Models 3 75
Paper XVI : 3 75
Paper XVII : Any three of the following Options: 3 75
Paper XVIII : 3 75
(i) Applied Stochastic Processes
(ii) Order Statistics
(iii) Bayesian Inference
(iv) Advanced Survey Sampling Theory
(v) Advanced Theory of Experimental Designs
(vi) Bio-Statistics
(vii) Operational Research
(viii) Nonparametric Inference
(ix) Actuarial Statistics
(x) Advanced Statistical Computing and Data Mining
Paper XIX: Practical-III: Problems based on Papers XI to XV to
be done on Computer using C language 4 75**
Paper XX : Practical-IV: Problems based on Papers XI to XV to
be done on Computer using Statistical
Software packages 4 75**
Internal Continual Assessment 250
Total 1000
Grand total of Part I and Part II: 1000 +1000 = 2000
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Note 1 :-It is recommended that two lectures per week and a fortnightly problem
Session will be devoted to each of papers I to VIII and XI to XVIII.
*Note 2 : Paper IX (Section B) with the following sub-divisions shall be of 55 marks:
Written 20 marks, Oral 15 marks, Record Book 20 marks.
**Note 3: Papers X, XIX and XX with the following sub-divisions shall each be of 75
marks:
Paper X : Written 50 marks, Oral 15 marks, Record Book 10 Marks.
Paper XIX : Written 40 marks, Oral 15 marks, Record Book 20 marks.
Paper XX : Written 40 marks, Oral 15 marks, Record Book 20 Marks.
INTERNAL CONTINUAL ASSESSMENT
M.A./M.Sc. PARTS I AND II
A. OCTOBER TEST PART I PART II
Two Written Papers 100 100
Practical 25 15
Viva-Voce - 10
B. JANUARY TEST
Two Written Papers 100 100
Practical 25 15
Viva-Voce - 10
C. MARCH TEST
Two Written Papers 100 100
Practical 25 15
Viva-Voce - 10
A. OCTOBER TEST
M.A./M.Sc. PART I
Paper P1 (50 marks) : Based on portions of Papers I to IV taught during the FIRST TERM.
Paper P2 (50 marks) : Based on portions of Papers V to VIII taught during the FIRST
TERM.
Practical PP1 (25 marks) : Based on Practicals I & II done during the FIRST TERM.
M.A./M.Sc. PART II
Paper F1 (50 marks): Based on portions of Papers XI to XIV taught during the FIRST
TERM.
Paper F2 (50 marks): Based on portions of Papers XV to XVIII taught during the FIRST
TERM.
Practical FP1 (15 marks): Based on Practicals III & IV done during the FIRST TERM.
Viva-voce FV1(10 marks) : Based on portions of Papers XI to XX taught during the
FIRST TERM.
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B. JANUARY TEST
M.A./M.Sc. PART I
Paper P3 (50 marks) : Based on portions of Papers I to IV taught during the SECOND
TERM.
Paper P4 (50 marks) : Based on portions of Papers V to VIII taught during the SECOND
TERM.
Practical PP2 (25 marks) : Based on Practicals I & II done during the SECOND TERM.
M.A./M.Sc. PART II
Paper F3 (50 marks): Based on portions of papers XI to XIV taught during the SECOND
TERM.
Paper F4 (50 marks): Based on portions of Papers XV to XVIII taught during the
SECOND TERM.
Practical FP2 (15 marks): Based on Practicals III & IV done during the SECOND TERM.
Viva-voce FV2 (10 marks) : Based on portions of Papers XI to XX taught during the
SECOND TERM.
C. MARCH TEST
M.A./M.Sc. PART I
Paper P5 (50 marks) : Based on portions of Papers I to IV taught during the THIRD
TERM.
Paper P6 (50 marks) : Based on portions of Papers V to VIII taught during the THIRD
TERM.
Practical PP3 (25 marks) : Based on Practicals I & II done during the THIRD TERM.
M.A./M.Sc. PART II
Paper F5 (50 marks): Based on portions of papers XI to XIV taught during the THIRD
TERM.
Paper F6 (50 marks): Based on portions of Papers XV to XVIII taught during the THIRD
TERM.
Practical FP3 (15 marks): Based on Practicals III & IV done during the THIRD TERM.
Viva-Voce FV3 (10 marks) : Based on portions of Papers XI to XX taught during the
THIRD TERM.
Each written paper will be of 3 hours duration and each of PP1, PP2 and PP3 will be
held in two sittings of 2 hours duration. Each of FP1, FP2 and FP3 will be held in one sitting
of 2 hours duration. Viva-Voce FV1, FV2 and FV3 will be held after the written papers and
practical.
Performance in the best two papers/practicals/viva-voce in each of the following
groups of papers will be added to the final score of the candidate:-
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M.A/M.Sc. Part I Marks M.A./M.Sc. Part II Marks
(a) P1,P3 and P5 100 F1, F3 and F5 100
(b) P2, P4 and P6 100 F2, F4 and F6 100
(c) PP1, PP2 and PP3 50 FP1, FP2 and FP3 30
____ FV1, FV2, and FV3 __20_
Total of Internal Assessment Tests : 250 Total of Internal Assessment Tests : 250
Eligibility: The students securing at least 30% of the total marks allotted to Internal
Continual Assessment Tests for each Part shall be eligible for taking the annual
examinations.
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Part I-Examination 2007 & Onwards
Paper I: Analysis
Monotone functions and functions of bounded variation. Real valued functions,
continuous functions, Absolute continuity of functions, standard properties. uniform
continuity, sequence of functions, uniform convergence, power series and radius of
convergence.
Riemann-Stieltjes integration, standard properties, multiple integrals and their
evaluation by repeated integration, change of variable in multiple integration. Uniform
convergence in improper integrals, differentiation under the sign of integral - Leibnitz rule.
Dirichlet integral, Liouville’s extension.
Introduction to n-dimensional Euclidean space, open and closed intervals
(rectangles), compact sets, Bolzano-Weierstrass theorem, Heine-Borel theorem.
Maxima-minima of functions of several variables, constrained maxima-minima of
functions.
Analytic function, Cauchy-Riemann equations, singularities, Statement of Cauchy
theorem and of Cauchy integral formula with applications, Residue and contour
integration.
Fourier and Laplace transforms, Mellin’s inversion theorem (without proof).
References:
1. Apostol, T.M. (1975): Mathematical Analysis, Addison- Wesley.
2. Bartle, R.G. (1976): Elements of Real Analysis, John Wiley & Sons.
3. Priestley, H.A. (1985): Complex Analysis, Clarenton Press Oxford.
4. Rudin, W. (1985): Principles of Mathematical Analysis McGraw Hill.
Paper II: Linear Algebra
Examples of vector spaces, vector spaces and subspace, independence in vector
spaces, existence of a Basis, the row and column spaces of a matrix, sum and intersection
of subspaces.
Linear Transformations and Matrices, Kernel, Image, and Isomorphism, change of
bases, Similarity, Rank and Nullity.
Inner Product spaces, orthonormal sets and the Gram-Schmidt Process, the Method
of Least Squares.
Basic theory of Eigenvectors and Eigenvalues, algebraic and geometric multiplicity
of eigen value, diagonalization of matrices, application to system of linear differential
equations.
Generalized Inverses of matrices, Moore-Penrose generalized inverse.
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Real quadratic forms, reduction and classification of quadratic forms, index and
signature, triangular reduction of a reduction of a pair of forms, singular value
decomposition, extrema of quadratic forms.
Jordan canonical form, vector and matrix decomposition.
References:
1. Biswas, S. (1997): A Text Book of Matrix Algebra, 2nd Edition, New Age
International Publishers.
2. Golub, G.H. and Van Loan, C.F.(1989): Matrix Computations, 2nd edition, John
Hopkins University Press, Baltimore-London.
3. Nashed, M.(1976): Generalized Inverses and Applications, Academic Press, New
York.
4. Rao, C.R.(1973): Linear Statistical Inferences and its Applications, 2nd edition, John
Wiley and Sons.
5. Robinson, D.J.S. (1991): A Course in Linear Algebra with Applications, World
Scientific, Singapore.
6. Searle, S.R.(1982): Matrix Algebra useful for Statistics, John Wiley and Sons.
7. Strang, G.(1980): Linear Algebra and its Application, 2nd edition, Academic Press,
London-New York.
Paper III: Probability Theory
Classes of sets, field, sigma field, minimal sigma field, Borel field, sequence of
sets, limits of a sequence of sets, measure, probability measure, Integration with respect
to measure.
Basic, Markov’s, Holder’s, Minkowski’s and Jensen’s inequalities.
Random variables, convergence of a sequence of random variables-convergence in
probability, almost surely, in the rth mean and in distribution, their relationship, Helly-
Bray theorem, monotone convergence theorem, Fatou’s lemma, dominated
convergence theorem, three-series criterion.
Characteristic function, uniqueness theorem, continuity theorem, inversion formula.
Laws of large numbers, Chebyshev’s and Khinchine’s WLLN, necessary and
sufficient condition for the WLLN, Kolmogorov and Hajek-Renyi inequalities, strong
law of large numbers and Kolmogorov’s theorem.
Central limit theorem, Lindeberg and Levy and Liapunov forms of CLT. Statement
of Lindeberg and Feller’s CLT and examples.
Definition and examples of Markov dependence, exchangeable sequences, mdependent
sequences, stationary sequences.
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References:
1. Ash, Robert B. (2000): Probability and Measure Theory (Second Ed.), Academic
Press, New York.
2. Bhat, B.R. (1999): Modern Probability Theory, 3rd Edition. New Age International
Publishers.
3. Billingsley, P. (1986): Probability and Measure, 2nd Edition. John Wiley & Sons.
4. Capinski, M. and Zastawniah (2001): Probability through problems, Springer.
5. Chung, K. L. (1974): A Course in Probability Theory, 2nd Edition. Academic Press,
New York.
6. Feller, W. (1968): An Introduction to Probability Theory and its Applications,3rd
Edition, Vol. 1, John Wiley & Sons.
7. Goon, A.M., Gupta, M.K. and Dasgupta. B. (1985): An Outline of Statstical
Theory, Vol. I, World Press
8. Laha, R. G. and Rohatgi, V. K.(1979): Probability Theory. John Wiley & Sons.
9. Loeve, M. (1978): Probability Theory, 4th Edition. Springer-Verlag.
10. Rohatgi, V. K. and Saleh, A.K. Md. E. (2005): An Introduction to Probability and
Statistics, Second Edn., John Wiley.
Paper IV: Statistical Methodology
Brief review of basic distribution theory. Symmetric Distributions, truncated and
compound distributions, mixture of distributions, Power series distribution, exponential
family of distributions, Characterization of distributions (Geometric, negative
exponential, normal, gamma), non-central chi-square, t and F distributions and their
properties, Concept of censoring. Approximating distributions, Delta method and its
applications. Approximating distributions of sample moments, limiting moment
generating function, Poisson approximation to negative binomial distribution. Order
statistics - their distributions and properties. Joint and marginal distributions of order
statistics. Extreme values and their asymptotic distributions (statement only) with
applications. Tolerance intervals, coverage of (X(r), X(s)). General theory of regression,
fitting of polynomial regression by orthogonal methods, multiple regression,
examination of regression equation. Development of robust procedures for estimation
and hypothesis testing, robustness of sample mean, sample standard deviation, chisquare
test and student’s t-test. Sample size determination for testing and estimation
procedures (complete and censored data) for normal, exponential, Weibull and gamma
distributions. .
References:
1. Arnold, B.C., Balakrishnan, N., and Nagaraja, H.N. (1992): A First Course in Order
Statistics, John Wiley & Sons.
2. David, H.A., and Nagaraja, H.N. (2003): Order Statistics, Third Edition, John
Wiley and Sons.
3. Dudewicz, E.J. and Mishra, S.N. (1988): Modern Mathematical Statistics, Wiley,
International Students’ Edition.
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4. Huber, P.J. (1981): Robust Statistics, John Wiley and Sons.
5. Johnson, N.L., Kotz, S. and Balakrishnan, N. (2000): Discrete Univariate
Distributions, John Wiley.
6. Johnson, N.L., Kotz, S. and Balakrishnan, N. (2000): Continuous Univariate
Distributions, John Wiley.
7. Rao, C.R. (1973): Linear Statistical Inference and Its Applications (Second
Edition), John Wiley and Sons.
8. Rohatgi, V.K. (1984): Statistical Inference, John Wiley and Sons.
9. Rohatgi, V.K. and Saleh, A. K. Md. E. (2005): An Introduction to Probability and
Statistics, Second Edition, John Wiley and Sons.
Paper V: Stochastic Processes
Poisson process, Brownian motion process, Thermal noise, shot noise, Two-valued
processes. Model for system reliability, Gieger counter. Mean value function and
covariance kernel of the Wiener and Poisson processes. Increment process of a Poisson
process, stationary and evolutionary processes. Equalizations and waiting times in
Bernoulli trials, Bernoulli trials with variable probabilities. Compound distributions, Total
progeny n branching processes, Recurrent events, Delayed recurrent events, Renewal
theory, Application to the theory of success runs, More general patterns for recurrent
events. One-dimensional, two-dimensional and three-dimensional random walks.
Gambler’s ruin problems. Classification of Markov chains. Higher transition probabilities
in Markov classification of states and chains, Irreducible ergodic chain. Homogeneous
birth and death processes. Martingales, Boob- Decomposition, Martingale Convergence
theorems, Stopping times, Optional Sampling Theorem.
References:
1. Bhat, B.R. (2000): Stochastic Models- Analysis and Applications, New Age
International Publishers.
2 Feller, William (1968) : An Introduction to Probability Theory and its Applications,
Vol. 1 (Third Ed.), John Wiley.
3. Hoel, P.G., Port, S.C.and Stone C.J. (1972) Introduction to Stochastic Processes,
Houghton Miffin & Co.
4. Karlin, S. and Taylor, H.M. (1975): A first course in Stochastic Processes (Second
Ed.), Academic Press
5. Medhi, J. (1994) : Stochastic Processes, 2nd Edition, Wiley Eastern Ltd.
6. Parzen, Emanuel (1962) : Stochastic Processes, Holden-Day Inc.
7. Ross, Sheldon M. (1983) : Stochastic Processes, John Wiley and Sons, Inc.
8. Takacs, Lajos (1967) : Combinatorial Methods in the Theory of Stochastic
Processes, John Wiley and Sons, Inc.
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Paper VI: Statistical Inference –I
Minimal sufficiency and ancillarity, Exponential families and Pitman families,
Invariance property of Sufficiency under one-one transformations of sample and parameter
spaces. Fisher Information for one and several parameters models. Lower bounds to
variance of estimators, necessary and sufficient conditions for MVUE.
UMP tests for simple null hypothesis against one-sided alternatives and for onesided
null against one-sided alternatives in one parameter exponential family. Extension of
these results to Pitman family when only upper or lower end depends on the parameters
and to distributions with MLR property, non existence of UMP tests for simple null against
two-sided alternatives in one parameter exponential family.
Interval estimation, confidence level, construction of shortest expected length
confidence interval, Uniformly most accurate one-sided confidence Interval and its relation
to UMP tests for one-sided null against one-sided alternative hypotheses.
References:
1. Ferguson, T.S. ( 1967). Mathematical Statistics, Academic Press.
2. Kale, B.K. (1999). A First Course on Parametric Inference, Narosa Publishing
House.
3. Lehmann, E.L. (1986). Theory of Point Estimation, John Wiley & Sons.
4. Lehmann E. L. (1986). Testing Statistical Hypotheses, John Wiley & Sons.
5. Rao, C.R. (1973). Linear Statistical Inference and Its Applications, Second Ed.,
Wiley Eastern Ltd., New Delhi.
6. Rohatgi, V.K. and Saleh, A.K. Md. E.(2005). An Introduction to Probability and
Statistics, Second Edition, John Wiley.
7. Zacks, S. (1971). Theory of Statistical Inference, John Wiley & Sons.
Paper VII: Survey Sampling
Basic ideas and distinctive features of sampling; Sampling designs and estimation;
Fixed (Design-based) and Superpopulation (model-based) approaches; Review of
important results in simple and stratified random sampling; Sampling with varying
probabilities (unequal probability sampling) with or without replacement –pps, pps and
non-pps sampling procedures and estimation based on them; Non-negative variance
estimation; Two-way (deep) stratification, post-stratification, controlled sampling;
Estimation based on auxiliary data (involving one or more auxiliary variables) under
design-based and model-based approaches; Double (two-phase) sampling with special
reference to the selection with unequal probabilities in at least one of the phases;
systematic sampling and its application to structured populations; Cluster sampling (with
varying sizes of clusters); Two-stage sampling (with varying sizes of first-stage units).
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Non-sampling errors with special reference to non-response, Warner’s and Simmons’
randomized response techniques for one qualitative sensitive characteristic.
References:
1. Cassel, C.M., Sarndal, C-E and Wretman, J.H. (1977). Foundations of Inference in
Survey Sampling, Wiley Inter-Science, New York
2. Chaudhari, A. And Vos, J.W.E. (1988). Unified Theory and Strategies of Survey
Sampling , North –Holland, Amsterdam.
3. Chaudhari, A. and Stenger, H. (2005). Survey Sampling Theory and methods, 2nd
Edn., Chapman and Hall.
4. Cochran, W.G. (1977). Sampling Techniques, John Wiley & Sons, New York
5. Hedayat, A.S, and Sinha, B.K. (1991). Design and Inference in Finite Population
Sampling, Wiley, New York.
6. Mukhopadhyay, Parimal (1997). Theory and Methods of Survey Sampling, Prentice
Hall of India, New Delhi.
7. Murthy, M.N. (1967). Sampling Theory and Methods, Statistical Publishing
Society, Calcutta.
8. Raj, D. and Chandhok, P. (1998). Sample Survey Theory. Narosa Publishing
House.
9. Sarndal, C-E., Swensson, B. and Wretman, J.H. (1992). Model Assisted Survey
Sampling, Springer-Verlag, New York.
10. Sukhatme, P.V., Sukhatme, B.V., Sukhatme, S. and Asok, C. (1984). Sampling
Theory of Surveys with Applications, Iowa State University Press, Iowa, USA.
11. Thompson, Steven K.(2002). Sampling, John Wiley and Sons, New York.
Paper VIII : Design of Experiments
Review of linear estimation and basic designs. ANOVA: Fixed effect models (2-way
classification with unequal and proportional number of observations per cell), Random and
Mixed effect models (2-way classification with m (>1) observations per cell).
Incomplete Block Designs, Concepts of Connectedness, Orthogonality and Balance
Intrablock analysis of General Incomplete Block design. B.I.B designs with and without
intrablock recovery.
Elimination of heterogeneity in two directions.
Symmetrical factorial experiments ( sm, s, a prime or a prime power), Confounding
in sm factorial experiments, sk-p fractional factorial where s is a prime or a prime power.
Split plot Experiments.
Finite field, Finite Geometry- Projective geometry and Euclidean geometry.
Construction of complete set of mutually orthogonal latin squares.
Construction of B.I.B.D. using MOLS, finite geometry and method of differences.
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References :
1. Chakrabarti,M.C. (1962 ) : Mathematics of Design and Analysis of Experiments,
Asia Publishing House , Bombay
2. Das,M.N. and Giri,N.C. (1986 ): Design and Analysis of Experiments . Wiley
Eastern Limited.
3. Dean, A. and Voss, D. (1999): Design and Analysis of Experiments, Springer. First
Indian Reprint 2006.
4. Dey,A. (1986 ) : Theory of Block Designs,John Wiley & Sons.
5. John, P.W.M. (1971): Statistical Design and Analysis of Experiments,Macmillan
Co., New York.
6. Kshirsagar, A.M. ( 1983 ) : A Course in Linear Models,Marcel Dekker,Inc.,N.Y.
7. Montgomery,D.C.(2005): Design and Analysis of Experiments, Sixth Edition, John
Wiley & Sons.
8. Raghavarao,D. (1970 ) : Construction and Combinatorial Problems in Design of
Experiments,John Wiley & Sons.
Paper IX: Practical-I
Data Structure & Statistical Computing
Section A (Theory)
Computer arithmetic: Representation of numbers, Errors-source and propagation. Review
of programming in C; Bit – Manipulations, Operators, Bit Fields; The C Preprocessor,
Macros. Conditional Compilation, make/Makefile utility; Files.
Stacks and their implementation; Infix, postfix and Prefix notations. Queues, Link list,
Dynamic Storage Management, Sparse matrix.
Trees– Binary trees, Binary trees representation, Tree traversal. Trees and their
applications. Graphs– Introduction, representation. Sorting– Introduction, Insertion sort,
Quicksort, Heapsort.
Random numbers: Pseudo-Random number generation, tests. Generation of non—uniform
random deviates– general methods, generation from specific distributions. Simulation-
Random Walk, Monte-Carlo integration, Applications.
Numerical Methods: Numerical root finding, Interpolation, Integration, Differentiation.
Section B (Practical)
Practicals based on Section A. Mathematical and Statistical problem-solving using
software package MATLAB: Matrix operations, Array operations, Vector and matrix
manipulation. Data analysis using MATLAB – statistical tools, Outliers, Regression and
curve fitting. Matrix functions – Triangular factorization, Orthogonal factorization,
Singular value decomposition, Eigenvalue decomposition. Control flow. M-files, their use
in MATLAB functions for numerical integration and Nonlinear functions/equations.
Graphics in 2-D and 3-D.
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References:
1. Kernighan, Brain W. and Ritchie, Dennis M. (1989). The C Programming Language,
Prentice Hall of India Pvt.Ltd., New Delhi.
2. Knuth, Donald E. (2002). The Art of Computer Programming, Vol. 2/Seminumerical
Algorithms, Pearson Education (Asia).
3. MATLAB User’s Guide: High-Performance Numeric Computation and Visualization
Software, The MathWorks, Inc.
4. Monahan, J.F. (2001). Numerical Methods of Statistics, Cambridge University Press.
5. Ross, S.M. (2002). Simulation, Third Edn., Academic press.
6. Rubinstein, R.Y. (1981). Simulation and the Monte Carlo Method, John Wiley & Sons.
7. Tenenbaum, Aaron M., Langsam, Yedidyah, and Augenstein, Moshe J. (1994). Data
Structures using C, Prentice-Hall of India Pvt.Ltd., New Delhi.
Paper X: Practical -II:-
Data analysis of problems from the following areas using Electronic
Calculators / Computers:
Statistical Methodology, Statistical Inference, Linear Algebra, Survey
Sampling and Design of Experiments.
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Part II-Examination 2008 & Onwards
Paper XI: Statistical Inference-II
Consistency and asymptotic relative efficiency of estimators. Consistent asymptotic normal
(CAN) estimator. Method of maximum likelihood, CAN estimator for one parameter
Cramer family, Cramer-Huzurbazar theorem. Solutions of likelihood equations, method of
scoring. Fisher lower bound to asymptotic variance. MLE in Pitman family and double
exponential distribution, MLE in censored and truncated distributions.
Similar tests, Neyman structure, UMPU tests for composite hypotheses, Invariance tests
and UMP invariant tests, Likelihood ratio test, Asymptotic distribution of LRT statistic,
Consistency of large sample test, Asymptotic Power of large sample test.
Sequential tests-SPRT and its properties, Wald’s fundamental identity, OC and ASN
functions. Sequential estimation.
Non- parametric methods-estimation and confidence interval, U-statistics and their
asymptotic properties, UMVU estimator, non parametric tests-single sample location,
location-cum-symmetry, randomness and goodness of fit problems; Rank order statistics,
Linear rank statistics, Asymptotic relative efficiency.
References:
1. Ferguson, T.S. ( 1967). Mathematical Statistics, Academic Press.
2. Gibbons, J. D. (1985). Non parametric Statistical Inference, 2nd Edition, Marcel
Dekker.
3. Kale, B.K. (1999). A First Course on Parametric Inference, Narosa Publishing
House.
4. Lehmann, E.L. (1986). Theory of Point Estimation, John Wiley & Sons.
5. Lehmann, E. L. (1986). Testing Statistical Hypotheses, John Wiley & Sons.
6. Randles, R.H. and Wolfe, D.S. (1979). Introduction to the Theory of Nonparametric
Statistics, John Wiley & Sons.
7. Rao, C.R. (1973). Linear Statistical Inference and Its Applications, Second Ed.,
Wiley Eastern Ltd.,
8. Rohatgi, V.K. and Saleh, A.K. Md.E. (2005). An Introduction to Probability and
Statistics, Second Edition, John Wiley.
9. Sinha, S. K. (1986). Probability and Life Testing, Wiley Eastern Ltd.
10. Zacks, S. (1971). Theory of Statistical Inference, John Wiley & Sons.
Paper XII : Multivariate Analysis
Multivariate normal distribution, its properties and characterization. Random
sampling from a multivariate normal distribution. Maximum likelihood estimators of
parameters. Distribution of sample mean vector. Inference concerning the mean vector
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when the covariance matrix is known. Matrix normal distribution. Multivariate central limit
theorem.
Wishart matrix __ its distribution and properties. Distribution of sample generalized
variance.
Hotelling’s T2 statistic __ its distribution and properties. Applications in tests on
mean vector for one and more multivariate normal populations and also on symmetry of
organs. Mahalanobis’D2.
Likelihood ratio test criteria for testing (1) independence of sets of variables, (2)
equality of covariance matrices, (3) identity of several multivariate normal populations, (4)
equality of a covariance matrix to a given matrix, (5) equality of a mean vector and a
covariance matrix to a given vector and a given matrix.
Distribution of the matrix of sample regression coefficients and the matrix of
residual sum of squares and cross products. Rao’s U-statistic, its distribution and
applications.
Classification and discrimination procedures for discrimination between two
multivariate normal populations __ sample discriminant function, tests associated with
discriminant functions, probabilities of misclassification and their estimation, classification
into more than two multivariate normal populations.
Principal components, canonical variables and canonical correlations. Elements of
factor analysis and cluster analysis.
Multivariate linear regression model __ estimation of parameters and their
properties. Multivariate analysis of variance [MANOVA] of one-and two-way classified
data. Wilk’s lambda criterion.
References:
1. Anderson, T.W. (1984) : An Introduction to Multivariate Statistical Analysis, 2nd
Ed., John Wiley & Sons.
2. Arnold, Steven F. (1981) : The Theory of Linear Models and Multivariate Analysis,
John Wiley & Sons.
3. Giri, N.C. (1977) : Multivariate Statistical Inference, Academic Press.
4. Johnson, R. A. and Wichern, D. W. (2001) : Applied Multivariate Statistical
Analysis, Fifth Edition, Prentice- Hall.
5. Kshirsagar, A.M. (1972) : Multivariate Analysis, Marcel Dekker.
6. Lawley, D. N. and Maxwell, A. E. (1971) : Factor Analysis as a Statistical Method,
2nd Ed., London Butterworths.
7. Muirhead, R. J. (1982) : Aspects of Multivariate Statistical Theory, John Wiley &
Sons.
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8. Rao, C. R. (1973) : Linear Statistical Inference and its Applications, 2nd Ed., John
Wiley & Sons.
9. Rencher, A. C. (2002) : Methods of Multivariate Analysis, 2nd Ed., John Wiley &
Sons.
10. Sharma, S. (1996) : Applied Multivariate Techniques, John Wiley & Sons.
11. Srivastava, M. S. and Khatri, C. G. (1979) : An Introduction to Multivariate
Statistics, North Holland.
Paper XIII : Demography, Statistical Quality Control and Reliability
Demography: Measures of mortality, description of life table, construction of
complete and abridged life tables, maximum likelihood, MVU and CAN estimators of life
table parameters.
Measures of fertility, models for population growth, intrinsic growth rate, stable
population analysis, population projection by component method and using Leslie matrix.
Quality control and Sampling Inspection: Basic concepts of process monitoring and
control, General theory and review of control charts, O.C and ARL of control charts,
CUSUM charts using V-mask and decision intervals, economic design of x- bar chart.
Review of sampling inspection techniques, single, double, multiple and sequential
sampling plans and their properties, methods for estimating (n, c) using large sample and
Bayesian techniques, curtailed and semi-curtailed sampling plans, Dodge’s continuous
sampling inspection plans for inspection by variables for one-sided and two-sided
specifications.
Reliability: Reliability concepts and measures, components and systems, reliability
function, hazard rate, common life distributions viz. exponential, gamma, Weibull,
lognormal, Rayleigh, bath tub, etc., Reliability and expected survivability of series,
parallel, mixed, maintained and non-maintained systems with and without redundancy,
preventive maintenance policy, preliminary concepts of coherent systems.
References:
1. Bain, L. J and Engelhardt, M. (1991): Statistical Analysis of Reliability and Life
Testing Models, Marcel Dekker.
2. Barlow, R. E. And Proschan, F (1985): Statistical Theory of Reliability and Life
Testing, Holt, Rinehart and Winston.
3. Biswas, S. (1988): Stochastic Processes in Demography and Applications, Wiley
Eastern Ltd.
4. Biswas, S. (1996): Statistics of Quality Control, Sampling Inspection and
Reliability, New Age International Publishers.
5. Chiang, C.L. (1968): Introduction to Stochastic Processes in Bio statistics, John
Wiley.
6. Keyfitz, N. (1971): Applied Mathematical Demography, Springer Verlag.
16
7. Lawless, J. F. (1982): Statistical Models and Methods of Life Time Data, John
Wiley & Sons.
8. Montgomery, D. C. (2005): Introduction to Statistical Quality Control, 5th Edn.,
John Wiley & Sons.
9. Spiegelman, M. (1969): Introduction to Demographic Analysis, Harvard University
Press.
10. Wetherhil, G. B. (1977): Sampling Inspection and Quality Control, Halsted Press.
Paper XIV: Econometrics and Time Series Analysis
Time series as discrete parameter stochastic process. Auto covariance and auto
correlation functions and their properties.
Detailed study of the stationary processes : Moving average (MA), Auto regressive
(AR), ARMA and ARIMA models. Box-Jenkins models. Discussion (without proof) of
estimation of mean, auto covariance and auto correlation functions under large sample
theory. Choice of AR and MA periods. Estimation of ARIMA model parameters.
Smoothing spectral analysis of weakly stationary process. Periodo gram and correlogram
analysis. Filter and transfer functions. Problems associated with estimation of spectral
densities.
Forecasting: Exponential and adaptive Smoothing methods
Econometrics: Review of G.L.M. and generalized least squares estimation, GLM with
stochastic regressors. Instrumental variables, estimation, consistency property, asymptotic
variance of instrumental variable estimators.
Bayesian analysis of G.L.M. with informative and non informative prior
distributions. Bayes estimation and testing of hypotheses of the regression coefficients.
Distributed lag models: Finite polynomial lags, determination of the degree of
polynomial. Infinite distributed lags, adaptive expectations and partial adjustment models,
determination of lag length. Methods of estimation.
Simultaneous equations models: Identification problem. Restrictions on structural
parameters-rank and order conditions. Restrictions on variances and covariances.
Estimation in simultaneous equations models. Recursive systems, 2SLS estimators,
Limited information estimators, k-class estimators, Instrumental variable method of
estimation. 3-SLS estimation.
References:
1. Johnston, J. (1984): Econometric Methods, Mc Graw Hill Kogakusha Ltd.
2. Judge, G.C., Hill, R,C. Griffiths, W.E., Lutkepohl, H. and Lee, T-C. (1988):
Introdutuction to the Theory and Practice of Econometrics (Second Edition), John
Wiley & Sons.
17
3. Kendall, M.G. and Stuart, A. (1968): The Advanced Theory of Statistics (Vol. III)
Second Edition, Charles Griffin.
4. Kmenta, J. (1986): Elements of Econometrics (Second Edition), Mac millan.
5. Medhi, J. (1994): Stochastic Processes (Second Edition), Wiley Eastern, New Delhi
6. Montgomery, D.C. and Johnson, L.A. (1976): Forecasting and Time Series
Analysis, Mc Graw Hill, New York .
Paper XV: Generalized Linear Models
Logistic and Poisson regression: logit model for dichotomous data with single and
multiple explanatory variables, ML estimation, large sample tests about parameters,
goodness of fit, analysis of deviance, variable selection, introduction to Poisson
regression, MLE for Poisson regression, Applications in Poisson regressions, Lack of
fit in Logistic regression.
Log linear models for contingency tables: interpretation of parameters, ML
estimation of parameters, likelihood ratio tests for various hypotheses including
independence, marginal and conditional independence, partial association.
Family of Generalized Linear Models: Exponential family of distributions, Formal
structure for the class of GLMs, Likelihood equations, Quasi likelihood, Link
functions, Important distributions for GLMs, Power class link function.
References:
1. Agesti, A. (1990). Categorical Data Analysis. Wiley.
2. Christensen, R. (1997). Log-linear Models and Logistic Regression, Second
Edition. Springer.
3. Green, P.J. and Silverman, B.W. (1994). Nonparametric Regression and
Generalized Linear Models. Chapman and Hall, New York.
4. Hasting, T.J. and Tibshirani, R.J. (1999). Generalized Additive Models. Second
Edition. Chapman and Hall, New York.
5. Hosmer, D.W. and Lemeshow, S. (2000). Applied Logistic Regression, Second
Edition. Wiley, New York.
6. McCullagh, P. and Nelder, J.A. (1999). Generalized Linear Models, Second
Edition. Chapman and Hall.
7. McCulloch, C.E. and Searle, S.R. (2001). Generalized, Linear, and Mixed
Models. John Wiley & Sons, Inc. New York.
8. Myers, R.H., Montgomery, D.C and Vining, G.G. (2002). Generalized Linear
Models With Applications in Engineering and the Sciences. John Wiley & Sons.
18
Papers XVI, XVII and XVIII: Any three of the following options:-
(i) Applied Stochastic Processes
Markov processes in continuous time. Poisson process, Kolmogorov equations.
Forward and backward equations for homogeneous case. Random variable technique,
Homogeneous birth & death processes. Divergent birth process. The effect of immigration.
The general birth and death process. Multiplicative processes. Simple non-homogeneous
processes. Polya process. The effect of immigration for non-homogeneous case. Queueing
processes. Equilibrium theory. Queues with many servers. First passage times. Diffusion.
Backward Kolmogorov diffusion equation. Fokker-Planck equation. Application to
population growth. Epidemic and Counter models. Supplementary variables. Embedded
Markov processes. Some multi-dimensional prey and predator and non-Markovian
processes, Renewal processes-ordinary, modified, equilibrium. Renewal functions. Integral
equation of renewal theory. Distribution of the number of renewals. The elementary
renewal theorem.
References:
1. Bailey, Norman T.J. (1964) : The Elements of Stochastic Processes, John Wiley and
Sons.
2. Bartlett, M.S. (1966) : An Introduction to Stochastic Processes, Cambridge
University Press.
3. Cox. D. R. and Miller, H. D. (1965) : The theory of Stochastic Processes, Mathuen
& C0., London.
4. Hoel, P.G., Port, S.C. and Stone, C.J. (1972) : Introduction to Stochastic Processes,
Houghton Miffein Company.
5. Karlin, S. and Taylor, H.M. (1975) : A First Course in Stochastic Processes (Second
Ed.), Academic Press.
6. Ross, S. M. (1983) : Stochastic Processes. John Wiley & Sons.
(ii) Order Statistics
Basic distribution theory. Order statistics for a discrete parent. Distribution-free
confidence intervals for quantiles and distribution-free tolerance intervals. Conditional
distributions, Order Statistics as a Markov chain and characterizations. Order statistics for
independently distributed variates. Moments of order statistics. Large sample
approximations to mean and variance of order statistics. Asymptotic distributions of order
statistics. Recurrence relations & identities. Distibution-free bounds for moments of order
statistics and of the range.
Order statistics for dependent variates, Bounds in the case of dependent variates.
Random division of an interval. Concomitants. Application to estimation and hypothesis
testing, Relation to Poisson Process. Order statistics from a sample containing a single
outlier.
Rank order statistics related to the simple random walk. Dwass’ technique. Ballot
theorem, its generalization, extension and application to fluctuations of sums of random
19
variables. Galton’s rank test statistics. Statistics of Kolmogorov-Smirnov type for two
samples.
References:
1. Arnold, B.C. and Balakrishnan, N. (1989) : Relations, Bounds and Approximations
for Order Statistics, Vol. 53, Springer-Verlag.
2. Arnold, B. C., Balakrishnan, N. and Nagaraja H. N. (1992) : A First Course in
Order Statistics, John Wiley & Sons.
3. David, H. A. and Nagaraja, H. N. (2003): Order Statistics, Third Edition, John
Wiley & Sons.
4. Dwass, M. (1967): Simple random walk and rank order statistics. Ann. Math.
Statist. 38, 1042-1053.
5. Gibbons, J.D. and Chakraborti, S. (1992): Nonparametric Statistical Inference,
Third Edition, Marcel Dekker.
6. Takacs, L. (1967) : Combinatorial Methods in the Theory of Stochastic Processes,
John Wiley & Sons.
(iii) Bayesian Inference
Subjective interpretation of probability in terms of fair odds; Subjective prior
distribution of a parameter; Bayes theorem and computation of posterior distribution.
Natural conjugate family of priors for a model. Conjugate families for exponential
family models, and models admitting sufficient statistics of fixed dimension. Mixtures
from conjugate family, Jeffreys’ invariant prior. Maximum entropy priors.
Utility function, expected utility hypothesis, construction of utility function, St.
Petersburg Paradox. Loss functions: (i) bilinear, (ii) squared error, (iii) 0-1 loss, and (iv)
Linex. Elements of Bayes Decision Theory, Bayes Principle, normal and extensive form of
analyses.
Generalized maximum likelihood estimation. Bayes estimation under various loss
functions. Evaluation of the estimate in terms of the posterior risk, Preposterior analysis
and determination of optimal fixed sample size. Linear Bayes estimates. Empirical and
Hierarchical Bayes Methods of Estimation.
Bayesian interval estimation: Credible intervals, HPD intervals, Comparison with
classical confidence intervals.
Bayesian testing of hypotheses, specification of the appropriate form of the prior
distribution for a Bayesian testing of hypothesis. Prior and posterior odds. Bayes factor for
various types of testing hypothesis problems. Lindley’s method for Significance tests, two
sample testing problem for the parameters of a normal population. Finite action problem
and hypothisis testing under “O-Ki” loss, function. Large sample approximation for the
posterior distribution. Lindley’s approximation of Bayesian integrals.
Predictive density function, prediction for regression models, Decisive prediction,
point and internal predictors, machine tool problem.
20
References:
1. Aitchison, J. and Dunsmore, I.R.(1975): Statistical Prediction Analysis, Cambridge
University Press.
2. Berger, J.O. (1985): Statistical Decision Theory and Bayesian Analysis, Springer
Verlag, New York.
3. Box, G.E.P. and Tiao, G.C. (1973): Bayesian Inference in Statistical Analysis,
Addison & Wesley.
4. De. Groot, M.H. (1970): Optimal Statistical Decisions, McGraw Hill.
5. Leonard, T. and Hsu, J.S.J. (1999): Bayesian Methods, Cambridge University Press.
6. Lee, P. M. (1997): Bayesian Statistics: An Introduction, Arnold Press.
7. Robert, C.P. (2001): The Bayesian Choice: A Decision Theoretic Motivation
(Second Edition), Springer Verlag, New York.
(iv) Advanced Survey Sampling Theory
Admissibility of estimators; Non-existence of UMV estimators; Estimation of
median; Sampling on two or more successive occasions (Repetitive surveys); Resampling
techniques for variance estimation- independent and dependent random groups, the
Jackknife and the Bootstrap; Small-area estimation; Estimation from multiple frames;
Double sampling for stratification; Non-sampling errors and double sampling; Nonresponse
problems, Randomized response techniques for one quantitative sensitive
characteristic.
Prediction of non-observed residuum under fixed (design-based) and super
population (model-based) approaches, different types of superpopulation models with
optimal strategies based on them; Robustness against model failures.
References:
1. Cassel, C.M., Sarndal, C-E and Wretman, J.H. (1977). Foundations of Inference in
Survey Sampling, Wiley Inter- Science.
2. Chaudhari, A. and Stenger, H. (2005). Survey sampling Theory and Methods, 2nd
Edn., Chapman and Hall.
3. Hedayat, A.S. and Sinha, B.K. (1991). Design and Inference in Finite Population
Sampling, Wiley Inter-Science.
4. Mukhopadhyay , P. (1996). Inferential Problems in Survey Sampling, New Age
International (P) Ltd.
5. Sarndal, C-E, Swensson, B. and Wretman, J.H. (1992). Model Assisted Survey
Sampling, Springer-Verlag.
6. Sukhatme, P.V., Sukhatme, B.V., Sukhatme, S. and Asok, C. (1984). Sampling
Theory of Surveys with Applications, Iowa State university Press, Iowa, USA.
7. Wolter, K.M.(1984). Introduction to Variance Estimation, Springer-Verlag.
(v) Advanced Theory of Experimental Designs
Partially balanced incomplete block designs. Resolvable and affine resolvable designs,
dual and linked block designs. Lattice Designs. General theory of Fractional factorial
Experiments. Optimum designs-various optimality criteria and their constructions.
21
Symmetrical and asymmetric orthogonal arrays and their constructions. Taguchi’s robust
designs. Response surface designs-orthogonality, rotatability and blocking. Weighing
designs. Mixture Experiments.
Construction of PBIB designs
References:
1. Chakrabarti,M.C. (1962) : Mathematics of Design and Analysis of
Experiments,Asia Publishing House.
2. Cornell, John A. (2002) : Experiments with Mixtures,John Wiley & Sons.
3. Das,M.N. and Giri,N.C. (1986) :Design and Analysis of Experiments,Wiley
Eastern Limited
4. Dey,A. (1986) : Theory of Block Designs, John Wiley & Sons.
5. Dey, A. and Mukerjee, R. (1999) : Fractional Factorial Plans,John Wiley & Sons.
6. Hedayat,A.S., Sloane, N.J.A. and Stufken,J. (1999) : Orthogonal Arrays:Theory
and Applications, Springer.
7. Montgomery,D.C. (2005) : Design and Analysis of Experiments, Sixth Edition,
John Wiley & Sons.
8. Myers,R.H. and Montgomery,D.C. (2002) ; Response Surface
Methodology: Process and Product Optimisation using Designed Experiments, John
Wiley & Sons.
9. Raghavarao,D. (1970) : Construction and Combinatorial Problems in Design of
Experiments, John Wiley & Sons.
10. Wu,C.F.J. and Hamada,M. (2000) : Experiments : Planning,Analysis and
Parameter Design Optimisation,John Wiley & Sons.
(vi) Bio-Statistics
Functions of survival time, survival distributions and their applications viz.
exponential, gamma, weibull, Rayleigh, lognormal, death density function for a distribution
having bath-tub shape hazard function. Tests of goodness of fit for survival distributions
(WE test for exponential distribution, W-test for lognormal distribution, Chi-square test for
uncensored observations). Parametric methods for comparing two survival distributions
viz. L.R test, Cox’s F-test.
Type I, Type II and progressive or random censoring with biological examples,
Estimation of mean survival time and variance of the estimator for type I and type II
censored data with numerical examples. Non-parametric methods for estimating survival
function and variance of the estimator viz. Acturial and Kaplan –Meier methods.
Competing risk theory, Indices for measure-ment of probability of death under competing
risks and their inter-relations. Estimation of probabilities of death under competing risks by
22
maximum likelihood and modified minimum Chi-square methods. Theory of independent
and dependent risks. Bivariate normal dependent risk model. Conditional death density
functions.
Stochastic epidemic models: Simple and general epidemic models (by use of random
variable technique).
Basic biological concepts in genetics, Mendels law, Hardy- Weinberg equilibirium,
random mating, distribution of allele frequency ( dominant/co-dominant cases), Approach
to equilibirium for X-linked genes, natural selection, mutation, genetic drift, equilibirium
when both natural selection and mutation are operative, detection and estimation of linkage
in heredity.
Planning and design of clinical trials, Phase I, II, and III trials. Consideration in
planning a clinical trial, designs for comparative trials. Sample size determination in fixed
sample designs.
References:
1. Biswas, S. (1995): Applied Stochastic Processes. A Biostatistical and Population
Oriented Approach, Wiley Eastern Ltd.
2. Cox, D.R. and Oakes, D. (1984) : Analysis of Survival Data, Chapman and Hall.
3. Elandt, R.C. and Johnson (1975): Probability Models and Statistical Methods in
Genetics, John Wiley & Sons.
4. Ewens, W. J. (1979) : Mathematics of Population Genetics, Springer Verlag.
5. Ewens, W. J. and Grant, G.R. (2001): Statistical methods in Bio informatics.: An
Introduction, Springer.
6. Friedman, L.M., Furburg, C. and DeMets, D.L. (1998): Fundamentals of Clinical
Trials, Springer Verlag.
7. Gross, A. J. And Clark V.A. (1975) : Survival Distribution; Reliability
Applications in Biomedical Sciences, John Wiley & Sons.
8. Lee, Elisa, T. (1992) : Statistical Methods for Survival Data Analysis, John Wiley
& Sons.
9. Li, C.C. (1976): First Course of Population Genetics, Boxwood Press.
10. Miller, R.G. (1981): Survival Analysis, John Wiley & Sons.
(vii) Operational Research
Definition and scope of Operational Research; phases in Operational Research;
different types of models, their construction and general methods of solution.
Linear Programming problems; duality theory; Transportation
Problems,Assignment Problems. Introduction to Integer Programming. Nonlinear
Programming, Unconstrained Problems and the Kuhn Tucher Conditions, Quadratic
Programming-Beale’s and Wolfe’s methods. Introduction to Dynamic Programming:
Bellman’s principle of optimality, general formulation, computational methods and
application of Dynamic Programming. Theory of Network - PERT and CPM.
23
Queueing Theory: Steady state solutions of queue length and waiting time for
M/M/1 and M/M/C. Method of stages for steady state solution of M/Ek/1 and Ek/M/1.
Simple design and control problems in queueing theory.
Inventory management: Characteristics of inventory systems. Classification of
items. Deterministic inventory systems with and without lead-time. All units and
incremental discounts. Single period stochastic models.
Introduction to Decision Analysis: Pay-off table for one-off decisions and
discussion of decision criteria, Decisions Trees. Replacement Analysis: Analysis of
replacement policies for equipment.
References:
1. Gross, D. and Harris, C.M. (1985): Fundamental of Queueing Theory, John Wiley
& Sons.
2. Hadley, G. and Whitin, T.M. (1963): Analysis of Inventory Systems, Prentice Hall.
3. Hadley, G. (2002): Linear Programming, Narosa Publishing House.
4. Hadley, G. (1964): Non-Linear and Dynamic Programming, Addison-Wesley
Publishing Company.
5. Hillier, F.S. and Lieberman, G.J. (2001): Introduction to Operations Research,
Seventh Edition, Irwin.
6. Taha, H. A. (1997): Operations Research: An Introduction, ( 6th Edition ), Prentice
Hall.
7. Wagner, B.M. (1975): Principles of OR, Englewood Cliffs, N.J. Prentice-Hall
8. Waters, Donald and Waters, C. D. J. (2003): Inventory Control and Management,
John Wiley & Sons.
(viii) Nonparametric Inference
Review of order statistics, Distribution-free statistics over a class, Counting
statistics, ranking statistics, Statistics utilizing counting and ranking, Asymptotic
distribution of U-statistics, Confidence interval for population quantile and scale
parameter, point estimation, Estimators associated with distribution free test statistics,
Exact small-sample and asymptotic properties of the Hodges-Lehmann location
estimators, Tests based on length of the longest run, runs up and down, Kolmogorov-
Smirnov two-sample statistic, rank order statistics: Correlation between ranks and
variate values, One sample, paired sample and two sample problems, distribution
properties of linear rank statistics, tests for equality of k independent samples: Kruskal-
Wallis one way ANOVA test, Measures of Association for bivariate samples: Kendall’s
Tau coefficient, Spearman’s coefficient of Rank correlation, relations between R and
T; E (R ), t and r. Measures of association in multiple classifications: Friedman’s twoway
ANOVA by ranks in a k x n table, the Coefficient of Concordance of k sets of
rankings of n objects, the Coefficient of Concordance of k sets of incomplete rankings.
Concept of power and robustness.
24
References:
1. David, H.A. and Nagaraja, H. N.(2003): Order Statistics, Third Edition, John Wiley
& Sons.
2. Gibbons, J.D. and Chakraborti, S. (1992): Nonparametric Statistical Inference,
Third Edition, Marcel Dekker.
3. Hettmansperger, T.P. (1984): Statistical inference Based on Ranks, John Wiley &
Sons.
4. Randles, R.H. and Wolfe, D.A.(1979): Introduction to the Theory of Nonparametric
Statistics, John Wiley & Sons.
5. Rohatgi, V.K. and Saleh, A.K. Md. E.(2005): An Introduction to Probability and
Statistics, Second Edition, John Wiley & Sons.
(ix) Actuarial Statistics
Section I- Probability Models and Life Tables
Utility theory, insurance and utility theory, models for individual claims and their
sums, survival function, curtate future lifetime, force of mortality.
Life table and its relation with survival function, examples, assumptions for
fractional ages, some analytical laws of mortality, select and ultimate tables.
Multiple life functions, joint life and last survivor status, insurance and annuity
benefits through multiple life functions evaluation for special mortality laws.
Multiple decrement models, deterministic and random survivorship groups,
associated single decrement tables, central rates of multiple decrement, net single
premiums and their numerical evaluations.
Distribution of aggregate claims, compound Poisson distribution and its
applications.
Section II- Insurance and Annuities
Principles of compound interest: Nominal and effective rates of interest and
discount, force of interest and discount, compound interest, accumulation factor,
continuous compounding.
Life insurance: Insurance payable at the moment of death and at the end of the year
of death-level benefit insurance, endowment insurance, diferred insurance and varying
benefit insurance, recursions, commutation functions.
Life annuities: Single payment, continuous life annuities, discrete life annuities, life
annuities with monthly payments, commutation functions, varying annuities, recursions,
complete annuities-immediate and apportionable annuities-due.
Net premiums: Continuous and discrete premiums, true monthly payment
premiums, apporionable premiums, commutation functions, accumulation type benefits.
Payment premiums, apportionable premiums, commutation functions, accumulation
type benefits.
25
Net premium reserves: Continuous and discrete net premium reserve, reserves on a
semicontinuous basis, reserves based on true monthly premiums, reserves on an
apportionable or discounted continuous basis, reserves at fractional durations, allocations
of loss to policy years, recursive formulas and differential equations for reserves,
commutation functions.
Some practical considerations: Premiums that include expenses-general expenses
types of expenses, per policy expenses.
Claim amount distributions, approximating the individual model, stop-loss
insurance.
References:
1. Atkinson, M.E. and Dickson, D.C.M. (2000) : An Introduction to Actuarial Studies,
Elgar Publishing.
2. Bedford, T. and Cooke, R. (2001): Probabilistic risk analysis,Cambridge.
3. Bowers, N. L., Gerber, H. U., Hickman, J. C., Jones D.A. and Nesbitt, C. J. (1986):
‘Actuarial Mathematics’, Society of Actuaries, Ithaca, Illinois, U.S.A., Second
Edition (1997)
4. Medina, P. K. and Merino, S. (2003): A discrete introduction : Mathematical
finance and Probability, Birkhauser.
5. Neill, A. (1977): Life Contingencies, Heineman.
6. Philip, M. et. al (1999): Modern Actuarial Theory and Practice, Chapman and Hall.
7. Rolski, T., Schmidli, H., Schmidt, V. and Teugels, J. (1998): Stochastic Processes
for Insurance and Finance, Wiley.
8. Spurgeon, E.T. (1972): Life Contingencies, Cambridge University Press.
9. Relevant Publications of the Actuarial Education Co., 31, Bath Street, Abingdon,
Oxfordshire OX143FF (U.K.)
(x) Advanced Statistical Computing and Data Mining
Random number generation: Review; Simulating multivariate distributions; Simulating
stochastic processes. Variance reduction. Stochastic differential equations: introduction,
Numerical solutions. Markov Chain Monte Carlo methods-Gibbs sampling; Simulated
annealing, cooling schedule, convergence, application. Non-linear regression: Method;
Estimation; Intrinsic and Parameter-effects curvature; application. EM algorithm and
applications. Smoothing with kernels: density estimation, choice of kernels.
Review of classification methods from multivariate analysis; classification and decision
trees. Clustering methods from both statistical and data mining viewpoints; Vector
quantization. Unsupervised learning; Supervised learning; Artificial neural networks:
Introduction, multilayer perceptron network, self-organizing feature map and radial basis
function network. Structural risk minimization, Introduction to support vector machine.
Overview of current applications.
26
References:
1. Bishop, C.M. Neural Networks for pattern Recognition, Oxford University Press.
2. Duda, R.O., Hart, P.E. and Strok, D.G. Pattern Classification, 2nd Edition, John Wiley
& Sons.
3. Gentle, J.E., Härdle W. and Mori Y., (2004). Handbook of computational statistics —
Concepts and methods, Springer-Verlag.
4. Han, J. and Kamber, M. (2000). Data Mining: Concepts and Techniques, Morgan
Kaufmann.
5. Hand, David, Mannila, Heikki, and Smyth, Padhraic, (2001). Principles of Data
Mining, MIT Press.
6. Haykin, S. Neural Networks-A Comprehensive Foundation, 2nd Edition, Prentics Hall.
7. McLachlan, G.J. and Krishnan, T. (1997). The EM Algorithms and Extensions,
Wiley.
8. Nakhaeizadeh, G. and Taylor G.C., (1997). Machine Learning and Statistics, John
Wiley & Sons.
9. Pooch, Udo W. and Wall, James A. (1993). Discrete Event Simulation (A practical
approach), CRC Press.
10. Rubinstein, R.Y. (1981). Simulation and the Monte Carlo Method, John Wiley &
Sons.
11. Simonoff, J.S. (1996). Smoothing Methods in Statistics, Springer.
Paper XIX:- Practical – III
Developing programs in C-language to analyse data from the following
areas: Multivariate Analysis, Statistical Inference, Econometrics,
Demography, Statistical Quality Control, Reliability Theory, Survival
Analysis, Time Series and Forecasting, General Linear Models.
Paper XX:- Practical - IV
Based on
(i) knowledge of statistically relevant Software,
(ii) application of Software for data analysis in the following areas:
Multivariate Analysis, Statistical Inference, Design of Experiments,
Econometrics, Demography, Statistical Quality Control, Reliability
Theory, Survival Analysis, Time Series and Forecasting, General
Linear Models.
For more details, visit http://www.du.ac.in/course/syllabi/mamsc-stats-syllabi.pdf
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