SYLLABUS for Ph.D. Statistics Entrance Examination

SYLLABUS for Ph.D. Statistics Entrance Examination

1.      Probability Theory

General probability space, various definition of probability, combinations of events, additive and multiplicative law of probability, conditional probability, Bayes’ theorem and its application. Random variables, Convergence in probability and convergence in distribution, Characteristic function, uniqueness theorem, continuity theorem, inversion formula. Laws of large numbers, Chebyshev’s and Khinchine’s WLLN, strong law of large numbers and Kolmogorov’s theorem. Central limit theorem, Lindeberg and Levy and Liapunov forms of CLT.

2.      Univariate Distributions

Binomial distribution, Poisson distribution, Geometric distribution, Hyper-geometric  distributions, Normal distribution, Exponential distribution, Lognormal distribution, Cauchy, truncated and compound distributions-binomial, poisson and normal, Power series distribution, Chi-Square, t and F distributions (central and non-central) and their applications. Order statistics: their distributions and properties, distribution of range.

3.      Inference

Definition and properties of a good estimator, Fisher Neyman factorization theorem, Cramer-Rao inequality, Rao-Blackwell theorem, Completeness and Lehmann-Scheffe theorem, Methods of Estimation: Maximum likelihood method, moments, minimum Chi-square, Properties of maximum likelihood estimator. Confidence Interval, Simple and composite hypothesis, procedure of testing of hypothesis, critical region, types of errors, level of significance, p-value, power of a test, most powerful test and Neyman-Pearson fundamental lemma. Likelihood ratio test.

Non-Parametric Tests: Sign tests, Kolmogorov-Smirnov test, Median test, Wilcoxon-Mann-Whitney U test

4.      Sample Survey

Concept of population, sample, sampling frame, sampling design and strategy. Simple random sampling with and without replacement, Stratified Sampling, Systematic sampling, Double sampling and its uses in ratio estimation, Population proportion to size with replacement and without replacement (PPSWR or PPSWOR) sampling, cumulative total and Lahiri’s method. Ratio and regression methods of estimation.

5.      ANOVA and Design of Experiments

Principles of design of experiments, layout and analysis of completely randomized design, randomized block design, Least square design. Analysis of covariance. Factorial Experiment 2n and 32, Split plot and Strip plot design, Partial and complete confounding.

6.      Multivariate Analysis

Multivariate normal distribution, Distribution of quadratic forms. Hotelling-T2 and its properties and applications, Mahanalobis D2. Wishart distributions and its properties. Classification problem and Fisher’s linear discriminant function, Concept and application of principle component analysis, factor analysis and cluster analysis,

7.      Econometrics

Introduction to econometric models, Ordinary least squares estimation and prediction, generalized least square estimation and prediction, Gauss Markoff theorem, Heteroscedasticity, Auto-correlation: Multicollinearity, Simultaneous equation model

8.      Populations Studies

Nature of demography, Sources of demographic data, Basic demographic measures, Complete Life table : Uses, assumptions and Construction, Stable and stationary population, Mortality and fertility Rates.

Definition of census and its features, methods of enumeration, Census in India, Basic highlights of census 2001 and 2011

9.      Statistical Quality Control

Quality and statistical quality control, Control limits, specification limits and tolerance limits, Control-Charts: Concept and construction of control charts for variables and attributes. Acceptance Sampling Plans for Attributes. Single and double sampling plans, Dodge and Romig tables. Cusum Chart

10.  Economic Statistics

Index Number : Price, Quantity and Value Indices. Meaning and uses, limitations, Construction and tests for Index numbers. Chain based Index numbers, Consumer price index.

Demand Analysis: Concept related to demand and supply, Theory and analysis of consumer demand, estimations of demand function. Demand and income elasticity. Pareto’s law of income distributions. Engle’s curve, curves of concentration.

Time Series: Definition and its different components, additive and multiplicative models. Different methods of determining trend and seasonal and cyclic fluctuations. Stationary time series, Box-Jenkins Model, Introduction to moving average (MA), auto-regressive (AR), ARMA, and AR integrated MA (ARIMA) models.

 

Books Recommended/Reference Books

  1. Goon, Gupta & Das Gupta. (2003): Outline of Statistical Theory. Vol. I, World Press.
  2. Gupta, S.C. and Kapoor ,V.K.: (2000) Fundamentals of Mathematical Statistics, S Chand  & Company, New Delhi
  3. Mood, Graybill and Boes. (1974): Introduction to the theory of Statistics 3rded, McGraw- Hill.
  4. Goon, A.M., Gupta, M.K. and Dasgupta, B. Das (2008): Fundamentals of Statistics, Volume II,
  5. Mukhopadhyay, P. (1996): Mathematical Statistics, New Central Book Agency (P) Ltd
  6. Shukhatme, P.V.et al(1984): Sampling Theory of Surveys in the Applications, Iawa State press & Ind.Soc. of Agri. Stat.
  7. Mukhopadhya, P.(1996): Inferencial Problems in Survey Sampling, New Age Intenational.
  8. Dey, Alok,(1988):Theory of Block Designs, John Wiley & Sons
  9. Montgomery, D.C. (2019): Introduction to Statistical Quality Control, 5th Ed. John Wiley.
  10. Croxton, F.E .& Cowden, DJ. (1979): Applied General Statistics, Prentice Hallof India.
  11. Gupta, S.C. and Kapoor, V.K.(2000): Fundamentals of Applied Statistics, S Chand & Company, New Delhi
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