Exam pattern, Syllabus for APPSC

15.STATISTICS

Probability Theory :

• Random experiment, Random event, Sample Space, Classes of sets, fields, sigma-fields, minimal sigma-fields, Borel sigma fields in R, Measure, Lebesque mesure, Lebesque-Stieltjes measures, Measurable functions, Borel function, induced sigma field, Probability Measure, Basic Properties of a Measure, conditional probability and Bayes Theorem. Caratheodory extension theorem (Statement only), measurable function, random variables, distribution function and its properties, expectation, statements and applications of monotone convergence theorem, Foatou’s lemma, dominated convergence theorem.

• Expectations of functions of rv’s, conditional expectation and conditional variance, their applications. Characteristic function of a random variable and its properties. Inversion theorem, uniqueness theorem (Functions which cannot be Characteristic functions). Levy’s continuity theorem (Statement only). Chebychev, Markov, Cauchy-Schwartz, Jenson, Liapunov, Holder’s and Minkowsky’s inequalities.

• Sequence of Random variables, convergence in Probability, convergence in distribution, almost sure convergence, convergence in quadratic mean and their interrelationships, Slutskey’s theorem, Borel-Cantelli lemma Borel 0-1 law, Kolm ogorov 0-1 law (Glevenko – Cantelli Lemma Statement only).

• Law of large numbers, Weak law of large numbers, Bernoulli and Khintchen’s WLLN’s, Kolomogorov Inequality, Kolmogorov SLLN for independent random variables and statement only for i.i.d. case and their applications, statements of three series theorem. Central Limit theorem : Demoviere – Laplace CLT, Lindberg- Levy CLT, Liapounou’ CLT, Statement of Lindberg-Feller CLT, simple applications.

• Introduction to stochastic processes; classification of stochastic process according to state-space and time-domain. Finite and countable state Markov chains; time- homogeneity; Chapman-Kolmogorov equations; marginal distribution and finite – dimensional distribution; classification of states of a Markov chain – recurrent, positive recurrent, null-recurrent and transient states.

Distribution Theory

• Standard discrete and continuous univariate distributions : Binomial, geometric, Poisson, Negative Binomial, Hyper-geometric, Uniform, Triangular, beta, exponential, gama, Weibull, Normal, Lognormal, and Cauchy distributions and their properties. Joint, Marginal and conditional pmf’s and pdf’s.

• Families of Distributions : Power series distributions, Exponential families of distributions. Functions of Random variables and their distributions (including transformation of rv’s). Bivariate Normal, Bivariate Exponential (Marshall and Olkins form), Compounding distributions using Binomial and Poisson. Truncated (Binomial, Poisson, Normal and Lognormal) and mixture distributions – Definition and examples.

• Sampling Distributions of sample mean and variance, independence of X and s2.

Central and Non-central Ï°2, t and F distributions. Order statistics Joint and marginal distributions of order statistics and Distribution of Range. Distributions of order statistics from rectangular, exponential and normal distributions. Empirical distribution function.

• Multinomial distribution. Multivariate normal, bi-variate as a particular case, moments, c.f., conditional and marginal distributions. Distributions of correlation coefficient, partial and multiple correlations, and inter relationships. Dimension reduction method : PCA, FA, Canonical Correlations an MDS. Discriminent analysis and cluster Analysis.

• Distributions of quadratic forms under normality and related distribution theory.

Statistical Inference :

• Point Estimation : Point Estimation Vs. Interval Estimation, Advantages, Sampling distribution, Likelihood function, exponential family of distribution. Desirable properties of a good estimator : Unbiasedness, consistency, efficiency and sufficiency – examples. Neyman factorization theorem (Proof in the discrete case only), examples. UMVU estimation, Rao-Blackwell theorem, Fisher Information, Cramer-Rao inequality and Bhattacharya bounds. Completeness and Lehmann- Scheffe theorem. Median and modal unbiased estimation.

• Methods of estimation : method of moments and maximum likelihood method, examples. Properties of MLE. Consistency and asymptotic normality of the consistent solutions of likelihood equations. Definition of CAN and BAN, estimation and their properties, examples. Interval estimation, confidence level CI using pivots and shortest length CI. Confidence intervals for the parameters for Normal, Exponential, Binomial and Poisson Distributions.

• Fundamental notions of hypothesis testing-Statistical hypothesis, statistical test, Critical region, types of errors, test function, randomized and non-randomized tests, level of significance, power function, Most powerful test, Neyman –Pearson fundamental lemma. MLR families and Uniformly most powerful tests for one parameter exponential families.

• Concepts of consistency, unbiased and invariance of tests. Likelihood Ratio tests, statement of the asymptotic properties of LR statistics with applications (including homogeneity of means and variances). Relation between confidence interval estimation and testing of hypothesis. Concept of robustness in estimation and testing with example.

• Concept of sequential estimation, sequential estimation of a normal population.

Notions of sequential versus fixed sample size techniques. Wald’s sequential probability Ratio test (SPRT) procedure for testing simple null hypothesis against simple alternative. Termination property of SPRT. SPRT procedures for Binomial, Poisson, Normal and Exponential distributions and associate OC and ASN functions. Statement of optimality of SPRT.

• Concepts of loss, risk and decision functions, admissible and optimal decision functions, Estimation and testing viewed as decision problems.

• Nonparametric methods : Nonparametric methods for one-sample problems based on sign test, Wilcoxon signed Rank test, run test and Kolmogorov – Smirnov test.

• Two sample problems based on sign test, Wilcoxon signed rank test for paired comparisons, Wilcoxon Mann-Whitney test, Kolmogorov – Smirnov Test, (Expectations and variance of above test statistics, except for Kolmogorov – Smirnov tests, Statements about their exact and asymptotic distributions), Wald- Wolfowitz Runs test and Normal scores test.

• Chi-Square test of goodness of fit and independence in contingency tables. Tests for independence based on Spearman’s rank correlation and Kendall’s Tau. Ansari-Bradley test for two sample dispersions. Kruskal – Wallis test for one-way layour (K-samples). Friedman test for two-way layout (randomised block).

• Asymptotic Relative Efficiently (ARE) and Pitman’s theorem. ARE of one sample,

paired sample and two sample locations tests.

Sampling Techniques

• Non – Sampling errors : Sources and treatment of non-sampling errors. Non –

sampling bias and variance.

• SRSWR / WOR, Stratified random sampling and Systematic Sampling.

• Unequal probability Sampling : ppswr / wor methods (including Lahiri’s scheme)

and related estimators of a finite population mean. Horowitz – Thompson, Hansen

– Horowitz and Yates and Grundy estimators for population mean / total and their variances.

• Ratio Method Estimation: Concept of ratio estimators, Ratio estimators in SRS, their bias, variance / MSE. Ratio estimator in Stratified random sampling – Separate and combined estimators, their variances / MSE.

• Regression method of estimation : Concept, Regression estimators in SRS with pre-assigned value of regression coefficient (Difference Estimator) and estimated value of regression coefficient, their bias, variance / MSE, Regression estimators in Stratified Random sampling – Separate and combined regression estimators, their variance / MSE.

• Cluster Sampling : Cluster sampling with clusters of equal sizes, estimator of mean per unit, its variance in terms of intracluster correlation, and determination of optimum sample and cluster sizes for a given cost. Cluster sampling with clusters of unequal sizes, estimator – population mean its variance / MSE.

• Sub sampling (Two – Stage only) : Equal first stage units – Estimator of population mean, variance / MSE, estimator of variance. Determination of optimal sample size for a given cost. Unequal first stage units – estimator of the population mean and its variance / MSE.

Design of Experiments

• Formulation of a linear model through examples. Estimability of a linear parametric function. Gauss-Markov linear model, BLUE for linear functions of parameters, relationship between BLUE’s and linear Zero-functions. Gauss-Markov theorem.

• Simple linear regression, examining the regression equation, Lack of fit and pure error. Analysis of Multiple regression models. Estimation and testing of regression parameters, sub-hypothesis. Introduction of residuals, overall plot, time sequence plot, plot against Yi, Predictor variables Xij, Serial correlation among the residual outliers. The use of dummy variables in multiple regression, Polynomial regressions – use of orthogonal polynomials. Derivation of Multiple and Partial correlations, tests of hypothesis on correlation parameters.

• Analysis of Covariance : One-way and Two-way classifications. Factorial experiments : Estimation of Main effects, interaction and analysis of 2k, factorial experiment in general with particular reference to k = 2, 3 and 4 and 32 factorial experiment. Multiple Comparisons : Fishers least significance difference (LSD) and Duncan’s Multiple Range test (DMR test).

• Total and Partial Confounding in case of 23, 24 and 32 factorial designs. Concept of balanced partial confounding. Fractional replications of factorial designs : One half replications of 23 and 24 factorial designs, one-quarter replications of 25 and 26 factorial designs. Resolutions of a design. Split – Plot design.

• Youdin design, intra block analysis. B.I.B.D., P.B.I.B.D., their analysis, estimation of parameters, testing of hypothesis.

**Statistics subject**Degree Lecturers 2017 in Government Degree Colleges in A.P. Collegiate: APPSC has given the Degree College Lecturers Recruitment 2017 notification and online applications are invited online from qualified candidates to the post of Degree College Lecturers in in Govt Degree Colleges in the State of Andhra Pradesh. The proforma Application will be available on Commission’s Website (www.psc.ap.gov.in) from 29/12/2016 to 28/01/2017 (Note: 27/01/2017 is the last date for payment of fee up- to 11:59 mid night). APPSC Degree College Lecturers Recruitment 2017 notification no.26/2016 and apply online now @**http://appscapplications17.apspsc.gov.in/**

__Scheme of Exam:__PART-A: Written ‘Examination (Objective Type) |
|||

Papers |
No. of
Questions |
Duration
(Minutes) |
Maximum
Marks |

Paper-1: General Studies & Mental Ability | 150 | 150 | 150 |

Paper-2: Statistics subject | 150 | 150 | 300 |

PART-B: Interview (Oral Test) |
50 | ||

TOTAL |
500 |
||

NEGATIVE MARKS: As per G.O.Ms. No.235, Finance (HR-I, Plg & Policy) Dept., Dt.
06/12/2016, for each wrong answer will be penalized with 1/3rd
of the marks prescribed for the question. |

**READ | APPSC Degree Lecturers Recruitment**

__Statistics Subject Syllabus:__15.STATISTICS

Probability Theory :

• Random experiment, Random event, Sample Space, Classes of sets, fields, sigma-fields, minimal sigma-fields, Borel sigma fields in R, Measure, Lebesque mesure, Lebesque-Stieltjes measures, Measurable functions, Borel function, induced sigma field, Probability Measure, Basic Properties of a Measure, conditional probability and Bayes Theorem. Caratheodory extension theorem (Statement only), measurable function, random variables, distribution function and its properties, expectation, statements and applications of monotone convergence theorem, Foatou’s lemma, dominated convergence theorem.

• Expectations of functions of rv’s, conditional expectation and conditional variance, their applications. Characteristic function of a random variable and its properties. Inversion theorem, uniqueness theorem (Functions which cannot be Characteristic functions). Levy’s continuity theorem (Statement only). Chebychev, Markov, Cauchy-Schwartz, Jenson, Liapunov, Holder’s and Minkowsky’s inequalities.

• Sequence of Random variables, convergence in Probability, convergence in distribution, almost sure convergence, convergence in quadratic mean and their interrelationships, Slutskey’s theorem, Borel-Cantelli lemma Borel 0-1 law, Kolm ogorov 0-1 law (Glevenko – Cantelli Lemma Statement only).

• Law of large numbers, Weak law of large numbers, Bernoulli and Khintchen’s WLLN’s, Kolomogorov Inequality, Kolmogorov SLLN for independent random variables and statement only for i.i.d. case and their applications, statements of three series theorem. Central Limit theorem : Demoviere – Laplace CLT, Lindberg- Levy CLT, Liapounou’ CLT, Statement of Lindberg-Feller CLT, simple applications.

• Introduction to stochastic processes; classification of stochastic process according to state-space and time-domain. Finite and countable state Markov chains; time- homogeneity; Chapman-Kolmogorov equations; marginal distribution and finite – dimensional distribution; classification of states of a Markov chain – recurrent, positive recurrent, null-recurrent and transient states.

Distribution Theory

• Standard discrete and continuous univariate distributions : Binomial, geometric, Poisson, Negative Binomial, Hyper-geometric, Uniform, Triangular, beta, exponential, gama, Weibull, Normal, Lognormal, and Cauchy distributions and their properties. Joint, Marginal and conditional pmf’s and pdf’s.

• Families of Distributions : Power series distributions, Exponential families of distributions. Functions of Random variables and their distributions (including transformation of rv’s). Bivariate Normal, Bivariate Exponential (Marshall and Olkins form), Compounding distributions using Binomial and Poisson. Truncated (Binomial, Poisson, Normal and Lognormal) and mixture distributions – Definition and examples.

• Sampling Distributions of sample mean and variance, independence of X and s2.

Central and Non-central Ï°2, t and F distributions. Order statistics Joint and marginal distributions of order statistics and Distribution of Range. Distributions of order statistics from rectangular, exponential and normal distributions. Empirical distribution function.

• Multinomial distribution. Multivariate normal, bi-variate as a particular case, moments, c.f., conditional and marginal distributions. Distributions of correlation coefficient, partial and multiple correlations, and inter relationships. Dimension reduction method : PCA, FA, Canonical Correlations an MDS. Discriminent analysis and cluster Analysis.

• Distributions of quadratic forms under normality and related distribution theory.

Statistical Inference :

• Point Estimation : Point Estimation Vs. Interval Estimation, Advantages, Sampling distribution, Likelihood function, exponential family of distribution. Desirable properties of a good estimator : Unbiasedness, consistency, efficiency and sufficiency – examples. Neyman factorization theorem (Proof in the discrete case only), examples. UMVU estimation, Rao-Blackwell theorem, Fisher Information, Cramer-Rao inequality and Bhattacharya bounds. Completeness and Lehmann- Scheffe theorem. Median and modal unbiased estimation.

• Methods of estimation : method of moments and maximum likelihood method, examples. Properties of MLE. Consistency and asymptotic normality of the consistent solutions of likelihood equations. Definition of CAN and BAN, estimation and their properties, examples. Interval estimation, confidence level CI using pivots and shortest length CI. Confidence intervals for the parameters for Normal, Exponential, Binomial and Poisson Distributions.

• Fundamental notions of hypothesis testing-Statistical hypothesis, statistical test, Critical region, types of errors, test function, randomized and non-randomized tests, level of significance, power function, Most powerful test, Neyman –Pearson fundamental lemma. MLR families and Uniformly most powerful tests for one parameter exponential families.

• Concepts of consistency, unbiased and invariance of tests. Likelihood Ratio tests, statement of the asymptotic properties of LR statistics with applications (including homogeneity of means and variances). Relation between confidence interval estimation and testing of hypothesis. Concept of robustness in estimation and testing with example.

• Concept of sequential estimation, sequential estimation of a normal population.

Notions of sequential versus fixed sample size techniques. Wald’s sequential probability Ratio test (SPRT) procedure for testing simple null hypothesis against simple alternative. Termination property of SPRT. SPRT procedures for Binomial, Poisson, Normal and Exponential distributions and associate OC and ASN functions. Statement of optimality of SPRT.

• Concepts of loss, risk and decision functions, admissible and optimal decision functions, Estimation and testing viewed as decision problems.

• Nonparametric methods : Nonparametric methods for one-sample problems based on sign test, Wilcoxon signed Rank test, run test and Kolmogorov – Smirnov test.

• Two sample problems based on sign test, Wilcoxon signed rank test for paired comparisons, Wilcoxon Mann-Whitney test, Kolmogorov – Smirnov Test, (Expectations and variance of above test statistics, except for Kolmogorov – Smirnov tests, Statements about their exact and asymptotic distributions), Wald- Wolfowitz Runs test and Normal scores test.

• Chi-Square test of goodness of fit and independence in contingency tables. Tests for independence based on Spearman’s rank correlation and Kendall’s Tau. Ansari-Bradley test for two sample dispersions. Kruskal – Wallis test for one-way layour (K-samples). Friedman test for two-way layout (randomised block).

• Asymptotic Relative Efficiently (ARE) and Pitman’s theorem. ARE of one sample,

paired sample and two sample locations tests.

Sampling Techniques

• Non – Sampling errors : Sources and treatment of non-sampling errors. Non –

sampling bias and variance.

• SRSWR / WOR, Stratified random sampling and Systematic Sampling.

• Unequal probability Sampling : ppswr / wor methods (including Lahiri’s scheme)

and related estimators of a finite population mean. Horowitz – Thompson, Hansen

– Horowitz and Yates and Grundy estimators for population mean / total and their variances.

• Ratio Method Estimation: Concept of ratio estimators, Ratio estimators in SRS, their bias, variance / MSE. Ratio estimator in Stratified random sampling – Separate and combined estimators, their variances / MSE.

• Regression method of estimation : Concept, Regression estimators in SRS with pre-assigned value of regression coefficient (Difference Estimator) and estimated value of regression coefficient, their bias, variance / MSE, Regression estimators in Stratified Random sampling – Separate and combined regression estimators, their variance / MSE.

• Cluster Sampling : Cluster sampling with clusters of equal sizes, estimator of mean per unit, its variance in terms of intracluster correlation, and determination of optimum sample and cluster sizes for a given cost. Cluster sampling with clusters of unequal sizes, estimator – population mean its variance / MSE.

• Sub sampling (Two – Stage only) : Equal first stage units – Estimator of population mean, variance / MSE, estimator of variance. Determination of optimal sample size for a given cost. Unequal first stage units – estimator of the population mean and its variance / MSE.

Design of Experiments

• Formulation of a linear model through examples. Estimability of a linear parametric function. Gauss-Markov linear model, BLUE for linear functions of parameters, relationship between BLUE’s and linear Zero-functions. Gauss-Markov theorem.

• Simple linear regression, examining the regression equation, Lack of fit and pure error. Analysis of Multiple regression models. Estimation and testing of regression parameters, sub-hypothesis. Introduction of residuals, overall plot, time sequence plot, plot against Yi, Predictor variables Xij, Serial correlation among the residual outliers. The use of dummy variables in multiple regression, Polynomial regressions – use of orthogonal polynomials. Derivation of Multiple and Partial correlations, tests of hypothesis on correlation parameters.

• Analysis of Covariance : One-way and Two-way classifications. Factorial experiments : Estimation of Main effects, interaction and analysis of 2k, factorial experiment in general with particular reference to k = 2, 3 and 4 and 32 factorial experiment. Multiple Comparisons : Fishers least significance difference (LSD) and Duncan’s Multiple Range test (DMR test).

• Total and Partial Confounding in case of 23, 24 and 32 factorial designs. Concept of balanced partial confounding. Fractional replications of factorial designs : One half replications of 23 and 24 factorial designs, one-quarter replications of 25 and 26 factorial designs. Resolutions of a design. Split – Plot design.

• Youdin design, intra block analysis. B.I.B.D., P.B.I.B.D., their analysis, estimation of parameters, testing of hypothesis.

**Exam pattern, Syllabus, Instructions for APPSC Degree Lecturers 2017**

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