# Virtual University of Pakistan

## STA631 : Inferential Statistics

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### Course Info

#### Course Category

Probability & Statistics

3

N/A

#### Instructor

Post Doctorate
École Polytechnique Fédérale de Lausanne (EPFL), Switzerland

### Course Contents

Statistical Inference and its importance Introduction to random sampling Statistic with examples Methods of drawing samples Sums of the random variables from a random sample Some results about Random Variable Moment generating function Exact sampling distribution: Chi-square distribution Exact sampling distribution: Normal distribution Exact sampling distribution: t-dsitribution Exact sampling distribution: F-dsitribution Application of Chi-Square t &F distribution Order statistics Joint distribution of ith and jth order statistics Application(joint distribution ) Chebychev's Inequality and its usage Convergence in probability Week law of large number Almost sure convergence Strong law of large number Convergence in distribution Central limit theory Generating a random sample Direct method with example Indirect method with example Direct Method:application Indirect Method:application Delta Methods Point estimation Methods of point estimation Likelihood function Regularity conditions Maximum Likelihood (ML) method MLE:Applications Regularity conditions:Assumptions Fisher Information Information for a location family Application(Fisher Information) Application(Chi-Sqaure) Properties of ML Theorem(MLE) Newton Raphson's method Method of Moments Applications of Method of Moments Properties of moment estimator Bayesian Method Conjugate Prior Other types of prior Application(Posterior dist) Application(Jeffreys prior ) Application(Bayes estimate ) Bayesian predictive distribution Properties of point estimate Unbiasdness Application(Unbiasedness) Consistency Efficiency Applications of Efficiency Asymptotically efficiency Mean Square error ApplicationQ7.12:MSE Best unbiased estimator Cramer-Rao Inequality Applications (Rao-Cramer lower bound) MLE are asymptotically efficent Sufficiency Applications of Sufficiency Neyman-Fisher factorization theorem Properties of sufficient statistics MVUE Sufficiency and Unbiasedness Rao-Blackwell theorem Uniqueness of Unbised estimator Continue(uniqueness of unbised estimator) UMVUE in Multiparameter Case Completeness Applications :Completeness Lehman-Scheffe theorem Applications: MVUE Applications: Completeness Application: MVUE Exponential families Theorem:Sufficiency Jointly sufficient Application:Jointly sufficient Minimal sufficiency Ancillary Statistics Location problems Scale problems Location-Scale problems Bayesian point estimate Application:Bayes Estimate Unbiasdness Bayes estimates Testing of Hypothesis Basic of testing p-value Best critical region Neyman-Pearson lemma Applications: Neyman-Pearson lemma Uniformly most powerful tests Important Remarks Likelihood ratio tests Likelihood Ratio Test for the exponential distribution Wilks Theorem Applications: Wilks Theorem LR for Binomial Proportion Sequential probability ratio test Application:Sequential probability ratio test Bootstrap testing procedures Bayesian Tests Interval Estimation Confidence interval for mean when population variance is known Confidence interval for mean when population variance is unknown Confidence interval for population variance when population mean is unknown Confidence interval for population proportion Application:Confidence interval for population proportion Bayesian interval