Research PhDTopics for the Statistics Curriculum

Topics for the Statistics Curriculum

Topics related to the basic preparation of the candidate

 

Data analysis

• Types of statistical survey
• Units and variables
• Classifications of variables
• Bar charts and histograms
• Empirical distribution function
• Median and quantiles
• Mean and properties
• Variance and properties
• Outliers and robustness

 

Elementary time series analysis

• Measures of change in a time series
• Simple index numbers
• Complex index numbers

 

Multivariate data analysis

• Scatter and scatter matrix
• Covariance, correlation and properties
• Covariance and correlation matrix and properties
• Least squares line
• Missing data
• Principal components
• Double and multiple contingency tables
• Conditional distributions
• Association indices for 2x2 tables
• Sample and structural zeros

 

Probability distributions

• Probability spaces
• Random variables (Binomial, Hypergeometric, Poisson, Uniform, Exponential, Normal)
• Expected value and moments from the origin
• Variance and central moments
• Indices of position, variance, skewness and kurtosis
• Chebyshev's inequality
• Law of large numbers
• Central limit theorem
• Multivariate distributions
• Mutivariate normal
• Multinomial

 

Sampling from finite populations

• Terminology: observation unit, target population, sampling unit, sampling frame
• Selection bias
• Measurement errors
• Sampling and non-sampling errors
• Random samples
• Simple random sampling
• Stratified sampling
• Cluster sampling

 

Plan of experiments

• Terminology of the plan of experiments: experimental combination, replication, randomisation, source of variability
• Role of factors
• Complete factor design
• Concept of interaction

 

Statistical models

• Parametric and non-parametric models
• Independent and identically distributed observations
• The normal model
• The Bernoulli model
• Independent but not identically distributed observations
• The analysis of variance model
• The simple linear regression model

 

Estimation methods

• Empirical distribution function
• Quantile-quantile graphs
• Method of moments
• Method of least squares
• Method of maximum likelihood

 

Frequentist inference

• Statistics and estimators
• Sample distributions in the normal model
• Non-bias
• Mean square error and its breakdown
• Efficiency and accuracy

 

Elementary asymptotic theory

• Consistency
• Asymptotic normality
• Asymptotic properties of maximum likelihood estimators and likelihood ratios

 

Confidence intervals

• Estimation intervals and coverage probabilities
• Pivot quantity method
• Likelihood intervals

 

Bayesian inference

• Conditional probability
• The generalised Bayes formula
• A priori parameter distributions
• A posteriori distribution
• Credibility intervals
• Comparison of confidence intervals and credibility intervals

 

Hypothesis testing

• Neyman-Pearson approach
• Test level
• Test statistics and level of significance of data (p-value)
• Uniformly most powerful tests
• Tests and confidence intervals
• Chi-square test of independence

 

Regression models

• Multiple linear regression
• Estimators of regression coefficients and standard errors
• Deviance analysis
• Future value prediction
• Residue diagnostics
• Weighted least squares
• Selection of variables
• Linear logistic regression model
• Goodness of fit in models with binomial data

 

Applications

• Experimental and observational studies
• Cohort and case-control studies
• Prevalence and incidence
• Risk measures
• Effect measures
• Duration data analysis
• Censored data
• Survival function and its estimation
• Mortality tables
• Risk function (force of mortality, failure rate)
• Diagnostic tests
• Sensitivity and specificity

Bibliography

Baldi, P. (2007). Calcolo delle probabilità. McGraw-Hill

Chiandotto, B. (2014). Inferenza statistica. Dispense, DISIA

Cicchitelli (2015). Statistica: principi e metodi. Pearson

Conti, P. L. & Marella, A. (2012). Campionamento da popolazione finite. Milan: Springer-Verlag Italia

Di Ciaccio, A. & Borra S. (2008). Statistica - metodologie per le scienze economiche e sociali. 2/ed. McGraw-Hill

Di Fonzo, T. Lisi, F. (2005). Serie storiche economiche. Carocci

Härdle, W. & Simar A. (2007). Applied Multivariate Statistical Analysis. Berlin: Springer

Liseo, B. (2008). Introduzione alla statistica bayesiana. Dispense

Montgomery, D. C. (2005). Progettazione e analisi degli esperimenti. McGraw-Hill

Pace, L. & Salvan, A. (2001). Introduzione alla statistica-II. Inferenza, verosimiglianza, modelli. Padua: CEDAM.

Pagano, M. & Gavreau, K. (2003). Biostatistica. Idelson Gnocchi

Santini, A. (2005). Appunti di analisi demografica. Dispense, DISIA.