Exam MAS-I

The Modern Actuarial Statistics I exam is a 4-hour computer-based CAS exam covering Stochastic Processes, Survival Models, Statistics, and Extended Linear Models as part of the ACAS credentialing pathway.

Prerequisite knowledge

The CAS Syllabus of Basic Education says MAS-I assumes prior knowledge from Exams 1 and 2. The CAS credits both through another actuarial society's equivalent exam, such as the SOA's Exams P and FM:

• Exam 1 (Probability) — Probability, Random Variables, Expected Value, Variance, Conditional Probability, and the named distributions
• Exam 2 (Financial Mathematics) — Interest and Present Value, which Struppeck's note uses to discount insurance and annuity payments

Learning Objectives

A. Probability Models (Stochastic Processes and Survival Models) (20–30%)

Candidates should be able to solve problems using Stochastic Processes and determine the probabilities and distributions associated with these processes.

1. Model claim frequencies using Poisson Processes
2. Calculate expected values, variances, and probabilities for any Poisson Process
3. Calculate Limited Expected Value
4. Perform Survival Model and Hazard Rate calculations
5. Perform Joint Life calculations
6. Calculate simple whole life or annuity problems

Poisson Processes

• Poisson Process — counts, interarrival times, thinning, and superposition
• Nonhomogeneous Poisson Process — a rate \lambda(t) that changes over time
• Compound Poisson Process — a random number of claims of random size
• Mixed Poisson Process — a random rate across the portfolio
• Distributions that arise: Poisson Distribution, Exponential Distribution, Gamma, Negative Binomial Distribution, Uniform Continuous Distribution

Markov Chains

• Markov Chain — the transition probability matrix, n-step transitions, and the stationary distribution (long-run proportions and limiting probabilities)
• Recurrent and Transient States — the classification of states
• Gambler's Ruin Problem
• Mean Time Spent in Transient States
• Branching Processes
• Time-Reversible Markov Chains
• Actuarial use: bonus-malus systems

Reliability Theory

• System Reliability — structure functions, minimal path and minimal cut sets, systems of independent components, system life as a function of component lives, and the expected system lifetime
• Inclusion-Exclusion Principle — bounds on the reliability function
• Combinatorics — counting the path and cut sets of a k-out-of-n system

Simulation

• Inversion Method — using the CDF to simulate a draw by inversion (the inverse transformation method)
• Rejection Method

Survival Models and Life Contingencies

• Survival Model and the Hazard Rate (force of mortality)
• Life Table — the tabular form of a survival model
• Joint Life statuses
• Whole Life Insurance and the Life Annuity — valued as an actuarial Present Value
• Limited Expected Value

Readings: Daniel · Ross · Struppeck

B. Statistics (20–30%)

Candidates should be able to apply the concepts typically covered in the second semester of a two-semester undergraduate sequence in Probability and Statistics.

1. Estimate the mean and variance given a sample
2. Estimate a Sufficient Statistic for a distribution
3. Test statistical hypotheses, including Type I and Type II errors
4. Test means and variances using critical values from a Sampling Distribution
5. Model insurance claim Frequency and Severity
6. Model insurance claims in aggregate
7. Calculate Order Statistics of a sample
8. Perform point estimation of statistical parameters using Maximum Likelihood Estimation (MLE) applying criteria to estimates such as Consistency, Unbiasedness, Sufficiency, Efficiency, Minimum Variance, Mean Square Error (e.g., accounting for Censoring and Truncation in the data)
9. Adjust calculations for the effect of missing data values, including Censoring and Truncation

Summarizing a Sample

• Sample Mean and Sample Variance, and the Standard Deviation
• Covariance between two variables
• Order Statistics and percentiles
• Cumulative Distribution Function (CDF)
• Sampling Distribution and the Central Limit Theorem

Distribution Theory

• Transformations of Random Variables — of one random variable and transformations for several random variables
• Linear Combinations of Random Variables
• Mixture Distributions, including mixtures of discrete and continuous type distributions
• Convergence in Probability

Estimation

• Maximum Likelihood Estimation, the Method of Moments, and Percentile Matching
• Sufficient Statistic and Sufficiency
• Rao-Blackwell Theorem — the properties of a sufficient statistic, and completeness and uniqueness
• Fisher Information and the Cramér–Rao lower bound
• Sampling models the estimators are built on, notably the Normal Distribution
• Criteria: Unbiasedness, Bias, Consistency, Efficiency, Minimum Variance, Mean Square Error
• Empirical Distribution and kernel estimation of the probability density function
• Incomplete data: Censoring and Truncation

Testing

• Hypothesis Testing, Type I Error, Type II Error, and the Power of a Test
• p-Value and the Confidence Interval
• Neyman-Pearson Lemma — most powerful tests and uniformly most powerful tests
• Likelihood Ratio Test
• Chi-Square Goodness-of-Fit Test and the Kolmogorov-Smirnov Test

Loss Models

• Frequency and Severity
• Claim-severity distributions: Exponential Distribution, Gamma, Weibull Distribution, and Pareto Distribution
• Tail Weight — the tail properties of claim severity
• Aggregate Loss Models
• Coverage modifications, notably the Deductible

Readings: Hogg, McKean, and Craig · Tse

C. Extended Linear Models (45–55%)

Candidates should be able to solve problems using extended linear models and determine when these models are appropriate to use.

1. Select the appropriate model for an extended linear model
2. Select the appropriate model structure for an extended linear model given the behavior of the data set to be modeled (e.g., appropriate Link Function and distribution for the dependent variable for GLM)
3. Evaluate models developed using an extended linear model approach
4. Interpret the extended linear model output from statistical software, such as parameter estimate tables and ANOVA tables
5. Distinguish among categorical, ordinal, and continuous predictors and their interactions, and how these relate to their usage in an extended linear model
6. Understand and apply control and offset variables in GLMs
7. Understand and calculate AIC, BIC, Deviance, and R-Squared
8. Analyze model diagnostic plots (e.g., residual, marginal model, QQ, and added variable plots) to assess quality of fit
9. Interpret Exploratory Data Analysis plots for various data types (e.g., box, univariate, histograms)

Model Families

• Linear Regression — the ordinary least-squares starting point
• Analysis of Covariance
• Linear Algebra — the design matrix, the hat matrix H = X(X^TX)^{-1}X^T, and fitted values
• Generalized Linear Model and the Exponential Family
• Logistic Regression for a binary response
• Nominal (multinomial) logistic regression and ordinal logistic regression
• Poisson Regression for claim counts
• Log-Linear Models for contingency tables
• Tweedie Distributions, whose variance is a power of the mean
• Quasi-Likelihood for overdispersion
• Naive Bayes and Quadratic Discriminant Analysis — generative models for classification

Moving Beyond Linearity

• Polynomial Regression and Step Functions
• Regression Splines and Smoothing Splines
• Local Regression
• Generalized Additive Models

Specifying the Model

• Model Structure, predictor types, and interactions
• Link Function and the Dispersion Parameter
• Offset and control variables
• Multicollinearity among predictors
• Regression Through the Origin

Evaluating the Model

• Deviance, R-Squared, and the Residual Sum of Squares
• AIC and BIC
• ANOVA and Parameter Estimate Tables
• Multiple Comparisons
• Chi-Squared Distribution, Student's t-Distribution, and F-Distribution — distributions related to the Normal distribution
• Noncentral chi-squared and F-distributions
• Supervised Learning and the Test Error
• Variable Selection, Cross-Validation, and the Bias-Variance Tradeoff
• Bootstrap

Shrinkage and Dimension Reduction

• Regularization — ridge regression and the lasso
• Principal Components Regression and Partial Least Squares — dimension reduction methods, with an overview of principal components analysis
• High-Dimensional Data

Diagnostic Plots

• Residuals and the Residual Plot
• QQ Plot
• Marginal Model Plot
• Added Variable Plot

Exploratory Data Analysis

• Exploratory Data Analysis
• Histogram
• Box Plot
• Univariate Plot
• Scatter Plot and Correlation

Readings: Dobson and Barnett · Hogg, McKean, and Craig · James et al. · Larsen

Source Material

Source Material

• Poisson Processes and Mixture Distributions (Daniel - 2008)
  • A1–A5 — CAS Study Note, June 2008; the practice problems noted in its Foreword are on the CAS website
• An Introduction to Generalized Linear Models (Dobson - 2018)
  • C1–C9 — 4th edition, Chapters 1–9, excluding 6.3.3, 6.7, 6.8 and 7.9
• Introduction to Mathematical Statistics (Hogg et al. - 2018)
  • B1–B8, C1–C9 — 8th edition. Domain B: Sections 1.7, 2.7, 2.8, 3.7, 4.1, 4.4, 4.5, 4.6, 4.7, 5.1, 6.1–6.5, 7.1–7.6 and 8.1–8.3. Domain C: Sections 4.4, 6.3, 6.5, 7.5 and 9.1–9.6
• An Introduction to Statistical Learning (James et al. - 2021)
  • C1–C9 — 2nd edition, Chapters 1, 2 (excluding Section 2.2.3), 3 (Sections 3.1 through 3.3, and the 3.6 labs on Sections 3.1 through 3.3), 4 (Sections 4.1 through 4.4, though the Linear Discriminant Analysis portion of Section 4.4 will not be tested), 5, 6 and 7
• Generalized Linear Models (Larsen - 2015)
  • C1–C9 — CAS Study Note, December 2015, revised June 2016
• Introduction to Probability Models (Ross - 2019)
  • A1–A6 — 11th, 12th or 13th edition (2014, 2019 or 2023), Sections 4.1–4.8, 5.2.1–5.2.4, 5.3, 5.4.1, 5.4.2, 9.1–9.6, 11.1, 11.2.1 and 11.2.2
• Life Contingencies (Struppeck - 2014)
  • A5–A6 — CAS Study Note, October 2014, revised September 2015
• Nonlife Actuarial Models (Tse - 2009)
  • B1–B4, B7–B9 — Sections 1.1, 2.2, 2.3, 2.4, 2.5, 10.1, 10.2, 11.1, 12.1, 12.3 and 13.2