CS2 - Part 3

Question 1

General formula for Cox proportional hazard (PH) model

Answer 1

Question 2

Ratio of hazards of lives with covariate vectors z1 and z2 (Cox PH model)

Answer 2

Question 3

Proportional hazards model: Likelihood estimator for beta vector

Answer 3

Question 4

Aims of graduation

Answer 4

• Produce smooth set of rates that are suitable for a particular purpose
• Remove random sampling errors
• Use the information available from adjacent ages

Question 5

Desirable features of graduation

Answer 5

• Smoothness
• Adherence to data
• Suitability to purpose to hand

Question 6

Degrees of freedom for Xi-Squared test

Answer 6

• Start with the number of groups
• If the groups form a set of mutually exclusive and exhaustive categories (probabilities add up to 1), subtract 1
• Subract further 1 for each parameter that has been estimated

Question 7

Distributions of D_x and mu~x

Answer 7

Question 8

Mortality experience: Deviation

Answer 8

Question 9

Mortality experience: Standardised deviation

Answer 9

Question 10

Degrees of freedom when comparing an experience with a standard table

Answer 10

Degrees of freedom = number of age groups

Question 11

Xi-squared failures: Standardised deviations test

Answer 11

To detect a few large deviations that the Xi-square test did not detect

Check if standardised deviations of mortality are following the standard normal distribution with Xi-Squared test

Question 12

Xi-squared failures: Signs test

Answer 12

To detect imbalance between negative and positive deviations

Binomial distribution

N number of negative deviations:

Check that 2*P(N <= x) > 5%

P number of positive deviations:

Check that 2*P(P >= x) > 5%

Question 13

Xi-squared failures: Cumulative deviations

Answer 13

Question 14

Xi-squared failures: Grouping of signs test

Answer 14

Detects ‘clumping’ of devations with the same sign.

Check ‘Grouping of signs test’ in tables.

If number of groups of positive (or negative) runs is lower or equal than the test statistic, we can reject the null hypothesis.

Question 15

Testing smoothness of graduation

Answer 15

Third difference (change in curvature) of the graduated quantities should

• Be small in magnitude compared with the quantities themeselves
• Progess regularly

Question 16

Methods of graduation

Answer 16

• Graduation by parametric formula
  
  • a1 + a2 exp(a3x + a4x^2+…)
  • well-suited to the production of standard tables from large amounts of data
• Graduation by reference to standard table
  
  • (a+bx) mu_x^s
  • Can be used to fit relatively small data sets where a suitable standard table exists
• Gradution using spline functions
  
  • Method is suitable for quite small experiences as well as very large experiences.

Question 17

Morality projection - Method based on expectation

Answer 17

Question 18

Autocovariance function

Answer 18

Question 19

Simplify:

Answer 19

Question 20

Autocorrelation function

Answer 20

Question 21

Correlation formula

Answer 21

Question 22

Autoregressive process of order p

AR(p)

Answer 22

Question 23

Moving average process of order q

MA(q)

Answer 23

Question 24

Autoregressive moving average

ARMA(p,q)

Answer 24

Question 25

Condition for stationarity of AR(p) process

Answer 25

Question 26

Conditions for invertibility of MA processes

Answer 26

Invertibility: White noise process e can be written explicitly in terms of X process

Question 27

Moving average model MA(q), in backwards shift notation

Answer 27

Question 28

ARMA(p,q) process defined in Backward operation notation

Answer 28

Question 29

Definition of an ARIMA process

Answer 29

Question 30

Features of MA(q) process

Answer 30

Question 31

Features of AR(p) process

Answer 31

Question 32

Features of ARMA (p,q) process

Answer 32

Question 33

Three possible causes of non-stationarity

Answer 33

1. Deterministic trend (e.g. exponential or linear growth) 2. Deterministic cycle (e.g. seasonal effect) 3. Time series is integrated

Question 34

Methods for compensating for trend/seaonality (6)

Answer 34

* Least squares trend removal (Tables p.24) * Differencing * Differencing d times will not only make I(d) series stationary but will also remove linear trend * Seasonal differncing * E.g. differencing 12 times for annual seasonality * Method of moving averages * Create transformation such that transformed time series is moving average of original time series * Method of seasonal means * Transformation of the data * E.g. take log

Question 35

Check if observed time series is stationary

Answer 35

Autocorrelation function should converge to 0 exponentially

Question 36

Identification of white noise

Answer 36

Option 1: * Check if values of the SACF or SPACF fall outside the range of * +-2/sqrt(n) --\> Approximated from +-1.96/sqrt(n) * Note that there is a chance of 1/20 that one value will fall out of the range (95% quantile) Option 2: * Portmanteau test (tables p. 42)

Question 37

Identification of MA(q)

Answer 37

Question 38

Identification of AR(q)

Answer 38

Question 39

Identification of appripriate order of differencing (d) of sample data

Answer 39

* Slowly decaing sample autocorrelation function indicates time series need to be differenced * Look for smallest sample variance for d=1,2,3,...

Question 40

Diagnostic checking for fitted ARIMA model

Answer 40

Question 41

Condition for stationarity of vector autoregressive process

Answer 41

Question 42

Calculate eigenvalues of matrix A

Answer 42

Values lambda, such that det (A-lambda\*I) = 0

Question 43

Two time series processes X and Y are called cointegrated if:

Answer 43

* X and Y are I(1) random processes * there exists a non-zero vector (a,b) such that aX+bY is stationary The vector (a,b) is called cointegration vector.

Question 44

Moment generating function (formula)

Answer 44

Question 45

Cumulant generating function

Answer 45

Question 46

Coefficient of skewness

Answer 46

Question 47

Kurtosis

Answer 47

* Fourth standardised moment * kurtosis = 3: mesokurtic (normal distribution) * kurtosis \>3 leptokurtic * more peaked, fatter tail * kurtosis \<3 platykurtic * broader peak, more slender tails

Question 48

Standardised moment

Answer 48

Question 49

Varying volatility over time

Answer 49

heteroscedacity

Question 50

Central limit theorem

Answer 50

Question 51

Generalized extreme value distribution

Answer 51

Question 52

GEV distributions: Different values of shape parameter gamma

Answer 52

Question 53

Rough criteria to chose family of GEV distributions

Answer 53

Question 54

Distribution of excess above u

Answer 54

Question 55

kth moment of a continuous positive-valued distribution with density function f(x)

Answer 55

Question 56

Measures of tail weight

Answer 56

Question 57

Coefficient of upper tail dependence

Answer 57

Question 58

Coefficient of lower tail dependence in terms of the copula function

Answer 58

Question 59

Coefficient of upper tail dependence in terms of the copula function

Answer 59

Question 60

Fundamental copulas

Answer 60

Question 61

Graphical representation of independence copula

Answer 61

Question 62

Graphical representation of comonotonous copula

Answer 62

Question 63

Graphical representation of counter-monotonic copula

Answer 63

Question 64

Gumbel copula

Answer 64

* Upper tail dependence determined by parameter alpha * No lower tail dependence

Question 65

Clayton copula

Answer 65

* Lower tail dependence determined by alpha * No upper tail dependence

Question 66

Frank copula

Answer 66

* Interdependence structure in which there is no upper or lower tail dependence

Question 67

Gaussian copula

Answer 67

Question 68

Archimedean copula

Answer 68

Question 69

Student's t copula

Answer 69

Question 70

Tail dependence of all copulas

Answer 70

Question 71

PDF of the reinsurer's claim amount under XOL with retention M

Answer 71

Question 72

Variance, mean and skewness of compound poisson process with parameter lambda

Answer 72

Question 73

Coefficient of skewness of compound poisson distribution

Answer 73

Question 74

Sum of independent compound Poisson random variables

Answer 74

Question 75

n choose k

Answer 75

Question 76

Machine Learning: Confusion matrices

Answer 76

Question 77

Machine Learning: Hyperparameters

Answer 77

Variables external to the model whose values are set in advance by the user. They are chosen based on the user’s knowledge and experience in order to produce a model that works well.

Question 78

Machine Learning: Parameters

Answer 78

variables internal to the model whose values are estimated from the data and are used to calculate predictions using the model.

Question 79

Machine Learning: Regularisation or penalisation

Answer 79

Question 80

Branches of Machine Learning

Answer 80

Question 81

Machine Learning: Stages of analysis

Answer 81

Question 82

Machine Learning: Data Types

Answer 82

Question 83

Machine Learning: Train-Validate-Test approach

Answer 83

Split data into * data for training (60%) * data for validation (20%) * data for testing (20%)

Question 84

Machine Learning: Requirements for analysis to be reproducible

Answer 84

* Data used should be fully described and available to other researchers * Any modification to the data should be clearly described * Selection of the algorithm and the development of the model should be desribed (including parameters and why they are chosen) * Ideally would provide computer code used * Specify seed value

Question 85

Machine Learning: Penalised general linear models

Answer 85

Maximize penalized likelihood

Question 86

Machine Learning: Naive Bayes Classification

Answer 86

Question 87

Machine Learning: Gini index of a final node in a decision tree

Answer 87

Question 88

Machine Learning: Gini index of a decision tree

Answer 88

Question 89

Machine Learning: K-means clustering advantages and disadvantages

Answer 89