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exam 7 > Taylor > Flashcards

Taylor Flashcards

1
Q

what is the purpose of the Taylor paper?

Taylor

A

showcase various stochastic models where the CL reserve happens to be the maximum likelihood forecast of the true loss reserve

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2
Q

what type of model are the stochastic models outlined?

Taylor

A

generalized linear models

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3
Q

what is the probability density function pi(y; theta, phi) of the Exponential Dispersion Family?

(Taylor)

A

ln(pi(y; theta, phi) = [(y*theta - b(theta)) / a(phi)] + c(y, phi)

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4
Q

what does theta represent in the EDF pdf?

Taylor

A

theta is a location parameter called the canonical parameter

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5
Q

what does phi represent in the EDF pdf?

Taylor

A

phi is a dispersion parameter called the scale parameter

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6
Q

what does b(theta) represent in the EDF pdf?

Taylor

A

b(theta) is the cumulant function, which determines the shape of the distribution

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7
Q

what does exp(c(y, theta)) represent in the EDF pdf?

Taylor

A

exp(c(y, theta)) is a normalizing factor producing unit total mass for the distribution

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8
Q

what is the expected value of an EDF distribution?

Taylor

A

E[Y] = mu = b’(theta)

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9
Q

what is the variance of an EDF distribution?

Taylor

A

Var(Y) = a(phi) * b’‘(theta)

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10
Q

what are three examples of EDF distributions?

Taylor

A
  • Poisson
  • binomial
  • gamma
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11
Q

what type of insurance data are the Poisson and binomial distributions useful for modeling?

(Taylor)

A

counts

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12
Q

what type of insurance data is the gamma distribution useful for modeling?

(Taylor)

A

amounts

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13
Q

what is b(theta), a(phi) and c(y, phi) for the Poisson distribution?

(Taylor)

A

b(theta) = exp(theta)
a(phi) = 1
c(y, phi) = -ln(y!)

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14
Q

what restriction on the EDF results in the Tweedie sub-family?

(Taylor)

A

restricting the variance function to:
V(mu) = mu^p
where p <= 0 or p >=1
where mu = [(1-p)*theta]^(1/(1-p))

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15
Q

what is V(mu) represent for the EDF, in terms of b(theta)?

Taylor

A

V(mu) = b’’((b’)^-1(mu))

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16
Q

what distributions does p in 0-3 represent in the Tweedie sub-family?

(Taylor)

A 
p=0: normal distr
p=1: over-dispersed Poisson
p=2: gamma
p=3: inverse Gaussian
1<=p<=2: compound Poisson distr with gamma severity distr
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17
Q

what informs the choice of p in a Tweedie distribution?

Taylor

A

heaviness of the tail indicated by the data: tail heaviness of Tweedie distributions increases as p increases

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18
Q

when might an increase in p be warranted when using the Tweedie distribution?

(Taylor)

A

residuals are more widely dispersed than is consistent with selected model

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19
Q

when is the ODP distribution useful?

Taylor

A

when little is known of the subject distribution

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20
Q

what are the response and linear response of a GLM?

Taylor

A

response: variate Y_i

linear response: x_i^Tbeta

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21
Q

what is the intent of the link function of a GLM?

Taylor

A

transform the mean of each observation into a linear function of the parameter vector beta

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22
Q

what is a weighted linear regression model?

Taylor

A

a standard linear regression where the errors are normally distributed with unequal variances

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23
Q

how would we generalized a weighted linear regression to get a GLM?

(Taylor)

A
  • allow a non-linear relationship between observations and predictors (ie-link function other than identity function)
  • allow non-normal errors
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24
Q

what four components does the selection of a GLM consist of?

Taylor

A

selection of:

  • cumulant function
  • index p
  • covariates x_i^T
  • link function
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25
what does selection of a cumulant function control? | Taylor
the model's assumed error distribution
26
what does selection of an index p control? | Taylor
relationship between the model's mean and variance
27
what are the covariates x_i^T in a GLM? | Taylor
the variables that explain mu_i
28
what does selection of a link function control? | Taylor
relationship between the mean mu_i and the associated covariates
29
how are parameters of a GLM often estimated? | Taylor
using maximum likelihood estimation
30
what are categorical covariates? | Taylor
predictors with discrete levels | ex: state
31
what are continuous covariates? | Taylor
predictors with continuous levels | ex: age
32
how is goodness-of-fit generally measured for GLMs? | Taylor
using scaled deviance - comparing the estimate of the model to the estimate of the saturated model
33
what is a 'saturated model'? | Taylor
a model with a parameter for every observation such that Y_hat = Y
34
what is the deviance formula, in simple words? | Taylor
- we find the difference between the saturated model and the actual model - small difference -> fitted values are close to actual values
35
what would we like our residuals to exhibit? | Taylor
unbiasedness (revolve around 0) and homoscedasticity (constant variance)
36
what is one consequence of Pearson residuals? | Taylor
they will reproduce any non-normality that exists in the observations (ex. skewed loss data)
37
what can be done if a residual plot exhibits heteroscedasticity? (Taylor)
weights can be used to correct it
38
what is the general rule about using weights to correct heteroscedasticity? (Taylor)
observations should be assigned weights that are inversely proportional to the variance of the residuals
39
what are outliers (in terms of residuals)? | Taylor
isolated observations with large residuals
40
how might outliers influence the regression? | Taylor
they shift the fitted values away from the main body of observations in favor of the outliers
41
how can we remove the influence of outliers? | Taylor
assign them weights of 0 in the model
42
what do we need to keep in mind when removing outliers? | Taylor
-if outlier is caused by a major catastrophe or other infrequent but possible event, we need to ensure that the cost of those events is captured somewhere - otherwise, model fails to recognize the possible impact of such an event
43
what is condition #1 that the Non-Parametric Mack model (1994) satisfies? (Taylor)
1. AYs are stochastically independent
44
what is condition #2 that the non-parametric Mack model satisfies? and what does it mean? (Taylor)
2. For each k=1,2,...,K, the X_kj form a Markov chain | - means that X_kj is only dependent on X_k,j-1
45
what is condition #3a that the non-parametric Mack model satisfies? (Taylor)
3a. For each k=1,2,...,K and j=1,2,...,J-1: | E[X_k,j+1] = f_j * X_kj for some parameter f_j > 0
46
what is condition #3b that the non-parametric Mack model satisfies? (Taylor)
3b. For each k=1,2,...,K and j=1,2,...,J-1: | Var[X_k,j+1] = sigma_j^2 *X_kj
47
what is result #1 derived from the non-parametric Mack model? (Taylor)
conventional CL estimates f_kj are unbiased AND minimum variance estimators **among estimators that are unbiased linear combinations of the f_kj**
48
what is result #2 derived from the non-parametric Mack model? (Taylor)
the conventional CL estimator R_k is unbiased
49
why is the non-parametric Mack model stochastic and non-parametric? (Taylor)
- stochastic because it considers the means AND variances of observations - non-parametric because it doesn't consider the distribution of the observations
50
how does the EDF Mack model change assumptions from the non-parametric Mack model? (Taylor)
- keeps 1-3a | - replaces 3b with the condition that Y_k,j+1 | X_kj ~ EDF
51
what is the first result of the EDF Mack model, assuming the data is a triangle? (Theorem 3.1) (Taylor)
-if M3b holds (Var[X_k,j+1 | X_kj] = sigma_j^2 * X_kj), then the MLEs of the f_i are the conventional (unbiased) CL estimators
52
what is the second result of the EDF Mack model, assuming the data is a triangle? (Taylor)
- if we are in the special case of the ODP Mack model AND dispersion parameters phi_kj are just column dependent (phi_kj = phi_j), then: - the conventional CL estimators are MVUEs - cumulative loss estimates X_kj and reserve estimates R_k are also MVUEs
53
how is Result 2 of the EDF Mack model stronger than Result 1 of the non-parametric Mack model? (Taylor)
- non-parametric CL estimates were MVUEs among all LINEAR COMBINATIONS of the f_kj - EDF estimates are MVUEs of ALL estimators
54
what are the assumptions for the EDF cross-classified model? | Taylor
- random variables Y_kj are stochastically independent - for each k=1,2,...,K and j=1,2,...,J: - Y_kj ~EDF(theta_kj,phi_kj,a,b,c) - E[Y_kj] = alpha_k*beta_j, for alpha, beta > 0 - summation[beta_j] from j=1 to J = 1
55
how does the EDF cross-classified model differ from the Mack model in terms of parameters? (Taylor)
- EDF cross-classified model includes explicit row (alpha_k) and column (beta_j) params - Mack only includes explicit column (f_j) params
56
what are additional assumptions (beyond the EDF cross-classified assumptions) of the ODP cross-classified model? (Theorem 3.2) (Taylor)
- Y_kj is restricted to an ODP distribution | - the dispersion parameters phi_kj are identical for all cells (phi_kj = phi)
57
what is the result of Theorem 3.2? | Taylor
-assumptions result in MLE fitted values and forecasts Y_kj that are the same as those given by the conventional CL method
58
what is theorem 3.3? | Taylor
- in general, MLEs Y_kj will not be unbiased - if we assume that the ODP cross-classified model assumptions (Theorem 3.2) apply AND that the fitted values Y_kj and R_k are corrected for bias: then they are MVUEs of Y_kj and R_k
59
why are theorems 3.2 and 3.3 more remarkable than theorem 3.1? (Taylor)
- they state that the forecasts from the ODP Mack and ODP cross-classified models are identical and the same as those from the conventional CL method, despite different formulations - forecasts can be obtained from the ODP cross-classified model without any explicit consideration of its parameters by working as if the model were the ODP Mack model
60
what observations might be given 0 weight in a GLM? | Taylor
- observations prior to the most recent m experience years | - outlier observations that the actuary wants to exclude

exam 7 (21 decks)

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