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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
Q

what does selection of a cumulant function control?

Taylor

A

the model’s assumed error distribution

26
Q

what does selection of an index p control?

Taylor

A

relationship between the model’s mean and variance

27
Q

what are the covariates x_i^T in a GLM?

Taylor

A

the variables that explain mu_i

28
Q

what does selection of a link function control?

Taylor

A

relationship between the mean mu_i and the associated covariates

29
Q

how are parameters of a GLM often estimated?

Taylor

A

using maximum likelihood estimation

30
Q

what are categorical covariates?

Taylor

A

predictors with discrete levels

ex: state

31
Q

what are continuous covariates?

Taylor

A

predictors with continuous levels

ex: age

32
Q

how is goodness-of-fit generally measured for GLMs?

Taylor

A

using scaled deviance - comparing the estimate of the model to the estimate of the saturated model

33
Q

what is a ‘saturated model’?

Taylor

A

a model with a parameter for every observation such that Y_hat = Y

34
Q

what is the deviance formula, in simple words?

Taylor

A
  • we find the difference between the saturated model and the actual model
  • small difference -> fitted values are close to actual values
35
Q

what would we like our residuals to exhibit?

Taylor

A

unbiasedness (revolve around 0) and homoscedasticity (constant variance)

36
Q

what is one consequence of Pearson residuals?

Taylor

A

they will reproduce any non-normality that exists in the observations (ex. skewed loss data)

37
Q

what can be done if a residual plot exhibits heteroscedasticity?

(Taylor)

A

weights can be used to correct it

38
Q

what is the general rule about using weights to correct heteroscedasticity?

(Taylor)

A

observations should be assigned weights that are inversely proportional to the variance of the residuals

39
Q

what are outliers (in terms of residuals)?

Taylor

A

isolated observations with large residuals

40
Q

how might outliers influence the regression?

Taylor

A

they shift the fitted values away from the main body of observations in favor of the outliers

41
Q

how can we remove the influence of outliers?

Taylor

A

assign them weights of 0 in the model

42
Q

what do we need to keep in mind when removing outliers?

Taylor

A

-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
Q

what is condition #1 that the Non-Parametric Mack model (1994) satisfies?

(Taylor)

A
  1. AYs are stochastically independent
44
Q

what is condition #2 that the non-parametric Mack model satisfies? and what does it mean?

(Taylor)

A
  1. 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
Q

what is condition #3a that the non-parametric Mack model satisfies?

(Taylor)

A

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
Q

what is condition #3b that the non-parametric Mack model satisfies?

(Taylor)

A

3b. For each k=1,2,…,K and j=1,2,…,J-1:

Var[X_k,j+1] = sigma_j^2 *X_kj

47
Q

what is result #1 derived from the non-parametric Mack model?

(Taylor)

A

conventional CL estimates f_kj are unbiased AND minimum variance estimators among estimators that are unbiased linear combinations of the f_kj

48
Q

what is result #2 derived from the non-parametric Mack model?

(Taylor)

A

the conventional CL estimator R_k is unbiased

49
Q

why is the non-parametric Mack model stochastic and non-parametric?

(Taylor)

A
  • stochastic because it considers the means AND variances of observations
  • non-parametric because it doesn’t consider the distribution of the observations
50
Q

how does the EDF Mack model change assumptions from the non-parametric Mack model?

(Taylor)

A
  • keeps 1-3a

- replaces 3b with the condition that Y_k,j+1 | X_kj ~ EDF

51
Q

what is the first result of the EDF Mack model, assuming the data is a triangle? (Theorem 3.1)

(Taylor)

A

-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
Q

what is the second result of the EDF Mack model, assuming the data is a triangle?

(Taylor)

A
  • 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
Q

how is Result 2 of the EDF Mack model stronger than Result 1 of the non-parametric Mack model?

(Taylor)

A
  • non-parametric CL estimates were MVUEs among all LINEAR COMBINATIONS of the f_kj
  • EDF estimates are MVUEs of ALL estimators
54
Q

what are the assumptions for the EDF cross-classified model?

Taylor

A
  • 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
Q

how does the EDF cross-classified model differ from the Mack model in terms of parameters?

(Taylor)

A
  • EDF cross-classified model includes explicit row (alpha_k) and column (beta_j) params
  • Mack only includes explicit column (f_j) params
56
Q

what are additional assumptions (beyond the EDF cross-classified assumptions) of the ODP cross-classified model?
(Theorem 3.2)

(Taylor)

A
  • Y_kj is restricted to an ODP distribution

- the dispersion parameters phi_kj are identical for all cells (phi_kj = phi)

57
Q

what is the result of Theorem 3.2?

Taylor

A

-assumptions result in MLE fitted values and forecasts Y_kj that are the same as those given by the conventional CL method

58
Q

what is theorem 3.3?

Taylor

A
  • 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
Q

why are theorems 3.2 and 3.3 more remarkable than theorem 3.1?

(Taylor)

A
  • 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
Q

what observations might be given 0 weight in a GLM?

Taylor

A
  • 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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