Publications by Jake
Binomial/Poisson Survival Rates
Binomal and Poisson Jake Warby 02/04/2022 Binomial Model Binomial without Censoring We observe \(N_x\) independent lives exactly aged \(x\) at the beginning of the year for one whole year We observe \(d_x\) deaths Each life has a probability \(q_x\) of death over that year (inital rate of mortality) Then the random variable \(D_x\), the number...
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Mortality Coefficients
Mortality Projection Variables Jake 01/05/2022 Lee Carter \(\alpha_x\) captures the age specific pattern of mortality \(\kappa_t\) captures the overall mortality trend \(\beta_x\) captures the differences in improvements in different ages \[ \ln(m_{xt})=\alpha_x+\beta_x\kappa_t\] Cairns-Blake-Dowd \(\kappa_t^{(1)}\) captures the intercept, w...
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Exposed To Risk
Exposed to Risk Jake 27/04/2022 Central Exposed to Risk This module focuses on the estimation of inital exposed to risk, which we can then use to approximate the inital exposed to risk (using the uniform distribution of deaths assumption): \[ E_x \approx E_x^c + \frac{1}{2}d_x\] Easy to calculate with complete data, but it is often complicate...
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Mortality Projection
Mortality Projection Jake 27/04/2022 Main Demographical Trends Expansion over time Regularization overtime Increasing trend over time in life expectancy Downward trend over time in death rates Dynamic Life Table Consider the mortality rates depending on both age and calender year \(q_x(t)\). Can read dynamic life tables in different ways: D...
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Non Parametric Survival
Non-Parametric Survival Estimation Jake Warby 31/03/2022 Parametric vs Non-Parametric Non-parametric models make no prior assumption about the shape or form of the underlying distribution. This leads to a relatively less smooth estimate but has no model risk Parametric models make an assumption about the shape or form of the underlying distr...
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Continuous Markov
Markov Models ACTL3141 Jake Warby 02/04/2022 Markov Model Notes Markov Processes can be used to model a random process with multiple steps Markov property governs the markov process, which states that only the most recent information (state) is needed for conditional probabilities \[ P(X_t\in A| \{X_u\}_{u\leq s}) = P(X_t\in A|X_s)\] Transiti...
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Graduation Tests Sheet
Graduation Test Cheat Sheet Jake 01/05/2022 Chi Squared Test \(H_0\): Observed number of deaths at each age are consistent with expected number predicted by graduated rates Test statistic: \[ X = \sum_x z_x^2\sim\chi^2_{n-p} \] General goodness of fitness test Does not tell us about direction of any bias or grouping of directional deviations....
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