Questions tagged [actuarial-science]

Questions relating to financial risk; often, but not limited to, insurance. This includes questions regarding stochastic distribution of cash flows, probability of ruin, financial payments above a threshold, etc.

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linear mixed model intrinsic aliasing

A bunch of us are studying for an upcoming actuarial exam and none of us understands item II. doesn't result in intrinsic aliasing. We've calculated the V = ZDZ' +R_1 matrix and as far as we can tell, ...
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Determining climate change effect on mortality rate time series

I am currently conducting research in determining the impact of climate change on mortality. I have got time series of the number of deaths for different age groups which dividing them by their ...
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1 answer
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How to setup the Panjer's recursion correctly?

I have a table of $k=(0,1,2,3,4,5,6)$ and $number=(40544,8082,1205,145,20,3,1)$ I need to fit data by a Compound Poisson-Gamma distribution and then make a discretization and compare results with ...
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Introduce a new variable to GLM

I have this one interview question regarding GLM model and would love some insights into method/product sense/common sense input. -Consider this car insurance pricing model: y (car price) = B1 * ...
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1 vote
1 answer
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Density of cost-per-loss random variable

Given a random variable $X$ with support $(a,b)$ and a value $d\in (a,b)$ (the "deductible"), the cost-per-loss random variable associated to $X$ is given by $$Y=\begin{cases}0&\text{if }...
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3 answers
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How to combine state-level COVID-19 vaccination rates with national demographic data at the individual level?

I’m investigating possible correlations between the COVID-19 vaccination rate in the United States and the results of a long-running survey of scientific personality traits like "Agreeableness&...
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1 answer
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Inferring properties of the sum of R.V.s from the copula

This is not a completely well defined question, so even help making it coherent will be useful. Setting: Suppose I know the marginal distributions of random variables describing the expected losses ...
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1 answer
65 views

calculate survival probability and expected payout

I have data taken from here. Let us look at one specific column qx for female up to the age of 10. qx represents the mortality rate between age x and (x+1), that is the probability that a person aged ...
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3 votes
4 answers
219 views

Covid-19 and life expectancy

A public figure in my home country said the following: "The life expectancy in our country is 82 years. The average age of those who die of covid-19 is also about 82 years. Hence, on average, ...
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Hilbert space of random VECTORS?

I'm trying to break through H. Buehlmann and Gisler's Cource in credibility theory. At the page 183 they write that we have to show that $$ \boldsymbol \mu (\Theta_i)-\widehat{\widehat {\boldsymbol \...
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1 answer
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Independence of censoring time $C$ and event time $T$ for randomised entry to a study

While reading through the textbook 'Modern Applied Statistics With S' by Venables and Ripley, I came across the following paragraph detailing the different types of censoring possible when dealing ...
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solvency capital requirement using copulas in R

I want to prove that using copulas the SCR for solvency 2 can be lower than using the standard formula. In the following code I simulate three different distributions, I calculate the scr and then I ...
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Estimating coefficients of a large categorical variable

I'm trying to fit a GLM model with a categorical variable with 400 categories, and I'd like to reduce the number of categories. There are some categories with a lot of data, and a lot of categories ...
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2 votes
1 answer
108 views

Do good Cross-Validation results imply good QQ-plot results?

In Edward Frees' book Predictive Modeling Applications in Actuarial Science, Volume 2 the first chapter goes over how to build a frequency GLM model (using a Poisson distribution) on sample auto-...
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1 vote
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Hazard rate from Weibull vs. cumulative hazard plot does not match

) My problem is that I have HR for a factor named "Selvrisiko" and I get the output estimates (HR) (Upper bound) (Lower bound) as: but when I plot (Kaplan - Meier) cumulative hazard for the ...
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1 answer
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Obvious demonstration of the posterior density of normal laws in bayesian inference

Given the observations' density $f(X|\theta) = \mathcal{N}(\theta,1)$ and the prior density $f(\theta) = \mathcal{N}(0,\sigma^2)$, it it is said that it is obvious that the posterior density is equal ...
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2 votes
0 answers
61 views

Lee Carter mortality model - OLS (SVD) estimators if errors are heteroscedastic

In the classical Lee-Carter model, central death rates are modelled as follows: $\log(m(x,t)) = a(x) + b(x)\kappa(t) +\varepsilon(x,t)$ for some $x=0,\ldots,\omega$ and $t=1,\ldots, T$ and where $a(...
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3 votes
3 answers
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Calculate $\mathbb{P}[Y=y|X=x]$ where $X$ is the number of claims reported during first year and $Y$ is ultimate number of claims

A property-casualty insurance company issues automobile policies on a calendar year basis only. Let $X$ be a random variable representing the number of accident claims reported during calendar year ...
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1 vote
1 answer
182 views

Copula partial derivates

I have some troubles with the demonstration of this theorem: Let C be a copula, for any v in I=[0,1] the partial derivative for u exists for almost (Lebesgue meaning) all u, and it is included between ...
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Remediation for distributions with infinite moments [closed]

A colleague of mine posed a question to me regarding certain distributions used in loss models. Naturally occurring distributions, such as inverse Pareto, do not have finite moments. But naturally ...
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1 vote
0 answers
144 views

R - matrix optimization problem (Lee-Carter with MLE parameters)

I am trying to reproduce one of models evolved on the base of Lee-Carter model by use of RStudio. Because I am still pretty fresh in this software, my question is not really sophisticated. Applying ...
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5 votes
1 answer
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Basic calculations with Order Statistics

I've come across the following problem, and I am tempted to delve into order statistics to solve this. I would greatly appreciate any help! Suppose you draw 6 independent samples from a continuous ...
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1 answer
60 views

Compound risk poisson models

I was just working through this question. A compound Poisson risk model is used to model the total claims S experienced by an insurance company over one year, of the form: $S = X_1 + ... + X_n$ ...
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-1 votes
1 answer
2k views

Is Maximum Likelihood Estimation the median? [closed]

I asked what maximum likelihood estimation to a friend of mine. He told me that it is the median which I don't understand.
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1 vote
2 answers
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Attempting to find mean of Weibull function in R

I am using a Weibull distribution in R, and know that: E(X) = 1000 and Var(X) = 500,000 Knowing: E(X^r) = ($\Gamma$(1+ (r/$\gamma$))) * 1/c^(r/$\gamma$)) I ...
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1 vote
0 answers
897 views

How to calculate life expectancy of the astronauts?

I'm trying to obtain a more authoritative answer to the question: Do astronauts live longer than earthlings? While the time dilation thing has been easily dismissed as it introduces negligible ...
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model for pure premium using GLM for frequency and severity [closed]

I want to estimate a model for frequency and for severity in order to obtain a pure premium, defined as frequency by severity. The frequency is calculated as claim count divided by exposures, this ...
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147 views

Maximal aggregate loss in Risk Process

I am studying the Crámer Lundberg Risk Process. This process is defined as follows: $$Y_t=r_0+ct-Z_t$$ Where $Z_t$ is a compound Poisson sum, it means that $Z_t=\displaystyle\sum_{i=0}^{N_t} X_i$, ...
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2 votes
1 answer
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Modeling bimodal time-to-event

Here is a plot of death registration frequencies by age for the UK in 1974. I see distributions like this quite often: there is some event (e.g. death) which happens either close to birth, or ...
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1 vote
1 answer
810 views

Compound Poisson Process with Weibull jumps

I need to simulate a compound Poisson Process in R, however I am not clear with the algorithm to generate it. I have conceptual gaps. I know by definition that: A compound Poisson process is the ...
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1 vote
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120 views

Monte Carlo approach in a distribution of a loss process

I am trying to estimate the next quantity using Monte Carlo method. I have the next well-known quantity called the Crámer-Lundberg risk process, given by the expression $$Y_t=x+ct-Z_t$$ where $Z_t=\...
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0 votes
2 answers
105 views

Correlation Between Years for Similar Population

A common problem in insurance is to predict the amount (number, severity, or aggregate total) of losses in the next year for a book of business (eg. all personal auto policies). To do this, one can ...
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2 votes
1 answer
52 views

What are good resources on nonlife actuarial science as it is being applied in industry?

I have already browsed my way through the book Nonlife Actuarial Models: Theory, Methods and Evaluation by Yiu-Kuen Tse and I am already familiar with nearly all of the underlying statistics in that ...
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62 views

What to do when you don't have a lot of data?

I understand that there won't necessarily be a great answer to this, but I'd like to hear what people would do in this situation. Here's the data situation: I have about 30K policy records of which ...
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1 vote
2 answers
4k views

What books are good for exam p for actuarial science

Hello I'm going to be a sophomore in college and planning on taking exam P for actuarial science. One of my weakest topic is probability. What "entry level" book would you recommend before I start ...
1 vote
1 answer
268 views

Is there a Poisson-Gamma-Gamma model?

An example to elucidate my problem: The total claim amount can be modelled by a Poisson-Gamma model as it is assumed that the events (e.g. accidents) are Poisson distributed and the claims are Gamma ...
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