Difference Statistical Distribution vs Stochastic Model

[GemmaHayes]

Hi,

Can some explain the difference between modelling stochastically and with a statistical distribution? Often both are mentioned as a next step.

Thanks

 

[Dar_Shan0209]

Hi,
They are for sure but you need to be clear in what context are you referring to these. For next steps, further investigation could be carried out into the following aspects:

• Data
• Parameters
• Model
• Output

So, you can have stochastic modelling for parameters if you have had, for example, one simple inflation rate or you can have stochastic modelling to enhance the underlying model you just carried out. The same applies to statistical distribution idea. So, a little health warning about these: While a lot of ideas of assumptions and next steps are generic and may help to generate ideas in the CP2 exams. These should not be copied as-is. For example, "modelling stochastically" could be mentioned like this: Model stochastically, eg to provide a distribution of output. Remember to state which variable you intended to make stochastic.

Trust this is helpful.

 

[GemmaHayes]

Hi,
They are for sure but you need to be clear in what context are you referring to these. For next steps, further investigation could be carried out into the following aspects:

• Data
• Parameters
• Model
• Output

So, you can have stochastic modelling for parameters if you have had, for example, one simple inflation rate or you can have stochastic modelling to enhance the underlying model you just carried out. The same applies to statistical distribution idea. So, a little health warning about these: While a lot of ideas of assumptions and next steps are generic and may help to generate ideas in the CP2 exams. These should not be copied as-is. For example, "modelling stochastically" could be mentioned like this: Model stochastically, eg to provide a distribution of output. Remember to state which variable you intended to make stochastic.

Trust this is helpful.

Thanks Darshan. Often we already have a distribution of outputs after performing a simulation with the random numbers from the Uniform distribution being an input. Are we talking here about another distribution for each of the simulations produced? It's not clear to me what the difference is between the stochastic modelling step and the statistical distribution (e.g changing the distribution used). What is the difference specifically between these two terms? Thanks

 

[Dar_Shan0209]

Hi,
IMO, to generate simulation random values, you would have assumed a distribution, for example, 100 simulations of random values assuming a Normal distribution with mean X and variance Y have been used as input. When talking about stochastic modelling for me in this particular example would be to model the input itself stochastically such that your simulations would be a probability distribution of potential results. Changing the distribution used would mean instead of assuming a Normal distribution, maybe you would want to run a Chi-Squared test to confirm whether the data conforms to the distribution, if not to amend it. One fitting example is that returns are assumed to have fat tails which a normal distribution would not capture.
Hope this helps.

 

[GemmaHayes]

Silly question perhaps but is a statistical distribution stochastic?

 

[mugono]

Silly question perhaps but is a statistical distribution stochastic?

No.
For example, the normal distribution has a 'certain' bell shaped curve.

The probability under the curve (P<0) of a standard normal random variable with a mean & standard deviation of 0 and 1 respectively is always 50%. At any point along the x-axis, the probability under the curve will not change: it is deterministic and not stochastic.

 

[GemmaHayes]

No.
For example, the normal distribution has a 'certain' bell shaped curve.

The probability under the curve (P<0) of a standard normal random variable with a mean & standard deviation of 0 and 1 respectively is always 50%. At any point along the x-axis, the probability under the curve will not change: it is deterministic and not stochastic.

Thanks, that makes sense but often we are using random number simulations from this statistical distribution so is the output stochastic then as it won't follow precisely the theoretical distribution? Thanks

 

[mugono]

Thanks, that makes sense but often we are using random number simulations from this statistical distribution so is the output stochastic then as it won't follow precisely the theoretical distribution? Thanks

There may be a number of actuaries on this forum who are closer to the underlying workings of stochastic models or modelling than me. Nevertheless I provide some comments below in the spirit of helpfulness.

- An insurer may model one or more inputs, e.g. rates, inflation etc stochastically.
- Each input may be simulated using a particular / different underlying theoretical distribution function. This could be done using a [0,1] random generator: convenient because a simulation between [0,1] can represent a probability. It is elementary to transform a probability to a point along the pdf of the selected probability distribution.
- For each input, you might expect there to be a sufficient number of simulations such that the distribution of the simulated data converges to the underlying theoretical probability distribution. Stability, stationarity etc. are desirable properties.
- The output would be a function of all of the inputs; and will give rise to a multivariate distribution. It's unclear (at least to me ) what the theoretical distrubution of the output will take; and could be influenced by factors including the choice of dependency structure.

The above is a first-principles explanation. Happy for more enlightened responses from the community .