forecasting mortality

[Anonymous2019]

Lately I have been studying forecasting mortality rates.

The following method is used for alpha and beta:

αx = ∑ [(1/t) – avg (1/t)] * ln ln {K*qx(t)/[1 – qx(t)]}
∑ [(1/t) – avg (1/t)]²

βx = ln ln {K * qx(2015)/[1 – qx(2015)]} – αx / 2015

and K = 10^6

I have attached a file, where I tried to simulate these numbers.
on sheat 1: my simulation for betas corresponding the above formula
on sheat 2: the mortality rates of the past on which the alpha and betas are based.

If anyone can help me simulating alpha.

For beta I already simulated (see sheat 1) and for the first three years, the calculated beta correspond with the beta. After the third year, it doesn't. Maybe there is a kind of smoothing formula which I couldn't find.

 

[tkhunny]

avg(1/t) doesn't seem to mean anything. Can you define it, please? An average requires at least two items. Perhaps an index would be helpful.

I presume your 'x's are supposed to be subscripts?

Do you recognize [math]\dfrac{q_{x}}{1-q_{x}}[/math]? [math]q_{x}[/math] is the annual mortality rate, clearly. What is this other expression? You'll need to know it to understand the formula. It has important implications concerning your assumption of the distribution of deaths during the course of a single year.

[math]\alpha_{x}[/math] appears to be a weighted average, assuming that oddly-formatted notation means division. The weights are related to that thing I was asking about in the previous comment.

"2015" seems rather odd. Why would the mortality be related to the year in which it is observed? Is this just an index of some sort? Maybe a superscript?

Feel free to learn just a little LaTex so you can communicate better.

 

[Anonymous2019]

My apologies.

α_x = ∑ [(1/t) – avg (1/t)] * ln ln {K*q_x(t)/[1 – q_x(t)]} / (∑ [(1/t) – avg (1/t)]²)

β_x = ln ln {K * q_x(2015)/[1 – q_x(2015)]} – α_x/2015

I have attached two files so you can see the formulas. In the second file you can see the used formula for the mortality forecasting.

2015 is the year in which the forecasting is been done.

 

[tkhunny]

You didn't answer my question. I'll need to know you have SOME background.

The use of "t" is quite awkward. It does appear to be more of an index than anything else, In particular [math]q_{x}(2004)\;vs\;q_{x}(2015)[/math] appears to be a projection from 2004 mortality to 2015 mortality. Thus, the t-index we seek is 15-4 = 1, 2, ..., 11. This may give some meaning to avg(1/t) but it still doesn't work for projections of only one year. Strange, I think, to have a projection that supposedly has some value for 2 and greater, but doesn't exist for t = 1. Does the author have a solution to this problem?​

 

[Anonymous2019]

"t" stands for time.

According to my sheat2: 1959-1963 = 1961; 1968-1972 = 1970; 1979-1982 = 1980,5; 1988-1990 = 1989; 1991-1993 = 1992; 1994 = 1994; etc.

in the original post I used the data of 2015 (which are known by me). In my second post I referred to the formula in the book, but the data in the book was limited to 2004. Hence the difference.

No explanation was given to the meaning of avg 1(/t), but I'll try to ask the author.

The reason why mortality rates of 2015 are used, is because the forecasting is set to 2016 and later in the future.