# Confidence Interval derivation

I am predicting a "yearly cumulative variable" from monthly results.

I use $Y_j = \sum_{i=1}^j (X_i) / f_i$ where $j$ is the current month.

I know the $f$ from history; e.g. January = .073, February = .070, March = .087, April = .076, May = .085, ...., December = .124)

So, $\sum (f_j) = 1.0$

Suppose X1 = 1000, then Y = 1000 / .073 = 13699

Suppose X2 = 950, then Y = (1000 + 950) / (.073 + .070) = 13636 etc

My question is how do I create a confidence / prediction interval on the prediction for Y? The prediction in January (11 months remaining) should be more variable than that in November (1 month remaining). Once December results are available, we are finished as Y is known and then we start over for the next year.

• Are your weights, $f$ and $X$ correlated or derived from the same data in any way? Dec 7, 2016 at 17:49
• the f are the historical average fraction of the ∑ (Xi) for each month compared to the eventual known total. So, the f and X are not correlated but the f and ∑ (Xi) are.
– Jeff
Dec 8, 2016 at 16:04

Another way to pose your question is to ask "What is the probability of making a year-end number given that we have cumulative sales of xxx so far ?" . At a baseball game , after 3 boring innings I NAIVELY try to use a ratio estimate to predict how long the game will take. Ratio estimates don't work out too well for me as the time to complete an inning is non-constant.

Procter & Gamble ( a NOT SO small company in Ohio ) asked us to add this feature so that as new data arises they can be more aware of what future totals will be and their associated uncertainty as they (and others !) had totally misread the downturn in the economy.

If you are making a forecast say for the next 6 months you can generate forecast confidence intervals (optionally allowing for anomalies/unusual values in the future) . Now these forecasts are cross-correlated and one can then compute the variance of a linear compound of these forecasts taking into account the 6 period forecast variances and their co-variances via convolution methods . See https://www.statlect.com/probability-distributions/normal-distribution-linear-combinations for more on this.

We see this question more frequently "can we make the month end goal ? " when daily data is in play . It is important (critical) in my opinion as there is "a time value of information" as well as " a time value of money ". What did you know and when did you know it ...is the issue or slightly differently "why did it take you so long to find out that numbers/goals were not going to be met so that we could make the forecasts wrong by taking remedial action ?".

I think that you should reword your question along the lines I have suggested as it might attract more attention ...just a thought ..