# Confidence % on a single point estimate

I am working on a business case problem for my company (CPG) and was asked to come up with a way to predict Customer Order Fill Rate (CFR) based on the amount of inventory we hold.

Now I have come up with a formula to estimate CFR based on the cases of inventory we hold using regression, however I was asked about the probability that my estimated CFR will be correct.

So if my regression formula suggests in the future that having 100 cases of product XYZ will result in an estimated CFR of 98%, the business team asked me what is probability of actually achieving that CFR.

But I was wondering if there is a way to give such a probability on a single point estimate? I mean I am aware that I could use a prediction interval (I guess is more accurate for this case than confidence interval) to say the I am 95% confident that the future point estimate will be within a range. But I wanted to confirm if there is any such thing (a confidence %) for a single point estimate.

EDIT (Clarification):

So my observed data set for building the regression is something like this: My inventory data set is from 0-100 cases (not unique values, I can have duplicate inventory values with different CFR).

For regression I plot from 0-100 cases (taking the average CFR for each point) and build a curve for it. Here's is the regression formula I determined: With this formula, the input is the inventory (number of cases) and the output is the projected CFR.

I want to know how I can determine (or if its even possible) to get a probability value of in the future actually achieving the CFR (point estimate).

My regression formula has an x (inventory (cases)) and y (CFR): the formula determines if x = 15 then y = 93.4%.

But what is the probability that my estimation of 93.4% CFR from x = 15 will be correct? Is there a way to give confidence % for the point estimate of 93.4% CFR?

Thank you.

• What do you mean by "single point estimate"? Is it that you observed only a single value and treat it as your estimate? – Tim Jul 26 at 16:57
• The information is not enough; description of model is unclear. – user158565 Jul 26 at 17:13
• Hi, I edited the question to try and make it more clear. Sorry if I seem to not be completely transparent with the data as it is confidential. – user254865 Jul 26 at 18:08
• What is assumed distribution of y? – user158565 Jul 27 at 16:44
• It is assumed to be normally distributed. – user254865 Jul 29 at 18:30

If these are means, the chance of doing better than them is going to be around 50%, on the proviso that the median is close to the mean. Probably you have a bit of negative skew, so the median might be a bit above the mean, giving you a bit better than 50%.

If you want more, it can be done but what you need is an explicit model of the order filling process.

The simplest way to do this is to let demand be random, and fill all orders where there is stock, and have excess demand lead to unfilled orders.

Your likely variables (some known and others not) will be the number of product lines, the distribution of the mean demand for each product, and the time variances of the individual product demands. In order to simplify the model, I would assume every product has an equal coefficient of variation for demand.

Now you can use Monte Carlo methods to generate estimated CFR from your model. Then you can adjust the parameters to match the real world CFR data. When the model is matching your data, you can then read off from it the variance of the CFR for each inventory level.

You will need to use nonlinear optimisation to do this - but in this case be sure to generate all your random numbers at the start - if you take a new set of random numbers each time a parameter changes, the GRG (generalised reduced gradient) optimisation problem will never converge.

You can do all of this in Excel if you have to - using the 'solver' add in to do the GRG.

• Hi, thank you for the answer. However I do not think I understand exactly what you said about the probability, because if the regression formula is giving an output I just was wondering if statistically we can get a probability of how accurate it will be. Are you suggesting that in reality the regression output will be correct 50% of the time? That seems strange. – user254865 Jul 29 at 18:31
• I assumed you want SE of CFR, so you can say the CRF for I= 20 is 95 %, AND that there is a x% chance of achieving a CFR of larger than eg. 90%. Now the probability of CRF>95% (doing better than the mean) is going to be close to 50%, on the proviso that the mean of the CRF (for inventory=20) is close to the median. If you have the data for each CFR event, you can directly estimate the SE of the CFR and the SE of the mean of the CFR by fitting a nonlinear model for CFR as a function of inventory. – Frustrated Jul 29 at 22:18
• Now if you want instead the SE of the means, but don't have the raw data, but know the number of observations is large, without any simulation you can say that the SE will be small. Indeed for the practical problem you are dealing with it might well be quite fine to assume the SE is zero. – Frustrated Jul 29 at 22:22
• Ah ok, I now understand from your second comment. Thank you. – user254865 Jul 30 at 14:33