# Suitable Distribution/Methodology for Estimating Data Points

I am trying to model future sales data for products which have very low sales volume. I am a programmer with a smattering in statistics, so I apologise in advance if this qustion is naive!

My question is what distribution is most suitable for my sales profile and is it possible to verify the distribution. The exact framing of the problem is that we might have a product that has say 6 units of inventory and we may sell 8 units a year with a "standard deviation" of 5 units (i.e. the sales are lumpy so we calculate a standard deviation, but its not really a normal distribution)...we want to say with a certain probability how many days inventory we have left. For high volume products we can assume the normal distribution and its pretty easy to back out the days inventory left. However for low volume products we can't assume normal distribution (as obviously it is bounded by the fact that we can't have less than 0 sales). I have looked at Poisson, but I am not sure if that is the most suitable.

Could someone point me to some resources that would help me identify the right model/technique to use?

Thanks,

Mike.

You are looking for a class of models known as 'purchase incidence'.

A poisson distribution with the rate of sales $\lambda$ such that $Y_{t}=\lambda t$ is the number of units sold during time period $t$ is a good place to start.

$$P(Y_{t}=y_{t})=\frac{e^{-\lambda t}(\lambda t)^{y_{t}}}{y_{t}!}$$

So, if you have 3 items left and 10 before restocking, the chance that you will have an item for each customer who wants to purchase one is $$1-P(Y_t=[0,1,2,3])$$

Additional parameters can be added as needed to model known processes that cause the distribution of sales to vary with time. For example, heterogeneity in $\lambda$ can be modeled as $\lambda\sim\text{Gamma}(a,b)$. You can find a description of this and related models in chapter 12 of Leeflang 2000 "Building models for marketing decisions"

• Thanks David...I will start with that. Another small question I had....is there some guidence on how I can interperate the notation you have used above? – Mike Hanrahan Jan 11 '11 at 20:56
• @mike $P(Y_{t}=y_{t})$ is the probability that you sell $y$ units in a time $t$; $P(Y_{t}=[0,1,2,3]$ is the probability that you sell 0, 1, 2, or 3 units, $\lambda$ is the poisson parameter, the average rate of sales such that the average units sold in time $t$ is $Y_t=\lambda t$. – David LeBauer Jan 11 '11 at 21:06
• Thanks..is there a general document somewhere on the notation you are using? Is it some statistical notation? For example, in "\frac{e^{-\lambda t}(\lambda t)^{y_{t}}}{y_{t}!}" I am not clear on the parameter precedence and do I interperate the ! as factorial, or does it have a special meaning there (and what does "frac" mean?). Thanks for the help. – Mike Hanrahan Jan 11 '11 at 21:27
• Ok...ignore that last message (and my first one!) (although I will leave them there in case anyone else has the same question). My browser was not resolving the notation into its MathJax image. – Mike Hanrahan Jan 11 '11 at 21:50
• @Mike thats okay. The programming language used by MathJax is $\LaTeX$ – David LeBauer Jan 12 '11 at 0:20