Definition of residuals versus prediction errors? I always thought the definition of residuals is the difference between the statistic and the observations.  And, the definition of the prediction error is the difference between the true value and the prediction.  
So, I was surprised when I saw on this wikipedia page it says 
"...while lower-case y in the definition of the residuals..."  under the section called "Residuals" and the sub-section called "Example".  It seems to be showing the error because the lower case "y" is the true value, but it calls them the "residuals".  Is that incorrect to call them "residuals"? 
 A: I find your post quite confusing, especially the part about the statistic and the example; how are they relevant here? Instead, let me provide my own understanding of [model] residuals and prediction errors.

A stochastic model includes an error term to allow the relationship between the variables to be stochastic (have some randomness to it) rather than deterministic (fixed, perfect). For example,
$$ y = \beta_0 + \beta_1 x + \varepsilon $$
implies a linear relationship between $y$ and $x$, up to some error $\varepsilon$. When the model is estimated, one gets the realized values of the model errors which are called [model] residuals (denoted $\hat\varepsilon$ or $e$):
$$ y = \hat\beta_0 + \hat\beta_1 x + \hat\varepsilon. $$
Now consider another expression which defines fitted values,
$$ \hat y := \hat\beta_0 + \hat\beta_1 x. $$
Together the above two expressions yield another expression for the [model] residuals; they are the difference between the actual and the fitted values of the dependent variable:
$$ \hat\varepsilon = y - \hat y. $$

Meanwhile, prediction errors arise in the context of forecasting. A prediction error is the difference between the realized value and the predicted value:
$$ e^{fcst} := y - y^{fcst}. $$
(Since the prediction $y^{fcst}$ is produced without (or before) having observed the realized value $y$, the prediction errors are generally not zero.)

Now to respond to your Wikipedia quote, let us look at it more closely:

<...> the capital letter Y is used in specifying the model, while lower-case y in the definition of the residuals; that is because the former are hypothesized random variables and the latter are actual data.

It only says that Y are hypothesized random variables and y are actual data. (If I had used this notation, I should have had capital Y in my first equation but lower-case y in my second equation and elsewhere.) The cited definition of residuals is five lines above the quoted text; there indeed is a formula including lower-case y and defining [model] residuals. If I interpret you correctly, you seem to have understood that y is called the residuals -- which it is not, if you read the Wikipedia quote carefully.
