What is the relation between Confidence Interval and prediction interval and, standard error for a point estimate? Please explain with an example.

As per my understanding, the standard error is a variation of point estimate (of a parameter) around the average estimation of the point estimate.

This standard error is accounted in confidence intervals.

And Prediction interval carries many such confidence intervals.

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#### Best Answer

A prediction interval is an interval that covers the future (or otherwise unknown) value of a random variable with some prespecified probability.

Prediction intervals are conceptually related to confidence intervals, but they are not the same. A prediction interval pertains to a *realization* (which has not yet been observed, but *will* be observed in the future), whereas a confidence interval pertains to a *parameter* (which is *in principle* not observable, e.g., the population mean).

In the time series context, prediction intervals are known as forecast intervals.

*(This is copied verbatim from the tag wiki for the prediction-interval tag.)*

The standard error is the estimated standard deviation of an estimate of a parameter. It is typically used to construct a confidence-interval for the parameter, together with the point estimate and usually a normal distribution, which is motivated by the Central Limit Theorem or similar.

As an example, let us simulate some weakly related data and fit a (correctly specified) model. In R:

`set.seed(1) nn <- 20 predictor <- runif(nn) response <- 3*predictor+rnorm(nn) model <- lm(response~predictor) xx <- seq(min(predictor),max(predictor),by=.01) plot(predictor,response,pch=19) lines(xx,predict(model,newdata=data.frame(predictor=xx))) ) `

We can now extract the quantities we are interested in:

The

*standard error*of (say) the parameter estimate for the regression slope is 0.7394:`> summary(model) ... truncated ... Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 0.9412 0.4593 2.049 0.0553 . predictor 1.4083 0.7394 1.905 0.0729 .`

This allows us to obtain a symmetric 95%

*confidence interval*for the regression slope parameter of $[-0.14,2.96]$:`> confint(model) 2.5 % 97.5 % (Intercept) -0.02380633 1.906270 predictor -0.14499836 2.961647`

Finally, we can get a 95%

*prediction interval*for a new observation. For that we need to specify at what value of the predictor we want to calculate this PI. For instance, we could use a predictor value of $0.5$, yielding a PI of $[-0.34,3.63]$:`> predict(model,newdata=data.frame(predictor=0.5),interval="prediction") fit lwr upr 1 1.645394 -0.3415354 3.632323`

We see how the standard error and the confidence interval pertain to *unobservable parameters*, while the prediction interval pertains to (yet unseen) *observations*.

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