conditional expected value

E(E(Y|X)=E(Y)
P(A|X)=E(1A|X)
P(A)=E[P(A)|X]
http://www.math.uah.edu/stat/expect/Conditional.html#prp2

rank and anti-rank

x = c(-1, 5, 0, 3, 1, -2)
> rank(x) # "classical" ranking
[1] 2 6 3 5 4 1
> order(x) # anti ranking
[1] 6 1 3 5 4 2

Controlling margins and axes with oma and mgp

January 27, 2009 bryan Articles, Help 4

When creating graphs, we’re usually most concerned with what happens near the center of our displays, as this is where most of the important information is generally held. But sometimes, either for aesthetics or clarity, we want to adjust what’s outside of the box – in the margins, labels or tick marks. The par() function offers several ways to do this and I’ll discuss two that deal primarily with spatial orientation – rather than content – below.

The oma, omd, and omi options

To control the width of the outer margins of your graph (the empty sections outside of the axes and labels) use either the oma, omd, or omi option of the par() function. All three of these options have the same effect and differ only in the units used to define the parameter. oma defines the space in lines,omd as a fraction of the device region, and omi specifies the size in inches. oma and omi take a four item vector where position one sets the bottom margin, position two the left margin, position three the top margin and position four the right margin. omd uses a four item vector where positions one and three define, in percentages of the device region, the starting points of the x and y axes, respectively, while positions two and four define the end points. Because these options all effect the same graph space, changing one also changes the remaining two. A few examples of code and the charts they produce are shown below. To help illustrate the different margin sizes, the blue area indicates the dimensions of the device display:


# generate some data
x<-log(seq(1:100))


# oma, omd, and omi defaults
par()


>...
>$oma
>[1] 0 0 0 0
>
>$omd
>[1] 0 1 0 1
>
>$omi
>[1] 0 0 0 0
>...


# plot using default margin settings
plot(x,pch=1, col = "red", ylab = "Y Label", xlab = "X Label")
title("Default")


# add four lines to bottom and top margins
par(oma = c(4, 0, 4, 0))
plot(x, pch=1, col = "red", ylab = "Y Label", xlab = "X Label")
title("oma = c(4, 0, 4, 0)")


# change via omd
par(omd = c(.15, .85, .15, .85))
plot(x, pch=1, col = "red", ylab = "Y Label", xlab = "X Label")
title("omd = c(.15, .85, .15, .85)")


# because oma, omd, and omi all affect the same graph space
# this doesn't make sense
par(omi = c(0, 0, 0, 0), omd = c(.10, .90, .10, .90))


# reset oma, omd, and omi to default by changing omi
par(omi = c(0, 0, 0, 0))

The mgp option

In addition to changing the margin size of your charts, you may also want to change the way axes and labels are spatially arranged. One method of doing so is the mgp parameter option. The mgp setting is defined by a three item vector wherein the first value represents the distance of the axis labels or titles from the axes, the second value is the distance of the tick mark labels from the axes, and the third is the distance of the tick mark symbols from the axes. As with the oma option discussed above, the distances are given in line widths. The defaults for the mgp setting are c(3, 1, 0). The examples below illustrate the effects of changing the various mgp values. Note: the mgp.axis() function in the Hmisc package can be used to change these settings for each axis individually.


# mgp default settings
plot(x, pch=1, col = "red", ylab = "Y Label", xlab = "X Label")


# move labels close to axes
par(mgp = c(0, 1, 0))
plot(x, pch=1, col = "red", ylab = "Y Label", xlab = "X Label")


# move tick labels out
par(mgp = c(0, 3, 0))
plot(x, pch=1, col = "red", ylab = "Y Label", xlab = "X Label")


# move tick lines out
par(mgp = c(0, 3, 2))
plot(x, pch=1, col = "red", ylab = "Y Label", xlab = "X Label")

Summary
The oma, omd, omi, and mgp parameter settings can be useful in defining and adjusting the outer regions of your charts. To arrange and size multiple graphing areas you may also find other par()settings such as fig, fin, or layout helpful.

Compute confidence intervals for percentiles in SAS

http://blogs.sas.com/content/iml/2013/05/06/compute-confidence-intervals-for-percentiles-in-sas.html

The plug-in principle

The plug-in principle is a technique used in probability theory and statistics to approximately compute or to estimate a feature of a probability distribution (e.g., the expected value, the variance, a quantile) that cannot be computed exactly. It is widely used in the theories of Monte Carlo simulation and bootstrapping.

Roughly speaking, the plug-in principle says that a feature of a given distribution can be approximated by the same feature of the empirical distribution of a sample of observations drawn from the given distribution. The feature of the empirical distribution is called a plug-in estimate of the feature of the given distribution. For example, a quantile of a given distribution can be approximated by the analogous quantile of the empirical distribution of a sample of draws from the given distribution.

https://www.statlect.com/asymptotic-theory/plug-in-principle