samp.dist: Animated representations of a statistic's sampling distribution

Description

Samples from a parent distribution without replacement with sample size = s.size R times. At each iteration a statistic requested in stat is calculated. Thus a distribution of R statistic estimates is created. This distribution is shown as an animated relative frequency histogram. Sampling distributions for up to four different statistics utilizing two different parent distributions are possible. Sampling distributions can be combined in various ways by specifying a function in func (see below).

Usage

samp.dist(parent, parent2 = parent, s.size = 1, s.size2 = s.size, 
n.seq = seq(1, 30),R = 1000, breaks = 30, stat = mean, stat2 = NULL, 
stat3 = NULL, stat4 = NULL, fix.n = TRUE, xlab = expression(bar(x)), 
ylab = "Relative frequency", ylim = NULL, func = NULL, show.n = TRUE, 
show.SE = FALSE, est.density = TRUE, col.density = 4, lwd.density = 2, 
est.ylim = TRUE, anim = TRUE, interval = 0.01, col.anim = "rainbow", 
digits = 3,...)

samp.dist.fixn(parent, parent2 = parent, s.size = 1, s.size2 = s.size, 
R = 1000, breaks = 30, stat = mean, stat2 = NULL, stat3 = NULL, stat4 = NULL, 
xlab = expression(bar(x)), ylab = "Relative frequency", func = NULL, 
show.n = TRUE, show.SE = FALSE, anim = TRUE, interval = 0.01, 
col.anim = "rainbow", digits = 3,...)

samp.dist.n(parent, R = 500, n.seq = seq(1, 30), stat = mean, 
xlab = expression(bar(x)), ylab = "Relative frequency", breaks = 30, func = NULL, 
show.n = TRUE, show.SE = FALSE, est.density = TRUE, col.density = 4, 
lwd.density = 2, est.ylim = TRUE, ylim = NULL, anim = TRUE, interval = 0.5, 
col.anim = NULL, digits = 3,...)

Arguments

parent

A vector containing observations from a parental distribution. For computational efficiency datasets exceeding 100000 observations are not recommended.

parent2

An optional second parental distribution, useful for construction sampling distributions of test statistics.

s.size

Sample size to be taken at each of R iterations from the parental distribution.

s.size2

An optional second sample size if a second statistic is to be calculated.

n.seq

A two element vector specifying the smallest and largest in a range of demonstrated sample sizes. Requires fix.n=FALSE.

The number of samples to be taken from parent distribution(s).

breaks

Number of breaks in the histogram.

stat

The statistic whose sampling distribution is to be represented. Will work for any summary statistic; e.g. mean, var, median

stat2

An optional second statistic. Useful for conceptualizing sampling distributions of test statistics. Calculated from sampling parent2.

stat3

An optional third statistic. The sampling distribution is created from the same sample data used for stat.

stat4

An optional fourth statistic. The sampling distribution is created from the same sample data used for stat2

fix.n

Logical indicating whether or not sample size should be held constant in demonstrations (see below).

xlab

X-axis label.

ylab

Y-axis label.

ylim

Limits for Y-axis. Specify using a two element vector.

func

An optional function used to compare the sampling distributions of two statistics. The function must have only sampling distributions, i.e. s.dist1, s.dist2, s.dist3, and/or s.dist as non-fixed argument

show.n

A logical command, TRUE indicates that sample size for parent will be displayed.

show.SE

A logical command, TRUE indicates that bootstrap standard error for the statistic will be displayed.

est.density

A logical command, if TRUE then a density line is plotted over the histogram. Only used if fix.n = true.

col.density

The color of the density line. See est.density above.

lwd.density

The width of the density line. See est.density above.

est.ylim

Logical. If TRUE Y-axis limits are estimated logically for the animation. Consistent Y-axis limits make animations easier to visualize. Only used if fix.n = TRUE.

anim

A logical command indicating whether or not animation should be used.

interval

Animation speed. Decreasing interval increases speed.

col.anim

Color to be used in animation. Three changing color palettes: rainbow, gray, heat.colors, or "fixed" color

digits

The number of digits to be displayed in the bootstrap standard error.

...

Additional arguments from plot.histogram.

Value

Returns a representation of a statistic's sampling distribution in the form of a histogram. } details{Sampling distributions of individual statistics can be created, or the function can be used in more sophisticated ways, e.g. to create sampling distributions of ratios of statistics, i.e. emph{t}*, emph{F}* etc. (see examples below). To provide pedagogical clarity animation for figures is provided. Two general uses of the function are possible. 1) One can demonstrate the accumulation of statistics for a single size with or without animation. This is useful because as more and more statistics are acquired the frequentist paradigm associated with sampling distributions becomes better represented (i.e the number of estimates is closer to infinity). This is elucidated by allowing the default code{fix.n = TRUE}. Animation will be provided with the default code{anim = TRUE}. This approach also allows specification of up two parent distributions, up to two sample sizes, and up to four distinct statistics (i.e. four distinct sampling distributions, representing four distinct estimators can be created). The arguments code{stat} and code{stat3} will be drawn from code{parent}, while code{stat3} and code{stat4} will be drawn from code{parent2}. These distributions can be manipulated and combined in an infinite number of ways with an auxiliary function called in the argument code{func} (see examples below). This allows depiction of sampling distributions made up of multiple estimators, e.g. test statistics. 2) One can provide a seamless animated demonstration of the effect of varying sample size on a sampling distribution by specifying code{n.fixed = FALSE}. If code{n.fixed = FALSE} is used, then a range of sample sizes (integers) must also be specified as a two element vector in code{n.seq}. Note if code{n.fixed = FALSE} then the arguments code{s.size} and code{s.size2} will be superfluous. In addition, multiple statistics and parent populations are not supported if code{n.fixed = FALSE}, although auxilary functions can still be called with code{func}. } seealso{code{plot.histogram}, code{hist}, code{bootstrap}.} author{Ken Aho} examples{ ###Not run ##Central limit theorem #Four sample sizes, one at a time exp.parent<-rexp(100000) samp.dist(parent=exp.parent, s.size=1, R=1000) ## n = 1 samp.dist(parent=exp.parent, s.size=5, R=1000) ## n = 5 samp.dist(parent=exp.parent, s.size=10, R=1000) ## n = 10 samp.dist(parent=exp.parent, s.size=50, R=1000)## n = 50 #All four at once par(mfrow=c(2,2),mar=c(4.4,4.5,1,0.5)) samp.dist(parent=exp.parent, s.size=1, R=300,anim=FALSE) ## n = 1 samp.dist(parent=exp.parent, s.size=5, R=300,anim=FALSE) ## n = 5 samp.dist(parent=exp.parent, s.size=10, R=300,anim=FALSE) ## n = 10 samp.dist(parent=exp.parent, s.size=50, R=300,anim=FALSE) ## n = 50 ##n not fixed -- sample mean exp.parent<-rexp(10000) samp.dist(parent=exp.parent, col.anim="heat.colors",fix.n=FALSE,interval=.3) ##n not fixed -- sample mean and sample median (both are consistent and unbiased, # but which is more efficient for mu?). # This will take a few seconds. parent<-rnorm(10000,sd=3) dev.new() samp.dist(parent, R=1000,col.anim="heat.colors",fix.n=FALSE,interval=.1, n.seq=seq(1,100),breaks=50,show.SE=TRUE) dev.new() samp.dist(parent, R=1000,col.anim="heat.colors",fix.n=FALSE,interval=.1,stat=median, n.seq=seq(1,100),xlab="Median",show.SE=TRUE,breaks=50) #How do the efficiency of the median and mean compare in a distribution with 10% #contamination? parent<-c(rnorm(9000),rnorm(1000,mean=10)) samp.dist(parent, col.anim="heat.colors",fix.n=FALSE,interval=.3, breaks=50,show.SE=TRUE) dev.new() samp.dist(parent, col.anim="heat.colors",fix.n=FALSE,interval=.3,stat=median, xlab="Median",est.ylim=TRUE,show.SE=TRUE,breaks=50) ##Distribution of t-statistics under valid and invalid assumptions #valid parent<-rnorm(100000) t.star<-function(s.dist1,s.dist2,s.dist3,s.dist4,s.size=6,s.size2=s.size){ MSE<-(((s.size-1)*s.dist3)+((s.size2-1)*s.dist4))/(s.size+s.size2-2) func.res<-(s.dist1-s.dist2)/(sqrt(MSE)*sqrt((1/s.size)+(1/s.size2))) func.res} samp.dist(parent, parent2=parent, s.size=6, R=1000, breaks=35,stat=mean,stat2=mean, stat3=var,stat4=var,xlab="t*", ylab="Relative frequency",func=t.star,show.n=FALSE) curve(dt(x,10),from=-6,to=6,add=TRUE,lwd=2) legend("topleft",lwd=2,col=1,legend="t(10)") #invalid; same means (null true) but different variances and other distributional #characteristics. parent<-runif(100000, min=0,max=2) parent2<-rexp(100000) samp.dist(parent, parent2=parent2, s.size=6, R=1000, breaks=35,stat=mean,stat2=mean, stat3=var,stat4=var,xlab="t*", ylab="Relative frequency",func=t.star,show.n=FALSE) curve(dt(x,10),from=-6,to=6,add=TRUE,lwd=2) legend("topleft",lwd=2,col=1,legend="t(10)") }