I have a problem at hand, which I am not being able to proceed. Can someone help me begin?

$Y_1<Y_2<Y_3$ :An order statistic of size 3 from distribution having pdf $$ f(x)=2x\ \ \ 0<x<1$$ Also, define $$U_1={Y_1\over Y_2} \ \ \text{and }\ \ \ U_2={Y_2\over Y_3}$$ The task is to compute the joint pdf of $U_1\ \&\ U_2$.

My work: I have found out the marginal of $U_1\ \&\ U_2$.

$$P(U_1\le u_1)=\int_0^1\int_0^{u_1y_2}f_{Y_1,Y_2}(y_1,y_2)dy_1dy_2$$ $$P(U_2\le u_2)=\int_0^1\int_0^{u_2y_3}f_{Y_2,Y_3}(y_2,y_3)dy_2dy_3$$ What step am I to do next?


2 Answers 2


Here is a guide to solving this problem (and others like it). I use simulated values to illustrate, so let's begin by simulating a large number of independent realizations from the distribution with density $f$. (All the code in this answer is written in R.)

n <- 4e4 # Number of trials in the simulation
x <- matrix(pmax(runif(n*3), runif(n*3)), nrow=3)

# Plot the data
for (i in 1:3) {
  hist(x[i, ], freq=FALSE, main=paste("i =", i))
  curve(f(x), add=TRUE, col="Red", lwd=2)

Histograms of original data

The histograms show $40,000$ independent realizations of the first, second, and third elements of the datasets. The red curves graph $f$. That they coincide with the histograms confirms the simulation is working as intended.

You need to work out the joint density of $(Y_1, Y_2, Y_3)$. Since you're studying order statistics, this should be routine--but the code gives some clues, because it plots their distributions for reference.

y <- apply(x, 2, sort)

# Plot the order statistics.
f <- function(x) 2*x
ff <- function(x) x^2
for (i in 1:3) {
  hist(y[i, ], freq=FALSE, main=paste("i =", i))
  k <- factorial(3) / (factorial(3-i)*factorial(1)*factorial(i-1))
  curve(k * (1-ff(x))^(3-i) * f(x) * ff(x)^(i-1), add=TRUE, col="Red", lwd=2)

Histograms of the order statistics

The same data have been reordered within each of the $40,000$ datasets. On the left is the histogram of their minima $Y_1$, on the right their maxima $Y_3$, and in the middle their medians $Y_2$.

Next, compute the joint distribution of $(U_1, U_2)$ directly. By definition this is

$$F(u_1, u_2) = \Pr(U_1 \le u_1, U_2 \le u_2) = \Pr(Y_1 \le u_1 Y_2, Y_2 \le u_2 Y_3).$$

Since you have computed the joint density of $(Y_1, Y_2, Y_3)$, this is a routine matter of doing the (triple) integral expressed by the right-hand probability. The region of integration must be $$0 \le Y_1 \le u_1 Y_2,\ 0 \le Y_2 \le u_2 Y_3,\ 0 \le Y_3 \le 1.$$

The simulation can give us an inkling of how $(U_1, U_2)$ are distributed: here is a scatterplot of the realized values of $(U_1, U_2)$. Your theoretical answer should describe this density.

u <- cbind(y[1, ]/y[2, ], y[2, ]/y[3, ])
plot(u, pch=16, cex=1/2, col="#00000008", asp=1)


As a check, we may look at the marginal distributions and compare them to the theoretical solutions. The marginal densities, shown as red curves, are obtained as $\partial F(u_1, 1)/\partial u_1$ and $\partial F(1, u_2)/\partial u_2$.

hist(u[, 1], freq=FALSE); curve(2*x, add=TRUE, col="Red", lwd=2)
hist(u[, 2], freq=FALSE); curve(4*x^3, add=TRUE, col="Red", lwd=2)

Histograms of U_1 and U_2

It is curious that $U_1$ has the same distribution as the original $X_i$.


Here is an exact symbolic solution which traces out the steps required ... here using automated tools to do the nitty gritties

Let $(X_1, X_2, X_3)$ denote a sample of size 3 from parent pdf $f(x)$:

enter image description here

Then, the joint pdf of the ordered sample $(X_{(1)}, X_{(2)}, X_{(3)})$ is say $g(x_1,x_2,x_3)$:

enter image description here

where I am using the OrderStat function form the mathStatica package for Mathematica.

The joint cdf of $(U_1, U_2)$ is $P\big(\frac{X_{(1)}}{X_{(2)}}<u_1, \,\frac{X_{(2)}}{X_{(3)}}<u_2\big)$:

enter image description here

The joint pdf of $(U_1, U_2)$ is derived by simply differentiating the cdf wrt $u_1$ and $u_2$:

enter image description here

Finally, as a quick Monte Carlo check, here is a comparison of:

  • the exact theoretical solution derived (the joint pdf - the orange surface)

  • plotted against an empirical Monte Carlo simulated joint pdf (3D histogram):

enter image description here


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