Question

Order Statistics: Let \(X_{1}, \ldots, X_{n}\) be i.i.d. from a continuous distribution \(F\), and let \(X_{(i)}\) denote the \(i\) th smallest of \(X_{1}, \ldots, X_{n}, i=1, \ldots, n\). Suppose we want to simulate \(X_{(1)}

Short answer

In summary, the hazard rate method can generate the ordered variables \(X_{(1)}, \ldots, X_{(n)}\) by simulating a random value from the Uniform(0, 1) distribution and finding the inverse of the hazard rate function. Alternatively, we can generate the ordered variables using the inverse CDF and Beta distribution. To generate \(U_{(1)}, \ldots, U_{(n)}\) without sorting, we can use i.i.d. exponentials. If \(U_{(n)} = y\), then the joint distribution of \(U_{(1)}, \ldots, U_{(n-1)}\) follows the order statistics of a set of \(n - 1\) uniform \((0, y)\) random variables. Finally, we can generate ordered uniforms using the result of part (d) without requiring sorting.

Step-by-step solution

(a) Generate the Variables Using Hazard Rate Method

For the hazard rate method, we generate the variables as follows: 1. Simulate a random value \(U\) from the Uniform(0, 1) distribution. 2. Find the inverse of the hazard rate function, and calculate the result as $$ X = \Lambda^{-1}(-\log(1 - U)). $$ 3. Repeat the above steps for all \(i = 1, \ldots, n\). This ensures that the ordering of \(X_1, \ldots, X_n\) is already determined without the need for sorting.

(b) Generate Variables Using Inverse CDF

If \(F^{-1}\) is easily computed, we can generate the ordered variables \(X_{(1)}, \ldots, X_{(n)}\) by simulating ordered \(U_{(1)}, \ldots, U_{(n)}\) and then setting $$ X_{(i)} = F^{-1}(U_{(i)}). $$ Since \(U_{(i)}\) are the order statistics of a Uniform(0, 1) distribution, they have the Beta distribution with parameters \(i\) and \(n + i + 1\). So, we can directly generate \(X_{(i)}\) from the Beta distribution as follows: $$ X_{(i)} = F^{-1}(\beta_i), $$ where \(\beta_i\) has a Beta distribution with parameters \(i\) and \(n + i + 1\).

(c) Generate Ordered Uniforms Using Exponentials

To generate \(U_{(1)}, \ldots, U_{(n)}\) without sorting, we can simulate i.i.d. exponentials \(Y_1, \ldots, Y_{n + 1}\) and set $$ U_{(i)} = \frac{Y_1 + \cdots + Y_i}{Y_1 + \cdots + Y_{n + 1}}, \quad i = 1, \ldots, n. $$ This is because, if we consider the times of all \(n + 1\) events of a Poisson process, it can be shown that the first \(n\) events are uniformly distributed over the interval \((0, \text{time of }(n+1)\text{th event})\). Here, the exponentials \(Y_i\) represent the inter-event times of a Poisson process, and their sum is the total time.

(d) Joint Distribution of Ordered Uniforms

If \(U_{(n)} = y\), then \(U_{(1)}, \ldots, U_{(n-1)}\) follow the same joint distribution as the order statistics of a set of \(n - 1\) uniform \((0, y)\) random variables. This is because \(U_{(n)} = y\) will set the maximum range as \(y\), and the remaining \(n - 1\) order statistics must be within that range while still being uniformly distributed.

(e) Generate Ordered Uniforms from the Result in (d)

Using part (d), we can generate \(U_{(1)}, \ldots, U_{(n)}\) as follows: Step 1: Generate random numbers \(U_1, \ldots, U_n\). Step 2: Set $$ \begin{aligned} U_{(n)} &= U_1^{1 / n}, \\ U_{(n-1)} &= U_{(n)}(U_2)^{1 / (n - 1)} \\ U_{(j-1)} &= U_{(j)}(U_{n-j+2})^{1 / (j - 1)}, \quad j = 2, \ldots, n - 1. \end{aligned} $$ This method generates the ordered uniforms without requiring sorting.