Question

Suppose it is relatively easy to simulate from \(F_{i}\) for each \(i=1, \ldots, n .\) How can we simulate from (a) \(\quad F(x)=\prod_{i=1}^{n} F_{i}(x) ?\) (b) \(F(x)=1-\prod_{i=1}^{n}\left(1-F_{i}(x)\right) ?\) (c) Give two methods for simulating from the distribution \(F(x)=x^{n}, 0

Short answer

To simulate from the product of multiple CDFs, use Inverse Transform Sampling by computing the inverses of the individual CDFs, drawing random samples from a uniform distribution, and multiplying the calculated values. For the CDF \(F(x) = 1 - \prod_{i=1}^{n}(1 - F_{i}(x))\), follow similar steps but compute \(x_i\) as \(F^{-1}_{i}(1-u_i)\) and compute the final simulated value as \(x = 1 - (1 - x)\). Two methods for simulating from the distribution with the CDF \(F(x) = x^n\) include Inverse Transform Sampling and simulating using exponential distribution.

Step-by-step solution

(a) Simulating from the product of CDFs)

To simulate from the product of multiple CDFs, we can use the following Inverse Transform Sampling technique: 1. For each distribution, compute the inverse of the CDF, denoted by \(F^{-1}_{i}(u)\). 2. Draw random samples \(u_{i}\), \(i = 1, \ldots, n\), from a uniform distribution in the range \([0,1]\). 3. Compute \(x_i = F^{-1}_{i}(u_i)\) for each \(i = 1, \ldots, n\). 4. Multiply the calculated \(x_i\) values: \(x = \prod_{i=1}^{n} x_i\). The resulting \(x\) values will follow the distribution \(F(x) = \prod_{i=1}^{n} F_{i}(x)\).

(b) Simulating from the given CDF)

To simulate from the CDF \(F(x) = 1 - \prod_{i=1}^{n} (1 - F_{i}(x))\), we use the following steps: 1. For each distribution, compute the inverse of the CDF, denoted by \(F^{-1}_{i}(u)\). 2. Draw random samples \(u_{i}\), \(i = 1, \ldots, n\), from a uniform distribution in the range \([0,1]\). 3. Compute \(x_i = F^{-1}_{i}(1 - u_i)\) for each \(i = 1, \ldots, n\). 4. Multiply the calculated \(1 - x_i\) values: \(1 - x = \prod_{i=1}^{n} (1-x_i)\). 5. Compute the final simulated value: \(x = 1 - (1 - x)\). The resulting \(x\) values will follow the distribution \(F(x) = 1 - \prod_{i=1}^{n} (1 - F_{i}(x))\).

(c) Two methods for simulating from \(F(x) = x^n\))

Here, we provide two methods for simulating from the distribution with a CDF given by \(F(x) = x^n\), \(0 < x < 1\): 1. Inverse Transform Sampling Method: a. Compute the inverse of the CDF: \(F^{-1}(u) = \sqrt[n]{u}\), where \(u\) is a random realization from a uniform distribution in the range \([0,1]\). b. Draw a random sample from a uniform distribution in the range \([0,1]\) and denote it as \(u\). c. Compute \(x = F^{-1}(u) = \sqrt[n]{u}\). The simulated \(x\) values will follow the distribution \(F(x) = x^n\). 2. Simulation using Exponential Distribution: a. Draw random samples \(y_i\), \(i = 1, \ldots, n\), from an exponential distribution with parameter \(\lambda = 1\). b. Calculate the sum of \(y_i\) values: \(y = \sum_{i=1}^{n} y_i\). c. Compute the simulated value: \(x = \frac{y}{1 + y}\). The resulting \(x\) values will follow the distribution \(F(x) = x^n\).