Question

Construct a rejection algorithm to simulate from \(f(x)=30 x(1-x)^{4}, 0 \leq x \leq 1\), using the \(U(0,1)\) density as the proposal function \(g\). Give its efficiency.

Short answer

Use rejection sampling with M=2.5 and efficiency is 40%.

Step-by-step solution

Define the target distribution

The target distribution given is \(f(x) = 30x(1-x)^4\), defined for \(0 \leq x \leq 1\). We need to simulate values from this distribution using a rejection sampling algorithm.

Choose the proposal distribution

The proposal distribution we will use is the uniform distribution, \(g(x) = 1\), for \(0 \leq x \leq 1\). This means any \(x\) drawn from \(U(0,1)\) is equally likely.

Determine the constant M

For rejection sampling, we need to find a constant \(M\) such that \(M \cdot g(x) \geq f(x)\) for all \(x\). Since \(g(x) = 1\), we require \(M \geq f(x)\). \(f(x)\) achieves its maximum value when \(x = \frac{1}{5}\), at which point \(f(\frac{1}{5}) = 30 \left(\frac{1}{5}\right) \left(1-\frac{1}{5}\right)^4 = 30 \times \frac{1}{5} \times \left(\frac{4}{5}\right)^4 = 30 \times \frac{1}{5} \times \frac{256}{625} = \frac{30 \times 256}{3125} = \frac{7680}{3125} \approx 2.4576.\) Thus set \(M = 2.5\).

Implement the rejection algorithm

1. Repeat the following steps until you obtain a required number of samples: - Sample \(x^{*}\) from the proposal distribution \(U(0,1)\). - Sample \(u\) from \(U(0,1)\). - Calculate the ratio: \(\frac{f(x^{*})}{M \cdot g(x^{*})} = \frac{f(x^{*})}{M} \). - Accept \(x^{*}\) if \(u \leq \frac{f(x^{*})}{M}\); otherwise, reject it and sample again.

Calculate the efficiency of the algorithm

The efficiency of the rejection sampling algorithm is given by \(\frac{1}{M}\) where \(M\) is the constant determined earlier. Therefore, the efficiency is \(\frac{1}{2.5} = 0.4\) or 40%.

Key concepts

Understanding Probability Distributions

Probability distributions serve as the backbone for statistical simulations, informing us of how likely different outcomes are to occur. In the case of the original problem, the target distribution is a beta-like distribution given by the formula:

• \(f(x) = 30x(1-x)^4\) for \(0 \leq x \leq 1\).

This specific distribution suggests that outcomes near 0 are likely less frequent than those near 1/5, due to the higher concentration of probability around this point.
Understanding these dynamics is crucial for simulations, as it dictates how samples are spread across the possible range of values. For example, the choice to use a uniform proposal distribution \(g(x) = 1\) in rejection sampling reflects the simplicity and range uniformity of uniform distributions, which perfectly covers the interval \([0,1]\). This makes it ideal for easy setup, although adjustments are necessary to match targets with more complex shapes.

Grasping the Concept of Algorithm Efficiency

Algorithm efficiency determines how well an algorithm performs its intended function with regard to computational resources like time or memory. It is critical to consider when choosing or creating algorithms.
In a rejection sampling scatter, efficiency means how many of the proposed samples are actually used as valid samples.
This is mathematically expressed as the reciprocal of the constant \(M\) chosen in step 3 of the step-by-step solution, i.e., \(\frac{1}{M}\). For the given problem, the constant was found to be \(M = 2.5\), so the efficiency is \(\frac{1}{2.5} = 0.4\) or 40%.
A less efficient algorithm requires more resource usage; here, a lower efficiency implies we retain only 40% of proposed samples. Thus, increasing the efficiency would speed up the sampling process without the need for additional trials.

The Role of Statistical Simulation

Statistical simulations, such as the one used in the exercise, allow us to model and understand complex distributions quickly and effectively. These simulations are invaluable in statistics, finance, and various scientific fields where analytical solutions are hard to come by.
The purpose of employing rejection sampling in statistical simulations is to generate random observations that follow a specified target distribution. To achieve this, we use a simpler distribution to guide the process of drawing samples.
In practical terms, simulations can provide insights into future trends or behaviors of systems. This particular simulation used a uniform distribution for ease and symmetry, combined with rejection sampling to incrementally build an approximation of more intricate distributions like the defined \(f(x) = 30x(1-x)^4\). Through repeated attempts and statistical logic, simulations turn theoretical distributions into practical understanding and predictions.