Question

Let \(X_{1}, X_{2}, \ldots, X_{n}\) be a random sample from a \(\Gamma(\alpha=3, \beta=\theta)\) distribution, \(0<\theta<\infty\). Determine the mle of \(\theta\).

Short answer

The maximum likelihood estimator (MLE) for the parameter \(\theta\) in a gamma distribution with known shape parameter \(\alpha=3\) and an unknown rate parameter \(\beta=\theta\) is \(\hat{\theta}_{MLE} = \frac{3n}{\sum_{i=1}^{n} x_i}\).

Step-by-step solution

Define Gamma distribution

The gamma distribution of the random variable X having shape \(\alpha\) and rate \(\beta\) is given by the probability density function: \(f(x;\alpha,\beta) = \frac{\beta^{\alpha}x^{\alpha-1}e^{-\beta x}}{\Gamma(\alpha)}\) for \(x > 0\), \(0 < \theta < \infty\), and where \(\Gamma(\alpha)\) is the gamma function.

Write out the likelihood function

The likelihood function \(L(\theta)\) of a sample \(\{x_1,x_2,...,x_n\}\) is the joint probability density function of the sample, based on the assumed distribution. For our sample from gamma distribution with known \(\alpha=3\) and unknown \(\beta=\theta\), the likelihood function is given by: \(L(\theta) = \prod_{i=1}^{n} f(x_i;3,\theta) = \prod_{i=1}^{n} \frac{\theta^3 x_i^{3-1}e^{-\theta x_i}}{\Gamma(3)}\).

Compute the log-likelihood

The log-likelihood function is the natural logarithm of the likelihood function, it often simplifies the process or removing products in favor of sums and squaring becomes multiplication. Here, for the given problem the log-likelihood \(\ell(\theta)\) simplifies as: \(\ell(\theta) = \ln L (\theta) = \sum_{i=1}^{n} \ln f(x_i;3,\theta) = \sum_{i=1}^{n} [3 \ln \theta + 2 \ln x_i - \theta x_i - \ln \Gamma(3)]\).

Derive log-likelihood function

To find the mle of \(\theta\), we need to take the derivative of this log-likelihood with respect to \(\theta\), and set it equal to zero (method of maximum likelihood). Then, differentiate \(\ell(\theta)\) with respect to \(\theta\): \(d \ell(\theta)/d\theta = \frac{3n}{\theta} - \sum_{i=1}^{n} x_i\).

Solve for theta

The maximum likelihood estimate (MLE) can be found by setting the derivative equal to zero i.e., \(d \ell(\theta)/d\theta = 0\) and solving for \(\theta\) gives us: \(\hat{\theta}_{MLE} = \frac{3n}{\sum_{i=1}^{n} x_i}\).