Question

Rao (page 368,1973 ) considers a problem in the estimation of linkages in genetics. McLachlan and Krishnan (1997) also discuss this problem and we present their model. For our purposes, it can be described as a multinomial model with the four categories \(C_{1}, C_{2}, C_{3}\), and \(C_{4}\). For a sample of size \(n\), let \(\mathbf{X}=\left(X_{1}, X_{2}, X_{3}, X_{4}\right)^{\prime}\) denote the observed frequencies of the four categories. Hence, \(n=\sum_{i=1}^{4} X_{i} .\) The probability model is $$ \begin{array}{|c|c|c|c|} \hline C_{1} & C_{2} & C_{3} & C_{4} \\ \hline \frac{1}{2}+\frac{1}{4} \theta & \frac{1}{4}-\frac{1}{4} \theta & \frac{1}{4}-\frac{1}{4} \theta & \frac{1}{4} \theta \\ \hline \end{array} $$ where the parameter \(\theta\) satisfies \(0 \leq \theta \leq 1\). In this exercise, we obtain the mle of \(\theta\). (a) Show that likelihood function is given by $$ L(\theta \mid \mathbf{x})=\frac{n !}{x_{1} ! x_{2} ! x_{3} ! x_{4} !}\left[\frac{1}{2}+\frac{1}{4} \theta\right]^{x_{1}}\left[\frac{1}{4}-\frac{1}{4} \theta\right]^{x_{2}+x_{3}}\left[\frac{1}{4} \theta\right]^{x_{4}} $$ (b) Show that the log of the likelihood function can be expressed as a constant (not involving parameters) plus the term $$ x_{1} \log [2+\theta]+\left[x_{2}+x_{3}\right] \log [1-\theta]+x_{4} \log \theta $$ (c) Obtain the partial derivative with respect to \(\theta\) of the last expression, set the result to 0 , and solve for the mle. (This will result in a quadratic equation that has one positive and one negative root.)

Short answer

The exercise involves three main steps: constructing the likelihood function, expressing the log-likelihood, and finding the mle of \(\theta\). The mle is obtained by differentiating the log-likelihood with respect to \(\theta\), equating to 0, and solving the resulting quadratic equation.

Step-by-step solution

Title: Compute the Likelihood Function

Given the probabilities \(p_i\) for the four categories \(C_i\), construct the likelihood function. The likelihood is constructed as the probabilities of obtaining the observed frequencies, \(x_i\). It's given by: \[ L(\theta \mid \mathbf{x})=\frac{n !}{x_{1} ! x_{2} ! x_{3} ! x_{4} !}\left[\frac{1}{2}+\frac{1}{4} \theta\right]^{x_{1}}\left[\frac{1}{4}-\frac{1}{4} \theta\right]^{x_{2}+x_{3}}\left[\frac{1}{4} \theta\right]^{x_{4}} \]

Title: Express the natural logarithm of the likelihood

The log-likelihood is often used as it is easier to differentiate. Taking the log of the likelihood function, you obtain: \[ x_{1} \log [2+\theta]+\left[x_{2}+x_{3}\right] \log [1-\theta]+x_{4} \log \theta \]

Title: Compute the Maximum Likelihood Estimation of the Parameter \(\theta\)

The maximum likelihood estimate of a parameter is obtained by taking the derivative of the log-likelihood function with respect to the parameter and setting it equal to 0. Doing this for \(\theta\) results in a quadratic equation. The solution to this equation yields the mle.