Home page for accesible maths 12.3 Likelihood Examples: continuous parameters Relative Likelihood intervals

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12.4 Likelihood Examples: discrete parameters

One case where differentiation is clearly not the right approach to use for maximisation is when the parameter of interest is discrete.

TheoremExample 12.4.1 Illegal downloads

A computer network comprises of $m$ computers. The probability of one of these computers to store illegally downloaded files is $0.3$ , independent for each computer. In a particular network it is found that exactly one computer contains illegally downloaded files. Our parameter of interest is $m$ .

What is a suitable model for the data?

What assumptions are being made?

Are these assumptions reasonable?

What is the likelihood of $m$ ?

Let $X\sim Bin(m,0.3)$ be the number of computers in the network that contains illegally downloaded files. Then $\Pr(\text{obs}|m)$ is

L(m)=\Pr(X=1|m)={m\choose 1}0.3^{1}\times 0.7^{m-1}=\frac{0.3}{0.7}0.7^{m}m.

Note that the possible values $m$ can take are $m=1,2,\ldots$ . We can sketch the likelihood for a suitable range of values:

⬇

> mrange<-0:20 # value for m=0 will be zero

> plot(mrange,dbinom(1,mrange,0.3),xlab="m",ylab="L(m)")

From the plot, we can see that the MLE for $m$ is $\hat{m}=3$ . Alternatively, from the likelihood we have

\frac{L(m+1)}{L(m)}=\frac{0.3^{1}\times 0.7^{m}(m+1)}{0.3^{1}\times 0.7^{m-1}m% }=\frac{0.7(m+1)}{m}.

The likelihood is increasing for $L(m+1)>L(m)$ , which is equivalent to $m<7/3$ .

To maximize the likelihood, we want the largest (integer) value of $m$ satisfying this constraint, i.e. $m=2$ , hence $\hat{m}=3$ .

Relative Likelihood intervals