The Value of Likelihood Function (Basic Question)

Question

Here is my question:

Can the value of 'likelihood' function take the value that is greater than 1? If yes, how can we mathematically show that?

[ I know that the likelihood function is not the probability density function and its value can be greater than 1, but I want to show my claim more properly.]
When we scale the data (such as the bond yields) by multiplying 100*data, how does this operation affect the 'log-likelihood' function comparing with the case of dividing the data by 100 ((1/100)*data)?

Thank you very much for your time and considerations. Sp

1. This reads rather like homework/coursework. Is it for some subject? 2. Your first question is addressed by several posts already on site, but follows directly from the definition of likelihood and facts you've already stated. — Glen_b, Sep 03 '20 at 05:10
@Glen_b Hi Glen, This is not homework. It is a comment from a referee. Some of them asked me why the value of log-likelihood is positive and that is why I am trying to find an example of this to verify my claim. Anyway, thank you for your reply. — SChatcha, Sep 03 '20 at 05:17
On the first question, see the links provided here, for example: https://stats.stackexchange.com/questions/319859/can-log-likelihood-funcion-be-positive -- searches turn up more. I suggest you edit to focus on the second question — Glen_b, Sep 03 '20 at 05:23
An example is easy! Take a normal distribution with small $σ$, like $σ=0.1$ say, and any sample size you like, and look at the likelihood for $\mu$. Or likelihood for $\theta$ in a uniform on $(0,\theta)$ where the largest observation is less that $\frac12$. — Glen_b, Sep 03 '20 at 05:26

score 0 · Answer 1 · answered Sep 03 '20 at 08:01

Yes. Likelihood is defined as being proportional to a probability and the scaling of a likelihood function is always arbitrary. There is no need for a mathematical demonstration because it is true by the definition of likelihood. (Sometimes definitions leave out the fact that likelihood is only defined up to an arbitrary multiplicative constant, but that just makes those definitions deficient.)
If the scaling of the data do not affect the probability of the data given the parameter values that make up the x-axis of the likelihood function then the likelihood function must be unaffected by such scaling.

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