Giving that Akaike Information Criterion (AIC) is as follow:
How can I Produce a 3D plot of AIC for suitable ranges of Lˆ and k.
In other words what could be a suitable ranges of L to try?
Moreover, what is the function of L^? I am struggling to find the equation to represent L^ so that I can plot it.
Thanks.
$\hat{L}$ is the value of the assumed likelihood function evaluated at $\hat{\theta}$, i.e. at its maximum value for the observed data. If our likelihood function is $L(\mathbf{X};\theta)$ then $\hat{L}=L(\mathbf{X};\hat{\theta})$. $k$ is the number of model parameters being estimated.
For comparing 2 models, the one with lower AIC is preferred. Higher values of the log-likelihood imply lower values of the AIC, holding $k$ constant, while fewer model parameters also imply lower values of AIC, holding $\hat{L}$ constant. The idea is to reward higher likelihood and penalize each time you add a parameter to the model, as you are losing degrees of freedom.
Assuming you are intending to construct a 3-D plot using triplets (AIC,$\hat{L},k$) from various models you have estimated, I'm not sure the plot will give you much insight beyond simply looking at AIC. The problem with creating a surface (as mentioned in the answer from @Lucas Farias) is that $k$ alone does not tell us which regressors we are including. For example $y=a_0+a_1x+a_2z$ and $y=b_0+b_1w+b_2v$ both have $k=2$, but will yield different values of $\hat{L}$ and AIC.
According to wiki, for the specification you presented:
Let $k$ be the number of estimated parameters in the model. Let $\hat{L}$ be the maximum value of the function for the model.
While $k$ is always non-negative, the range and shape of the model likelihood function $\hat{L}$ is different for each problem, since it depends on the densities and data you are working with.
For this reason, even if you can create the surface you want for different specifications of the same model, it's impossible to obtain an AIC surface that is representative for all types of models.
