Plotting log likelihood function of Pareto distribution inR How can I plot log likelihood function with type "I" Pareto distribution function $f(x|\alpha,k)=\alpha k^\alpha/x^{\alpha+1}$ in R?  The problem for me is there are two parameters, how can I plot the log likelihood with two unknown parameters? Do I need to give a value of $\alpha$ and then plot the log likelihood function with respect to single parameter $k$?
 A: It is crucial that you get the PDF right.  It is not $\alpha k^{\alpha} / x^{\alpha+1}$; rather, it is equal to that expression only when $x \ge k$.  Otherwise it equals zero.  This makes an enormous difference: in effect, the log likelihood is defined only for those values of $k$ not exceeding the smallest of all the data values.
At this point there are many standard ways to plot $\log(f)$.  We tend to think in three dimensions and so will favor a pseudo-3D plot of $\log(f)$ against $k$ and $\alpha$:

It can be difficult to read such plots quantitatively, though; for such purposes, people favor contour plots:

The lighter shading corresponds to larger values of log likelihood: evidently the maximum is reached near $k=3.03$ and $\alpha \approx 1.35$.  The contour lines are drawn for integral values of $\log(f)$.
Sometimes, for more detailed evaluation it does help to look at "profile" log likelihoods.  Here, we look at different slices of $k$ ranging from the minimum value of $x$ (green curve) and then decreasing by $0.01$ with each slice.  These correspond to some of the mesh lines seen in the first figure:


The Mathematica code which generated these data and drew these plots is:
x = RandomReal[ParetoDistribution[3, 4/3], 100];
logLikelihood[x_, {k_, a_}] := 
  Sum[If[y >= k, Log[a] + a Log[k] - (1 + a) Log[y], -Infinity], {y, x}];
ContourPlot[logLikelihood[x, {k, a}], {k, Min[x] - 10/Length[x], Min[x]}, {a, 1, 5/3},
  FrameLabel -> {k, a}]
Plot3D[logLikelihood[x, {k, a}], {k, Min[x] - 10/Length[x], Min[x]}, {a, 1, 5/3},
  AxesLabel -> {k, a, Log[f]}, Boxed -> False, PlotRegion -> {{0.05, .95}, {0, 1}}]
Plot[Evaluate@
  Table[logLikelihood[x, {k, a}], {k, Min[x] - 12/Length[x], Min[x], 4/Length[x]}], 
  {a, 1, 5/3}, PlotStyle -> Thick, AxesLabel -> {a, Log[f]}]

A histogram of the data is
Histogram[x]


The methods in R are similar but take more effort to make them this nice.  Look at curve and graphics::filled.contour.  I am not aware of anything really comparable to the first (pseudo 3D) plot; there are some crude 3D utilities.
