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2

Not sure what is going on with the different results of Integrate and NIntegrate. This does not mean that the analytic form of $C$ is erroneous. Note that plotting the likelihood function (using the expression of $C$ provided by Integrate and the parameter values you used) over a resticted range of $s$ (instead of $[0,1]$) clearly shows that the ...


3

The problem is numerically unstable for some parameter ranges. We shall show a simple example. Your normalized distribution is given by p[q_, n_, \[Mu]_, \[Nu]_, s_] := Exp[4 n s q] q^(4 n \[Nu] - 1) (1 - q)^(4 n \[Mu] - 1)/(Gamma[4 n \[Mu]] Gamma[ 4 n \[Nu]] Hypergeometric1F1Regularized[4 n \[Nu], 4 n (\[Mu] + \[Nu]), 4 n s]) Check ...


4

If a random vector $x= \{x_1, x_2,...,x_k\}$ has Dirichlet distribution of order k>2 with parameters $\alpha_1,\alpha_2,...,\alpha_k$ its kth component is determined by the condition $x_k= 1- x_1 - x_2 - \cdots- x_{k-1}$ (see Wikipedia > Dirichlet Distribution). So you need to append 1-Total[rand] to rand to get the required parameter for the ...


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Use Normalize[rand, Total] I believe I have asked this before in here. In my case, MMA is very misleading as my probabilities do sum to 1.


3

First some analysis. The slowness is due to the time it takes to compute InverseSurvivalFunction[dist, u]. InverseSurvivalFunction is a general purpose function and perhaps not optimized for mixture of normal distributions. An improvement in speed can be obtained by using some calculus to reduce the number of times we have to compute ...


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You can easily find the list of supported distributions using: ?? *Distribution. Then click on the StudentTDistribution to get the support page, where you'll read: StudentTDistribution[\[Mu],\[Sigma],\[Nu]] represents a Student t distribution with location parameter \[Mu], scale parameter \[Sigma], and \[Nu] degrees of freedom. As such, it is a simple ...


7

Plot the objective function f[x,y] for 100 values of x: plt = Plot[ y InverseSurvivalFunction[dist, # + y] & /@ Range[0, 1., .01], {y, 0, 1}, Evaluated -> True, ImageSize -> 400, PlotRange -> {{0, 1}, {-.1, 1}}]; Post-proces the previous plot to mark points corresponding to the global maximum on each of the 100 curves: plt /. ...



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