LNN Parameter Fitting

We present a fast method to estimate parameters to the LNN.  It is based on expectations, the symmetry of the distribution, and the fact that the lognormal component must always be positive.  For the following the variables, {c, s, d} will be the estimates for {γ, σ, δ} of the LNN.  The estimate, d is simple, it is just the mean.

d=(1/n) (∑_(i=1)^n)(Subscript[X, i])

After we determine d we adjust our sample such that for i = 1 to n:

(Subscript[Y, i])=(Subscript[X, i])-d

Next we note, where E is the expectation operator and LNNRV is an LNN sample, LNRV is a lognormal sample, and NRV is a normal sample.

E(log(|LNNRV|))=E(log(LNRV))+E(log(|NRV|))

Since the NRV always has the same parameters we calculate that:

ExpectedValue[Log[Abs[x]],NormalDistribution[0, Sqrt[2]],x]
%//N

Thus we find that

log(c)=(1/n)(∑_(i=1)^n)log(|(Subscript[Y, i])|)+0.28860783245076643

Obviously

c=(e^(log(c)))

Since NRV and LNRV are independent random variables we can write:

E(|LNNRV|)=E(LNRV) * E(|NRV|)

We calculate:

ExpectedValue[Abs[x],NormalDistribution[0, Sqrt[2]],x]

ExpectedValue[x,LogNormalDistribution[Log[c],s],x]

Thus

c (e^((s^2)/2))=(((Sqrt[π]) (∑_(i=1)^n)|(Subscript[Y, i])|)/(2 n))

And for the final parameter.

s=(Sqrt[2 (log(((Sqrt[π])/2))-log(c)+log((((∑_(i=1)^n)|(Subscript[Y, i])|)/n)))])

In the LNN package this algorithm is implemented and we also have a maximum likelihood method that takes these parameters as the initial guesses.  The maximum - likelihood algorithm may run out of memory for large samples and these estimates will be excellent for large samples.

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