# lnreSgam: generalized inverse Gauss-Poisson (GIGP, gamma free)

`lnreSgam (-h -mW -kX -KY -EZ -H -eR -sS -Nn -Vv -S) text.spc`

The standard LNRE model based on the generalized inverse Gauss-Poisson distribution. Since there are no closed-form expressions for estimating the parameters of the full generalized inverse Gauss-Poisson model with free gamma, the program asks the user whether to use downhill simplex minimization or to provide interactive user-guided minimization. In the case downhill simplex minimization is selected, the program calculates E[V(N)] and E[V(1,N)] for the simpler model with gamma=-0.5. The user is offered the choice between using these parameters as starting point for minimization, or to specify another starting point. By default, cost function C_1 is used, but cost function C_{2}(r) can be selected as well using the -e option.

input

`text.spc:` frequency spectrum

options

`-h:` display on-line help

`-mW:` number of ranks in fit is set to W (default: 15)

`-kX:` number of chunks for interpolation is set to X (default: 20)

`-KY:` number of chunks for extrapolation is set to Y (default: 20)

`-EZ:` extrapolation sample size is set to Z (default: 2N_0)

`-H:` input files do not have a header (default: header is presupposed)

`-eR:` use cost function C_{2}(r) with r=R

`-sS:` calculate only the expected spectrum for S ranks, output on textG.fsp

`-Nn:` force N to equal n (in case of a partial spectrum)

`-Vv:` force V(N) to equal v (in case of a partial spectrum)

`-S:` calculate Good-Turing estimates (output in textG.str)

output

`text_G.spc:` observed and expected frequency spectrum

`m:` m (frequency)

`Vm:` V(m,N) (frequency at sample size N)

`EVm:` E[V(m,N)] (expected frequency at sample size N)

`text_G.fsp:` expected frequency spectrum

`m:` m (frequency)

`EVm:` E[V(m,N)] (expected frequency at sample size N)

`text_G.sp2:` expected frequency spectrum at

`m:` m (frequency)

`EVm2N:` E[V(m,2N)] (expected frequency at sample size 2N)

`text_G.ev2:` vocabulary size statistics

`V:` V(N) (observed vocabulary size at N)

`EV:` E[V(N)] (expected vocabulary size at N)

`EV2N:` E[V(2N)] (expected vocabulary size at 2N)

`text_G.int, textG.ext:` interpolation and extrapolation statistics

`N:` N (number of tokens)

`E[V(N)]:` E[V(N)] (expected number of types)

`Alpha1:` E[alpha(1)] (E[V(1,N)]/E[V(N)])

`EV1-5:` E[V(1-5,N)] (expected spectrum elements)

`GV:` E[V(N+1)] - E[V(N)] (token-unit growth rate)

`text_G.sum:` summary statistics and estimated parameters

`N:` N (number of tokens)

`V(N):` V(N) (observed number of types)

`E[V(N)]:` E[V(N)] (expected number of types)

`V(1,N):` V(1,N) (observed number of hapax legomena)

`E[V(1,N)]:` E[V(1,N)] (expected number of hapax legomena)

`S:` S (population number of types)

`b:` b (parameter)

`c:` c (parameter)

`Z:` Z = 1/c (parameter)

`gamma:` gamma (parameter)

`text_G.str:` Good-Turing estimates based on the GIGP fit

`m:` the frequency spectrum

`mstar:` the corresponding Good-Turing estimates

technical details

The Bessel function K_{v}(z) of real order v,``

K_{v}(z) = frac{pi}{2} frac{I_{-v}(z) - I_{v}(z)}{sin(v pi)},

is itself defined in terms of the simpler function

I_{v}(z) = sum_{n=0}^{infty} frac{(z/2)^{v+2n}}{ n! Gamma(v+n+1) },

which is calculated up to the point where two successive terms of the sum differ by less than 1.0e-9. The downhill simplex minimization method is used for parameter estimation, using the subroutine amoeba of Press et al. (1988).

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