/usr/share/openturns/validation/ValidEstimate.txt is in openturns-validation 1.5-7build2.
This file is owned by root:root, with mode 0o644.
The actual contents of the file can be viewed below.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 | > restart:
> mu:=alpha*GAMMA(1+1/beta);
> var:=alpha^2*(GAMMA(1+2/beta)-GAMMA(1+1/beta)^2);
> V:=alpha^2*(GAMMA(1+1/beta)-GAMMA(1+1/beta)^2);
1
mu := alpha GAMMA(1 + ----)
beta
2 / 2 1 2\
var := alpha |GAMMA(1 + ----) - GAMMA(1 + ----) |
\ beta beta /
2 / 1 1 2\
V := alpha |GAMMA(1 + ----) - GAMMA(1 + ----) |
\ beta beta /
> with(plots):
> p1:=plot(V/var,beta=0...30,color='red'):
> expr1:=evalf(convert(series(V/var,beta=infinity,3),polynom)):
> p2:=plot(expr1,beta=0...30,color='blue'):
> x:=fsolve(diff(V/var,beta),beta):
> expr:=convert(series(V/var,beta=x,3),polynom):
> p3:=plot(expr,beta=0...2,color='green'):
> display({p1,p2,p3});
> expr1;
0.3509050462 beta
> solve({mu=m,k*beta*V=v},{alpha,beta});
Warning, solutions may have been lost
> M:=map(factor,<<diff(mu,alpha)|diff(mu,beta)>,<diff(var,alpha)|diff(va
> r,beta)>>);
M :=
[ beta + 1 beta + 1 ]
[ alpha Psi(--------) GAMMA(--------)]
[ beta + 1 beta beta ]
[GAMMA(--------) , - -----------------------------------]
[ beta 2 ]
[ beta ]
[ / beta + 2 beta + 1 2\ 2 /
[-2 alpha |-GAMMA(--------) + GAMMA(--------) | , 2 alpha |
[ \ beta beta / \
beta + 2 beta + 2
-Psi(--------) GAMMA(--------)
beta beta
beta + 1 2 beta + 1 \ / 2]
+ GAMMA(--------) Psi(--------)| / beta ]
beta beta / / ]
> help(Psi);
> r:=(log(k)-Psi(k)-S)/(Psi(1,k)-1/k);
ln(k) - Psi(k) - S
r := ------------------
Psi(1, k) - 1/k
> p:=expand(evalf(convert(series(r,k),polynom)));
2 2 2 3
p := k + k ln(k) + 1.577215665 k - 1. S k - 1.712652469 k
3 3 4 4
+ k ln(k) - 1. S k - 0.700897539 k - 0.644934068 k ln(k)
4 5 5
+ 0.644934068 S k + 1.578815873 k + 0.114245670 k ln(k)
5
- 0.114245670 S k
> series(solve(log(k)-Psi(k)-S,k),_Z);
RootOf(-ln(_Z) + Psi(_Z) + S)
> n:=1;evalf(series(solve(convert(series(log(k)-Psi(k)-S,k,n),polynom),k
> ),S));
n := 1
- 0.03381041981 + 0.6131875243 I + (0.2506324443 + 0.1887666633 I)
2
S + (0.1410530934 - 0.01305786841 I) S +
3
(0.04710451564 - 0.03604051692 I) S +
4
(0.01169159185 - 0.02083727925 I) S +
5 6
(0.002334197122 - 0.009323936491 I) S + O(S )
>
|