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Calculates maximum likelihood estimates for the parameters of one of several univariate probability distributions.
X Array containing the data. (Input)
IPDF Specifies the probability density function. (Input)
Distribution |
IPDF |
size(PARAM) |
Discrete uniform |
0 |
2 |
Bernoulli |
1 |
1 |
Binomial |
2 |
1 |
Negative binomial |
3 |
1 |
Poisson |
4 |
1 |
Geometric |
5 |
1 |
Continuous uniform |
6 |
2 |
Beta |
7 |
2 |
Exponential |
8 |
1 |
Gamma |
9 |
2 |
Weibull |
10 |
2 |
Rayleigh |
11 |
1 |
Extreme value |
12 |
2 |
Generalized extreme value |
13 |
3 |
Pareto |
14 |
2 |
Generalized Pareto |
15 |
2 |
Normal |
16 |
2 |
Log-normal |
17 |
2 |
Logistic |
18 |
2 |
Log-logistic |
19 |
2 |
Inverse Gaussian |
20 |
2 |
PARAM Array of length p containing the parameter values, where p denotes the number of parameters (see IPDF table above). On input, the values of PARAM are used as starting values, when USEMM = .false.. On output, final parameter estimates replace the starting values. (Input/Output)
NOBS Number of
observations to use in the analysis.
(Input)
Default: NOBS = size(X).
PARAMLB
Array of length p containing the lower bounds of the
parameters. (Input)
Default: The default lower bound
depends on the range of the parameter. That is, if PARAM(i)
is positive, PARAMLB(i)
= 0.01. If PARAM(i)
is non-negative (≥ 0) , then PARAMLB(i)
= 0.0. If
PARAM(i)
can be any real value, then
PARAMLB(i) = -10000.00. Exceptions are
PARAMLB(i) = 0.25 for the
scale parameter of the extreme value distribution, PARAMLB(i) = -5.0 for the
shape parameter of the generalized Pareto distribution, and PARAMLB(i) = -10.0 for the
shape parameter of the generalized extreme value distribution.
PARAMUB Array
of length p containing the upper bounds of the parameters.
(Input)
Default: PARAMUB(i) = 10000.00.
Exceptions are PARAMUB(i) = 5.0 for the shape
parameter of the generalized Pareto distribution and PARAMUB(i) = 10.0 for
the shape parameter of the generalized extreme value distribution
USEMM Logical.
If .true.,
starting values are set to the method of moments estimates.
(Input)
If USEMM = .false., PARAM values are
used.
Default: USEMM = .true..
XSCALE Array of
length p containing the scaling factors for the parameters. XSCALE is used in the
routine BCONF
mainly in scaling the gradient and the distance between two points. See
BCONF in the
Math Libray, Chapter 8, Optimization for details.
Default: XSCALE =1.0.
MAXIT Maximum
number of iterations. (Input)
Default: MAXIT = 100.
MAXFUN Maximum
number of function evaluations. (Input)
Default: MAXFUN = 400.
MAXGRAD Maximum
number of gradient evaluations. (Input)
Default: MAXGRAD = 400.
IPRINT Printing option (Input)
IPRINT |
Action |
0 |
No printing |
1 |
Print final results only |
2 |
Print intermediate and final results |
Default: IPRINT = 0.
MLOGLIKE Minus log-likelihood evaluated at the parameter estimates. (Output)
SE Array of length p containing the standard errors of the parameter estimates. (Output)
HESS Array of size p by p containing the Hessian matrix. (Output)
Generic: CALL MLE (X,IPDF,PARAM [, ])
Specific: The specific interface names are S_MLE and D_MLE.
Routine MLE calculates maximum likelihood estimates for the parameters of a univariate probability distribution, where the distribution is one specified by IPDF and where the input data X is (assumed to be) a random sample from that distribution.
Let represent a random sample
from a probability distribution with density function
, which depends on a vector
containing the values of the parameters of the distribution. The
values in
are fixed but unknown
and the problem is to find an estimate for
given the sample data.
The likelihood function is defined to be the product
The estimator
That is, the estimator that maximizes L also
maximizes log L and is the maximum likelihood estimate, or MLE for
.
The likelihood problem is in general a constrained
non-linear optimization problem, where the constraints are determined by
the permissible range of . In a few situations,
the problem has a closed form solution. Otherwise, MLE
uses the numerical method as documented in routine BCONF
(see
Chapter 8, Optimization for details) to solve the likelihood
problem. If USEMM
is .true.
(the default), method of moments estimates serve as starting values of the
parameters. In some cases, method of moments estimators may not exist,
such as when certain moments of the true distribution do not exist; thus it is
possible that the starting values are not truly method of moments
estimates. If USEMM
is set to .false.,
input values of PARAM
are used as starting values.
Upper and lower bounds, when needed for the optimization, have default values for each selection of IPDF (defaults will vary depending on the allowable range of the parameters). It is possible that the optimization will fail. In such cases, the user may try adjusting upper and lower bounds using the optional parameters PARAMLB, PARAMUB, or adjusting up or down the scaling factors in XSCALE, which can sometimes help the optimization converge.
Standard errors and covariances are supplied, in most
cases, using the asymptotic properties of ML estimators. Under some
general regularity conditions, ML estimates are consistent and
asymptotically normally distributed with variance-covariance equal to the
inverse Fishers Information matrix evaluated at the true
value of the parameter, :
MLE approximates the asymptotic variance using the negative inverse Hessian evaluated at the ML estimate:
The Hessian is approximated numerically for all but a few cases where it can be determined in closed form.
In cases when the asymptotic result does not hold, standard errors may be available from the known sampling distribution. For example, the ML estimate of the Pareto distribution location parameter is the minimum of the sample. The variance is estimated using the known sampling distribution of the minium, or first order-statistic for the Pareto distribution.
For further details regarding the properties of the estimators and the theory of the maximum likelihood method, see Kendall and Stuart (1979). The different probability distributions have wide coverage in the statistical literature. See Johnson & Kotz (1970a, 1970b, or later editions).
Parameter estimation (including maximum likelihoood) for the generalized Pareto distribution is studied in Hosking and Wallis (1987) and Giles and Feng (2009), and estimation for the generalized extreme value distribution is treated in Hosking, Wallis, and Wood (1985).
1. The location parameter is not estimated for the generalized Pareto distribution (IPDF=15). Instead, the minimum of the sample is subtracted from each observation before the estimation procedure.
2. Only the probability of success parameter is estimated for the binomial and negative binomial distributions, (IPDF=2,3). The number of trials and the number of failures, respectively, must be provided in PARAM(1) on input.
3. MLE issues an error if missing or NaN values are encountered in the input data. Missing or NaN values should be removed before calling MLE.
4. Informational errors
Type Code
3 1 The Hessian is not calculated for the negative binomial distribution.
3 2 Hessian is not used to calculate the standard errors of the estimates for the continuous uniform distribution.
3 3 The Hessian is not used to calculate the standard errors of the estimates for the Pareto distribution.
3 4 For the Pareto distribution, the Hessian cannot be calculated because the parameter estimate is 0.
The data are N = 100 observations generated from the
logistic distribution with location parameter and scale parameter
.
use mle_int
implicit none
integer, parameter :: ipdf=18, npar=2
real(kind(1e0)) :: param(npar), stderr(npar), hess(npar,npar)
real(kind(1e0)) :: fval
! Logistic distribution mu = 0.85, sigma=0.5
real(kind(1e0)) :: log1(100)
data log1 /&
2.020394, 2.562315, -0.5453395, 1.258546, 0.7704533, &
0.3662717, 0.6885536, 2.619634, -0.49581, 2.972249, &
0.5356222, 0.4262079, 1.023666, 0.8286033, 1.319018, &
2.123659, 0.3904647, -0.1196832, 1.629261, 1.069602, &
0.9438083, 1.314796, 1.404453, -0.5496156, 0.8326595, &
1.570288, 1.326737, 0.9619384, -0.1795268, 1.330161, &
-0.2916453, 0.7430826, 1.640854, 1.582755, 1.559261, &
0.6177695, 1.739638, 1.308973, 0.568709, 0.2587071, &
0.745583, 1.003815, 1.475413, 1.444586, 0.4515438, &
1.264374, 1.788313, 1.062330, 2.126034, 0.3626510, &
1.365612, 0.5044735, 2.51385, 0.7910572, 0.5932584, &
1.140248, 2.104453, 1.345562, -0.9120445, 0.0006519341, &
1.049729, -0.8246097, 0.8053433, 1.493787, -0.5199705, &
2.285175, 0.9005916, 2.108943, 1.40268, 1.813626, &
1.007817, 1.925250, 1.037391, 0.6767235, -0.3574937, &
0.696697, 1.104745, -0.7691124, 1.554932, 2.090315, &
0.60919, 0.4949385, -2.449544, 0.668952, 0.9480486, &
0.9908558, -1.495384, 2.179275, 0.1858808, -0.3715074, &
0.1447150, 0.857202, 1.805844, 0.405371, 1.425935, &
0.3187476, 1.536181, -0.6352768, 0.5692068, 1.706736/
param = 1.0
stderr = 0.0
hess = 0.0
call mle(log1,ipdf,param,iprint=2,usemm=.true., &
mloglike=fval,se=stderr,hess=hess)
end
Univariate Statistics from UVSTA
Variable Mean Variance Std. Dev. Skewness Kurtosis
1 0.9068 0.8600 0.9274 -0.6251 0.9725
Variable Minimum Maximum Range Coef. Var. Count
1 -2.4495 2.9722 5.4218 1.0227 100.0000
Variable Lower CLM Upper CLM Lower CLV Upper CLV
1 0.7228 1.0908 0.6629 1.1606
Maximum likelihood estimation for the logistic distribution
Starting estimates: 0.90677 0.51128
Initial -log-likelihood: 132.75304
-Log-likelihood 132.61487
MLE for parameter 1 0.95341
MLE for parameter 2 0.50944
Std error for parameter 1 0.08845
Std error for parameter 2 0.04364
Hessian
1 2
1 -127.9 -5.7
2 -5.7 -525.4
The data are N = 100 observations generated from the
generalized extreme value distribution with location parameter , scale parameter
, and shape parameter
.
use mle_int
implicit none
integer, parameter :: ipdf=13, npar=3
real(kind(1e0)) :: param(npar), stderr(npar), &
hess(npar,npar)
real(kind(1e0)) :: fval
! Generalized Extreme Value
! oc = 0, scale =1, shape = =-0.25
real(kind(1e0)) :: gev(100)
data gev/ &
0.7688048, 0.1944504, -0.2992029, -0.3853738, -1.185593, &
0.3056149, -0.4407711, 0.5001115, 0.3635027, -1.058632, &
-0.2927695, -0.3205969, 0.03367599, 0.8850839, 1.860485, &
0.4841038, 0.5421101, 1.883694, 1.707392, 0.2166106, &
1.537204, 1.340291, 0.4589722, 1.616080, -0.8389288, &
0.7057426, 1.532988, 1.161350, 0.9475416, 0.4995294, &
-0.2392898, 0.8167126, 0.992479, -0.8357962, -0.3194499, &
1.233603, 2.321555, -0.3715629, -0.1735171, 0.4624801, &
-0.6249577, 0.7040129, -0.3598889, 0.7121399, -0.5178735, &
-1.069429, 0.7169358, 0.4148059, 1.606248, -0.4640152, &
1.463425, 0.9544342, -1.383239, 0.1393160, 0.622689, &
0.365793, 0.7592438, 0.810005, 0.3483791, 2.375727, &
-0.08124195, -0.4726068, 0.1496043, 0.4961212, 1.532723, &
-0.1106993, 1.028553, 0.856018, -0.6634978, 0.3573150, &
0.06391576, 0.3760349, -0.5998756, 0.4158309, -0.2832369, &
-1.023551, 1.116887, 1.237714, 1.900794, 0.6010037, &
1.599663, -0.3341879, 0.5278575, 0.5497694, 0.6392933, &
0.592865, 1.646261, -1.042950, -1.113611, 1.229645, &
1.655998, 0.6913992, 0.4548073, 0.4982649, -1.073640, &
-0.4765107, -0.8692533, -0.8316462, -0.03609102, 0.655814/
! initialize
param=1.0
stderr=0.0
hess = 0.0
call mle(gev, ipdf, param, iprint=2, usemm=.true., &
mloglike=fval, se=stderr, hess=hess)
end
Univariate Statistics from UVSTA
Variable Mean Variance Std. Dev. Skewness Kurtosis
1 0.3805 0.7484 0.8651 0.05492 -0.6240
Variable Minimum Maximum Range Coef. Var. Count
1 -1.3832 2.3757 3.7590 2.2738 100.0000
Variable Lower CLM Upper CLM Lower CLV Upper CLV
1 0.2088 0.5521 0.5769 1.0100
Maximum likelihood estimation for the generalized extreme value distribution
Starting estimates: -0.00888 0.67451 0.00000
Initial -log-likelihood: 135.43820
-Log-likelihood 126.09403
MLE for parameter 1 0.07500
MLE for parameter 2 0.85115
MLE for parameter 3 -0.27960
Std error for parameter 1 0.09467
Std error for parameter 2 0.07007
Std error for parameter 3 0.06695
Hessian
1 2 3
1 -141.1 -51.3 -112.8
2 -51.3 -337.0 -241.0
3 -112.8 -241.0 -438.8
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