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Estimate Menu
The Estimate Menu contains entries to apply parameteric estimation
procedures to data sets. The options presented depend on the selected
mode and domain, so you will not see all of the options within one
particular mode. The following
options are provided
by the Estimate Menu:
Univariate Mode
Multivariate Mode
MAX domain: Gumbel-McFadden (EV) Marshall-Olkin (EV) Huesler-Reiss (EV) POT domain: Gumbel-McFadden (GP) Huesler-Reiss (GP)
Univariate D(iscrete) Domain
MLE(Poisson)
The maximum likelihood estimator for parameter lambda is given by the
sample mean. Additionaly, the estimation procedure will give
the p-value for the
Chi-square Goodness-of-Fit test in the Poisson model (recall that the
p-value is the smallest significance level for which the null
hypothesis is rejected).
See Dialog options for detailed
information.
Bayes(Poisson)
Compute the Bayes estimator of lambda in the
Poisson model based on a gamma prior with shape parameter s and
reciprocal scale parameter d (see Statistical Analysis,
page 114). The default prior distribution is the standard exponential
distribution.
Plot the prior and the posterior density.
MLE(Negative Binomial)
The maximum likelihood estimator for the Negative Binomial distribution
is calculated numerically.
See Dialog options for detailed
information.
Moment(Negative Binomial)
Given a sample x[i] of nonnegative integers, the moment
estimates r[n] and p[n] of the parameters r and p are the solutions
to the equations
r(1-p) = x und r(1-p)/p**2 = s**2 ,
where x and s**2 denote the sample mean and the sample variance. We
have
p[n] = x/s**2 und r[n] = x**2/(s**2 - x) .
See Dialog options for detailed
information.
Univariate SUM Domain
MLE(Gaussian)
The maximum likelihood estimator (MLE) in the Gaussian model is given
by the sample mean and sample standard deviation.
See Dialog options for detailed
information.
MDE(Gaussian)
Let h[n](x) be a histogram of the data set. The MDE minimizes the L2
distance between h[n](x) and the Gaussian density
phi(mu, sigma),
i.e. it returns mu and sigma such that the function
D(mu,sigma) :=
Integral ( phi(mu,sigma)(x) - h[n](x) )**2 dx
is minimized.
Note that the MHDE (Gaussian)
utilizes the Hellinger distance H, instead.
If the estimator is applied to a grouped data set, then the histogram is
constructed using the partition stored in the data set. In case of
univariate data, a histogram is constructed using the following partition:
t[1] := x[1:n]
t[20] := x[n:n]
t[i] := t[1]+(i-1)*19*(t[n]-t[1]), 2 <= i <= 18 .
See Dialog options for detailed information.
MHDE(Gaussian)
The Gaussian MHDE is a minimum distance
estimator that utilizes the Hellinger distance H
H(mu,sigma) :=
( 2 ( 1 -
Integral(phi(mu,sigma)(x)h[n](x))**(1/2)dx )**(1/2)
instead of the L2 distance. For construction of the pertaining histograms,
see MDE (Gaussian).
See Dialog options for detailed
information.
Quick(Gaussian)
See Statistical Analysis, page 95.
Least Squares(Gaussian)
The location and scale parameters mu and sigma are estimated
by a trimmed least squares method.
See Dialog options for detailed information.
MLE(Student)
The shape parameter alpha > 0 and the scale parameter
sigma > 0 of a Student distribution with location parameter
equal to zero (see Statistical Analysis, page 94) are
computed.
MLE(Non-central Student)
MLE(Stable)
The tail-index parameter alpha > 0 in ( 0 , 2], the
skewness parameter beta in ( -1 , 1 ), a real
location parameter mu, and a scale parameter
sigma > 0 in the continuous parameterization (see
Statistical Analysis, Section 6.3) are computed.
The subsequent McCulloch estimator is taken as an initial
estimator. This procedure is taken from the STABLE program.
McCulloch(Stable)
The same parameter as in the preceding lines are computed
based on sample quantiles.
AR and ARMA Series Estimators
Consider data (t,x[t]) of type Xtremes Time Series,
where x[t] are realizations of a zero-mean, stationary
series { X[t] }. First subtract the sample mean, if this is
necessary. X[t] will be defined by means of a white noise
process { Z[t] }.
AR(p): Yule-Walker
The Yule-Walker estimator computes estimates for the
coefficients phi[1], ... , phi[p] of the AR(p) process
X[t] = phi[1] X[t-1] + ... + phi[p] X[t-p] + Z[t]
and the white noise variance. The estimation is based on the
Yule-Walker equations. Parameter estimates are
obtained by replacing the theoretical autocovariances by
their sample counterparts in the Yule-Walker equations.
In the Estimation dialog box, one must specify the
order p of the
AR polynomial. The order of the MA polynomial must be zero.
Select Yule-Walker AR(p) and execute Estimate.
ARMA(p,q): Hannan-Rissanen
The Hannan-Rissanen algorithm uses linear regression to
establish estimates for the parameters and the white noise
variance of an ARMA(p,q) process.
X[t] - phi[1] X[t-1] - ... - phi[p] X[t-p] =
Z[t] + theta[1] Z[t-1] + ... + theta[q] Z[t-q]
.
For this purpose, estimates of the unobserved
white noise values z[t], ... , z[t-q] are computed.
In the Estimation dialog box, one must specify the
orders p and q of the AR and MA polynomials.
ARMA(p,q): Innovations Algorithm
One obtains estimates of
the parameters and the white noise variance of a
causal ARMA(p,q) process
X[t] - phi[1] X[t-1] - ... - phi[p] X[t-p] =
Z[t] + theta[1] Z[t-1] + ... + theta[q] Z[t-q]
.
In the Estimation dialog box one must specify
the orders p and q of the AR and MA polynomials.
One must also enter a value for the maximum autocovariance
time lag m which is needed to compute the parameter estimates.
ARMA(p,q): MLE
To obtain MLEs of the parameters of a causal
ARMA(p,q) process, one must first
apply one of the preceding estimators.
The MLEs are numerically computed by means of a Newton-Raphson
procedure. Press the MLE ARMA(p,q) button in the
dialog box. If the initially estimated process is not causal,
the MLE cannot be applied and an error message appears.
Literature: Brockwell and Davis (1996)
GARCH Series Estimators
This is the Quasi MLE of the three parameters in the
GARCH(1,1)-model.
The Yule-Walker-Estimator delivers the initial values.
Afterwards the Newton-Raphson procedure is employed to
determine solutions of the likelihood-equations. If this
procedure fails, the gradient procedure is utilized.
Univariate MAX Domain
MLE(EV0)
The MLEs mu(n) and sigma(n) of the location and scale
parameters are evaluated numerically.
See Dialog options for detailed
information.
MLE(EV1/EV2)
MLEs for the shape and scale parameters alpha and sigma with
a fixed location parameter mu = 0 are evaluated for the
Frechet (EV 1) (respectively, the Weibull (EV 2)
model) whenever all data are positive (respectively,
all data are negative).
See Dialog options for further information.
MLE(EV)
The MLE (EV) is numerically evaluated by an iteration procedure with
the LRSE (EV) as initial value.
The MLE is the location of a local maximum of the likelihood
function if the iterated values remain in the region
gamma > -1.
It is not clarified what is actually
computed if the data are generated under gamma < -1 .
See Dialog options for detailed information.
MDE(EV)
Let h[n](x) be a histogram of the active data. The MDE minimizes the
Hellinger distance between h[n](x) and an EV
density g(gamma, mu, sigma), i.e. it
returns gamma, mu and sigma such that
H(mu,sigma) :=
[ 1-Integral (g(gamma,mu,sigma)(x) h[n](x))**(1/2)
dx]**(1/2)
is minimized.
If the estimator is applied to grouped data, then the histogram is
constructed using the partition given by the data set. For
univariate data, a histogram is constructed using the partition
t[1] := x[1:n]
t[20] := x[n:n]
t[i] := t[1]+(i-1)*19*(t[n]-t[1]), 2 <= i <= 18 .
See Dialog options for detailed information.
LRSE(EV)
The LRSE is a linear combination of ratios of spacings (RS's)
r = (x([nq2]:n)-x([nq1]:n))/(x([nq1]:n)-x([nq0]:n)) ,
with q0=q, q1=q**a, q2=q**(2 a). Then, one obtains an
estimator of the shape parameter by
gamma(n) = log r/log (1/a) .
The LRSE returns the mean of gamma(n) for q = i/n,
i = 1, ... , [n/4], and
a= ( log((n+i)/n)/log(i/n) )**(1/2) .
See Dialog options for detailed information.
Univariate POT Domain
MLE(GP0)
This is the maximum likelihood estimator for
parameters sigma and mu based on
the k largest values of a data set in the exponential (GP 0)
model.
See Dialog options for detailed information.
Hill(GP1/GP2)
The Hill estimator is based on the k
largest data generated under Pareto (GP1) or a beta (GP2)
distribution. The second modeling is applied if all data
are negative. The Hill estimator corresponds to a MLE in the
restricted Pareto model.
See Dialog options for detailed information.
M(GP1)
The M-estimate for the shape parameter alpha is based on the k largest
values of data sets generated under a Pareto distribution. It is obtained as
a solution of the M-equation (see Statistical Analysis, p. 141).
See Dialog options to get detailed information
about basic options for the M-estimator. Additionaly, a value for the
parameter b of the M-function must be entered in the
corresponding edit field.
Bayes(GP1)
This option and the pertaining dialog box are the most ambitious ones
in the menu system. First study carefully the corresponding passages
in Statistical Analysis, particularly the pages 143-145 about
the alpha, eta parameterization.
The upper part of the dialog box (cf. Fig. 5.6 on page 162 in
Statistical Analysis) is related to the usual estimator
dialog boxes in the univariate pot domain. The output is given in the
usual alpha or gamma parameterization.
The lower part concerns the alpha, eta parameterization.
Notice that the internal computations and the specification of the
prior and posterior distributions concern this parameterization.
The priors for alpha and eta are gamma and,
respectively, reciprocal gamma distributions.
See Dialog options to get detailed information
about basic options for the Bayes estimator.
Grouped Data MLE(GP1)
Maximum likelihood estimate for parameters alpha and
sigma in case
of grouped data related to the Pareto (GP 1) distribution. In analogy to
the MLE
for Xtremes Univariate Data which is based on the k upper order
statistics
of the sample, the MLE for Xtremes Grouped Data is based on the k
upper cells of the grouped data set.
See Dialog options for detailed information.
MLE(GP)
Maximum likelihood estimator for parameters gamma and
sigma
based on data sets governed by a generalized Pareto (GP) distribution.
See Dialog options for detailed information.
Moment(GP)
Moment estimate for parameter gamma of generalized Pareto
distributions (a better name would be Log-Moment estimate).
The estimate is related to the Hill estimate and computed
with the k largest values of the active data set.
See Dialog options for detailed information.
L-Moment(GP)
This is the L-Moment estimator in the full GP model as
described in Statistical Analysis, Section 11.4.
Take care that the true shape parameter gamma
is smaller than 1.
See Dialog options for detailed information.
Drees-Pickands(GP)
Drees-Pickands estimate of the parameter gamma of generalized
Pareto distributions. The estimate is a mixture of Pickands
estimates based on the k largest values of the underlying sample.
See Dialog options for detailed information.
Slope(GP)
The slope beta of the least squares line, fitted to the mean
excess function right of the k largest observations, is close
to gamma/(1-gamma), and, therefore,
gamma[k] = beta/(1+beta) is
a plausible estimate of gamma.
See Dialog options for detailed information.
Grouped Data MLE(GP)
Maximum likelihood estimate for parameters gamma and
sigma in case
of grouped data governed by generalized Pareto (GP) distributions. In
analogy to the MLE
for Xtremes Univariate Data which is based on the k upper order
statistics
of the sample, the MLE for Xtremes Grouped Data is based on the k
upper cells of the grouped data set.
See Dialog options for detailed information.
Multivariate MAX Domain
Gumbel-McFadden (EV)
First consider a bivariate data set. It is assumed that the
data in the first and second component are governed by an EV
distribution (taken in the gamma-parametrization).
The parameters in the univariate marginals are estimated by the
MLE(EV). After the transformation (9.15) in Statistical Analysis,
using the estimated parameters,
assume that the data are governed by a standard Gumbel-McFadden
df L[lambda] with dependence parameter lambda. The
estimate of lambda is based on the sample
correlation coefficient.
Thus, we use the PTE (piecing-together) method to obtain a
parametric estimate. Such a PTE may also serve as an initial value
of an iteration procedure to evaluate the MLE in the full bivariate
model.
Generally, the pairwise PTE is evaluated. The pairwise dependence
parameters are displayed in a matrix. First, make sure that
there are names attributed to the different columns.
Marshall-Olkin (EV)
In contrast to the estimation in the Gumbel-McFadden and
Huesler-Reiss models, it is assumed that the given data are governed
by a standard Marshall-Olkin distribution. The estimator
in Statistical Analysis, formula (9.35), is applied to
each pair. The output consists of a single dependence
parameter lambda (if two columns are given) or a matrix of
pairwise evaluated dependence
parameters lambda. First, make sure that names are attached
to the different columns of the data set (use the data editor).
Huesler-Reiss (EV)
First consider a bivariate data set. It is assumed that the
data in the first and second component are governed by an EV
distribution (taken in the gamma-parametrization).
The parameters in the univariate marginals are estimated by the
MLE(EV). After the transformation (9.34) in Statistical Analysis,
using the estimated parameters,
assume that the data are governed by a standard Huesler-Reiss
df H[lambda] with dependence parameter lambda. It remains
to construct an estimate of lambda. Two methods are available:
Thus, we use the PTE (piecing-together) method to obtain a
parametric estimate. Such a PTE may also serve as an initial value
of an iteration procedure to evaluate the MLE in the full bivariate
model.
Generally, the pairwise PTE is evaluated. The pairwise dependence
parameters are displayed in a matrix.
First, make sure that
there are names attributed to the different columns.
Multivariate POT Domain
Gumbel-McFadden (GP)
Not implemented yet!
Huesler-Reiss (GP)
Estimate the dependence parameter lambda
in the original parameterization or theta
in the canonical parameterization.
Choose the exponential or full GP model for the
univariate margins.
Also plot the pertaining parametric canonical dependence
function.
For example, apply this facility to data generated
under a Huesler-Reiss (EV) distribution with the
number k of univariate extremes being sufficiently
small.
Basic options for estimators in POT domain
Each dialog box of an estimator in the
POT domain provides the
following items:
The local menu is not available for discrete models.
Basic options for estimators in MAX domain
Each dialog box of an estimator in the
MAX domain provides the following items:
Options for MLE(EV 0) and estimators in SUM domain
Estimates for the location and scale
parameter mu and sigma are displayed.
Select one of the buttons on the right-hand side to plot either the
density, the quantile function (qf) or
the distribution function (df) of the pertaining
distribution.
To draw a Q-Q-plot with
respect to the standard distribution, mark the
corresponding box.
A local menu is available by means of a rightclick within the dialog
box. It provides the following options:
Options of the local menu are not available within discrete
models.
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