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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

Discrete domain: MLE(Poisson) Bayes(Poisson)
MLE(Negative Binomial) Moment(Negative Binomial)
SUM domain: MLE(Gaussian) MDE(Gaussian)
MHDE(Gaussian) Quick(Gaussian)
Least Squares(Gaussian) MLE(Student)
MLE(Non-central Student) MLE(Stable)
McCulloch(Stable) AR and ARMA Series Estimators
GARCH Series Estimator
MAX domain: MLE(EV0) MLE(EV1/EV2)
MLE(EV) MDE(EV)
LRSE(EV)
POT domain: MLE(GP0) Hill(GP1/GP2)
M(GP1) Bayes(GP1)
Grouped Data MLE(GP1) MLE(GP)
Moment(GP) L-Moment(GP)
Drees-Pickands(GP) Slope(GP)
Grouped Data MLE(GP)

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:
Moment Estimate:
Take an estimate of lambda based on the sample correlation coefficient.
Maximum Likelihood Method:
Deduce the likelihood equation and calculate the MLE numerically. The moment estimate serves as the initial value of the iteration procedure.
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:
Sample
This item provides information about the active data set. Xtremes calculates estimates of parameters etc. using these active data.
Estimation
The estimated shape, location and scale parameters alpha, mu and sigma are displayed. In the gamma-mode, alpha is replaced by gamma.
Number of Extremes
Specify a number k of extremes. The estimation will be based on the k largest values of the active data set in the case of Xtremes Univariate Data. In the case of Xtremes Grouped Data, the k upper cells of the grouped sample will be utilized. If one marks the field Automatic Selection, Xtremes will calculate the optimal number of extremes according to the "end of plateau method". Be aware that this is a crude method if it is carried out automatically (see Statistical Analysis, p. 149 for details).
Estimate
Click here to re-evaluate the estimation. This will be necessary after a new selection of parameters.
Plot of functions with estimated parameters:
One may plot the density, the quantile function (qf), the distribution function (df), the median excess function and the mean excess function (with trimming parameter p) of the pertaining distribution. For instance, if one selects the MLE(GP0), Xtremes plots an estimated exponential density etc.
Q-Q-Plot
Draws a Q-Q-plot depending on the estimated shape parameter. The Q-Q-plot is independent of the estimated location and scale parameters. Select Solid line to get the single points connected.
Active parametrization
Estimation is carried out in the model (GP 0 etc.) for which the estimator is defined. By means of Active Parametrization one may choose the alpha- or gamma-parametrization. Select GP 1,2 (alpha) to get an estimate for the alpha- or GP (gamma) for the gamma-parametrization.
Diagram
This option provides a plot of gamma(k), alpha(k), mu(k) or sigma(k) against the number k of upper extremes. Choose gamma, alpha, mu or sigma to get the diagram for the parameter.
Close before plotting
Mark this box to let the dialog box disappear when the curve is plotted.
Local menu of dialog box
The local menu becomes available by a rightclick into the dialog box. It provides the following options:
T-year value
Calulate the 1-1/T -quantile for the estimated distribution.
Bootstrap Confidence Interval/MSE
Calculate a confidence interval and the mean squared error (MSE) for the estimated parameters by using a parametric bootstrap procedure. One must select the respective parameter and specify the number of simulations and confidence level.
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:
Plot of functions with estimated parameters:
One may plot the density, the quantile function (qf), the distribution function (df ) of the pertaining distribution. For instance, if one selects the MLE (EV 1/EV 2), Xtremes will either plot the estimated Frechet- or the Weibull density etc.
Q-Q-Plot
Draws a Q-Q-plot for the estimated shape parameter. The Q-Q-plot is independent of the estimated location and scale parameters. Select Solid line to get the single points connected.
Active parametrization
Estimation is carried out within the model for which the estimator is defined. By means of Active Parametrization one may choose the alpha- or gamma-parametrization.
Close before plotting
Mark this field to let the dialog box disappear when the curve is plotted.

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:
T-year value
Calulate the 1-1/T -quantile for the estimated distribution.
Bootstrap Confidence Interval/MSE
Calculate a confidence interval and the mean squared error (MSE) for the estimated parameters by using a parametric bootstrap procedure. One must select the respective parameter and specify the number of simulations and confidence level.

Options of the local menu are not available within discrete models.

Estimation of discrete distributions

The estimates for the different parameters of the Poisson and Negative Binomial distributions are displayed in this dialog box, calculated by the corresponding MLE's or the Negative Binomial Moment Estimator.
Select the button Histogram an the right-hand side to plot an histogram of the distribution, using the estimated parameters. The other buttons are marked grey when they are disabled. Mark Close before plotting to close the dialog box before the histogram is plotted.
A local menu is not available within discrete models.

© 2005
Xtremes Group · updated Jun 21, 2005