APTECH SYSTEMS, INC. INTRODUCES TWO NEW GAUSS APPLICATIONS, AND A MAJOR
UPDATE TO ONE OF IT'S MOST POPULAR GAUSS APPLICATIONS

Constrained Maximum Likelihood - NEW
Constrained Optimization - NEW
Maximum Likelihood 4.0 - UPDATED

Important New Software for Statistical Inference in Constrained and
Unconstrained Models

JANUARY 24, 1995 -- Constrained Maximum Likelihood solves general maximum
likelihood problems by incorporating linear and nonlinear, equality and
inequality constraints on parameters. But what is more important, this
module incorporates new techniques for statistical inference in maximum
likelihood models, constrained or unconstrained models.

For constrained models, the correct confidence limits may now be computed -
a capability that hasn't been available before in any other software
package. Most, if not all maximum likelihood estimations contain
parameters that are at least implicitly bounded. Frequently, "tricks" were
used, e.g., squaring variances, to keep them out of undefined territory
(there are no tricks, however, to keep covariance matrices positive
definite - but now this can be done with Constrained Maximum Likelihood).
Statistical inference, whether using tricks or when ignoring the problem,
has never been properly handled. This is doubly unfortunate because in
many cases the correct confidence limits that take into account bounds and
constraints on parameters may be smaller than the incorrect limits.

Now Aptech Systems is making available software for correctly computing
confidence limits for the general constrained maximum likelihood problem
with the new Constrained Maximum Likelihood module.

In recent years an awareness has spread in statistical and econometric
fields of the inadequacy of the usual methods of statistical inference in
maximum likelihood models. In response to this, bootstrapped estimation
and heteroskedastic-consistent standard errors were developed. Several
years ago Aptech Systems introduced heteroskedastic-consistent standard
errors for general maximum likelihood estimation in the Maximum Likelihood
module. This year, we are introducing bootstrapping for general maximum
likelihood estimation in the upgrade to the Maximum Likelihood module as
well as the new Constrained Maximum Likelihood module. This includes
graphical presentation for when the distribution of the coefficients is
not adequately described by its mean and variance.

When models are nonlinear, the distributions of the coefficients may not be
known and may be non-Gaussian. Newly introduced in the Constrained Maximum
Likelihood module and the upgraded Maximum Likelihood 4.0 module are
important diagnostic methods that are generalizations of techniques
developed by Bates and Watts in Nonlinear Regression Analysis: likelihood
profile traces and profile t plots. These methods depict the marginal
univariate and bivariate distributions of the coefficients
non-parametrically, providing assistance in evaluating your coefficient
estimates.

--Constrained Maximum Likelihood--
Available: March 1995

Maximum likelihood models often contain estimates of variances or
covariance matrices. Negative variances and covariance matrices that are
not positive definite produce undefined log-likelihood functions and the
estimation fails, usually with an attempt to take the logarithm of a
negative number. With Constrained Maximum Likelihood you can constrain a
variance or the minimum eigenvalue of a covariance matrix to be greater
than some small number.

Features:

 * Solves general maximum likelihood problem using Sequential
   Quadratic Programming method
 * Linear and nonlinear constraints on parameters
 * Equality and inequality constraints on parameters
 * Confidence limits for constrained problems
 * Bootstrapping with histogram and surface plot output
 * Likelihood profile trace plots
 * Profile t plots
 * Weight observations
 * Fix selected parameters

Applications:

 * Constrain covariance matrices to be positive definitive
 * Bound coefficients to avoid undefined regions
 * Incorporate known information about coefficients
 * Correct statistical inference for constrained problems
 * Diagnose estimation problems with profile trace plots

Constrained Maximum Likelihood will be available for both PC and UNIX
versions of GAUSS. Requires GAUSS version 3.2 or later.

--Constrained Optimization--
Available: March 1995

Constrained Optimization will solve the important Markowitz asset
allocation model - which minimizes portfolio variance, s2 = x'Sx, subject
to x'm = r, where x is a vector of proportions and 0 <= x <= 1, sum(x) =
1, r is the portfolio return, m is the vector of the means and S the
covariance matrix of the observed returns of the portfolio securities.
More significantly, however, Constrained Optimization can easily handle
recent extensions of the Markowitz model that incorporate third and fourth
moments of the observed returns, as well as providing the capability of
adding nonlinear constraints to the model.

Features:

 * Solves standard Nonlinear Programming Problem using Sequential
   Quadratic Programming method
 * Linear and nonlinear constraints on parameters
 * Equality and inequality constraints on parameters

Applications:

 * Solve the Markowitz asset allocation model
 * Solve nonparametric extensions of Markowitz model incorporating
   higher moments
 * Bound Coefficients to avoid undefined regions

Constrained Optimization will be available for both PC and UNIX versions of
GAUSS. Requires GAUSS version 3.2 or later.

--Constrained Maximum Likelihood and
Constrained Optimization Compared--

Both of these new GAUSS Applications perform optimization with general
constraints on parameters.

Constrained Maximum Likelihood solves the general maximum likelihood
problem subject to general constraints on the parameters, that is, it
minimizes:

      sum log P_i(X_i|b)
       i

where a function to compute log P_i(X_i|b) is provided by the user.
Constrained Optimization allows you to provide any twice differentiable
function for optimization.

The essential difference between Constrained Optimization and Constrained
Maximum Likelihood is that the function to be minimized in Constrained
Maximum Likelihood is specialized to the maximum likelihood function and
is designed to handle a data set. Thus, Constrained Optimization is more
general, but if your model incorporates a data set, Constrained Maximum
Likelihood will greatly simplify the programming. Constrained Maximum
Likelihood also contains re-sampling and likelihood profile trace and
profile t plots.

Both Constrained Optimization and Constrained Maximum Likelihood produce
estimates of q subject to equality constraints,

      Ab = B,    G(b) = 0

inequality constraints,

      Cb >= D,    H(b) >= 0

and bounds,

      b_low <= b <= b_up

Linear constraints are separated from the nonlinear constraints because
their derivatives are known and easy to compute, speeding up the
optimization especially if the model contains only linear constraints.
Constrained Optimization and Constrained Maximum Likelihood use the
Sequential Quadratic Programming method. At each iteration the new
direction is computed as the solution of a quadratic programming problem
involving the Hessian (or estimated Hessian) of the function to be
minimized, the gradient of the function, the constraints, and the
gradients of the constraints. A line search is conducted for a step length
that minimizes a "merit" function. The Hessian is next updated by any one
of several methods - either directly or using a choice of the BFGS, DFP,
or BHHH methods.

Bootstrapping

The newly introduced Constrained Maximum Likelihood as well as the update
to the original Maximum Likelihood modules contain functions that produce
mean vectors and covariance matrices, histograms, and bivariate surface
plots of the resampled coefficients. Global variables control the size of
the re-sampling as well as the granularity of the histograms and surface
plots. With default values for the globals, all this information can be
generated by merely replacing the call to Maximum Likelihood or
Constrained Maximum Likelihood with calls to MAXBOOT, NLPMAXBOOT, or
MAXHIST, NLPMAXHIST.

Profile t Plots and Likelihood Profile Trace Plots

The usual standard errors computed from the inverse of the Hessian have
been shown to be potentially misleading in nonlinear models. To determine
the extent of any problems with them, the module functions MAXPROFILE and
NLPMAXPROFILE are provided. The profile t plot provides exact likelihood
intervals for the coefficients, revealing how nonlinear the estimation is.
For linear models, the plotted line for a coefficient is a straight line
through the origin with unit slope. For nonlinear models, the amount of
curvature is diagnostic of the nonlinearity of the estimation. High
curvature suggests that the usual statistical inference using the
t-statistic will be hazardous.

The likelihood profile trace plots provide information about the bivariate
likelihood surfaces. For nonlinear models the profile traces are curved,
showing how the coefficient estimates affect each other and how the
projection of the likelihood contours onto the b_i, b_j plane might look.
If the likelihood surface contours are long and thin, indicating the
coefficients to be collinear, the profile traces will be close together.
If the contours are fat, indicating the coefficients to be more
uncorrelated, the profile traces will tend to be perpendicular. And if the
contours are nearly elliptical, the profile traces will be straight. The
surface contours for a linear model would be elliptical and thus the
profile traces would be straight and perpendicular to each other.
Significant departures of the profile traces from straight, perpendicular
lines, therefore, indicate difficulties with the usual statistical
inference.

--Update - Maximum Likelihood Version 4.0--
Available: March 1995

Since it's introduction over six years ago, the GAUSS Application - Maximum
Likelihood has been an important tool for researchers in many fields. This
latest version has been significantly updated with many new features,
including graphics.

For those times when even the heteroskedastic-consistent standard errors
are not enough, the updated Maximum Likelihood package will automatically
generate bootstrapped estimates along with histograms of their
distributions and surface plots of their bivariate distributions.

New Features:

 * Re-sampling with histogram and surface plot output
 * Likelihood profile trace plots
 * Profile t plots
 * Weighting
 * Fix selected parameters
 * Improved descent and line search

Maximum Likelihood will be available for both PC and UNIX versions of
GAUSS.Requires GAUSS version 3.2 or later.

GAUSS and GAUSS Applications

The GAUSS Mathematical and Statistical System is a matrix-based programming
language with built-in mathematical functions, code debugger, publication
quality graphics, and foreign language interface for incorporating C and
FORTRAN routines into GAUSS programs. GAUSS can be used in either command
mode (interactively) or in edit mode. In command mode; one-line commands,
or small screen-resident programs, can be issued and the results of
calculations seen immediately. In edit mode you can write complex programs
and store them in files.

GAUSS Applications are essentially programs written in the GAUSS language
for performing specialized data analysis. Each GAUSS Application comes
with extensive documentation and GAUSS source code.

For more information contact:

APTECH SYSTEMS, INC.
23804 S.E. Kent-Kangley Road
Maple Valley, WA 98038
Tel: (206) 432-7855
Fax: (206) 432-7832
email: info@aptech.com

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