How to perform MLE calibration on t location scale with boundaries on the parameters?

I need to calibrate a t-locationScale distribution over a set of data of historical returns on financial data.

Using mle(), it returns with parameters that are not consistent with my assumptions: I need the degrees of freedom "nu" to be higher or equal to 3.

If I try to use LowerBound, UpperBound or fmincon as additional inputs the function doesn't consider these input without data truncation as reported in the helper (but I don't need to truncate or censor data).

In RStudio is possible to fit this data with these assumptions using fitdist with mle using the method "lbfgs". Is it possible to replicate also in matlab?

Thanks

Answer 1

[praguna manvi]

Hi @Gabriele Guglielmi,

According to the documentation for "mle", it is possible to enforce upper and lower bound constraints on the "nu" variable when a custom pdf is defined. Please refer to the sections of "LowerBound" / "UpperBound" under the "Other Options" link below :

https://www.mathworks.com/help/stats/mle.html#mw_fa1b2dab-d83f-48a9-8615-995d9e24131e

It’s possible to define a pdf for the “tLocationScale” distribution using “tpdf”” which implements student's t-distribution pdf as follows:

custompdf = @(x, mu, sigma, nu) tpdf((x - mu) / sigma, nu) / sigma;
% pass this as an argument with other constraints to mle as follows:
[paramEstimates, paramCIs] = mle(data, 'pdf', custompdf, 'start', start, 'lower', lb, 'upper', ub, 'options', ...
    statset('MaxIter', 1e5, 'MaxFunEvals', 1e5));

Answer 2

[Srinivas]

% Example data

data = randn(100, 1); % replace with your actual data

% Initial parameter estimates

start = [mean(data), std(data), 5]; % Example: mean, std, degrees of freedom

% Parameter bounds

lb = [-Inf, 0, 1]; % mu can be any value, sigma > 0, nu > 0

ub = [Inf, Inf, Inf]; % no upper limits for mu, sigma, nu

% Custom PDF function

custompdf = @(x, mu, sigma, nu) tpdf((x - mu) / sigma, nu) / sigma;

% MLE estimation

[paramEstimates, paramCIs] = mle(data, 'pdf', custompdf, 'start', start, ...

'lower', lb, 'upper', ub, 'options', statset('MaxIter', 1e5, 'MaxFunEvals', 1e5));

% Display results

disp('Parameter Estimates:');

disp(paramEstimates);

disp('Confidence Intervals:');

disp(paramCIs);

% Example data

data = randn(100, 1); % replace with your actual data

% Initial parameter estimates

start = [mean(data), std(data), 5]; % Example: mean, std, degrees of freedom

% Parameter bounds

lb = [-Inf, 0, 1]; % mu can be any value, sigma > 0, nu > 0

ub = [Inf, Inf, Inf]; % no upper limits for mu, sigma, nu

% Custom PDF function

custompdf = @(x, mu, sigma, nu) tpdf((x - mu) / sigma, nu) / sigma;

% MLE estimation

[paramEstimates, paramCIs] = mle(data, 'pdf', custompdf, 'start', start, ...

'lower', lb, 'upper', ub, 'options', statset('MaxIter', 1e5, 'MaxFunEvals', 1e5));

% Display results

disp('Parameter Estimates:');

disp(paramEstimates);

disp('Confidence Intervals:');

disp(paramCIs);

Answer 3

[Srinivas]

% Example data

data = randn(100, 1); % replace with your actual data

% Initial parameter estimates

start = [mean(data), std(data), 5]; % Example: mean, std, degrees of freedom

% Parameter bounds

lb = [-Inf, 0, 1]; % mu can be any value, sigma > 0, nu > 0

ub = [Inf, Inf, Inf]; % no upper limits for mu, sigma, nu

% Custom PDF function

custompdf = @(x, mu, sigma, nu) tpdf((x - mu) / sigma, nu) / sigma;

% MLE estimation

[paramEstimates, paramCIs] = mle(data, 'pdf', custompdf, 'start', start, ...

'lower', lb, 'upper', ub, 'options', statset('MaxIter', 1e5, 'MaxFunEvals', 1e5));

% Display results

disp('Parameter Estimates:');

disp(paramEstimates);

disp('Confidence Intervals:');

disp(paramCIs);