Formula Used
The calculator maximizes the selected model’s log-likelihood. Closed-form estimators are used whenever valid. Numerical methods handle custom and difficult models.
How to Use
- Select a probability distribution.
- Enter raw observations, frequencies, summary statistics, or binomial counts.
- Choose an estimation method and uncertainty settings.
- Add optional initial values and parameter bounds.
- Run the calculation and inspect estimates, diagnostics, charts, and comparisons.
- Copy, print, or export the results.
Example Data
| Distribution | Example input | Estimated parameter |
|---|---|---|
| Bernoulli | 1, 0, 1, 1, 0 | p = success proportion |
| Poisson | 2, 4, 1, 3, 2 | λ = sample mean |
| Normal | 2.4, 3.1, 2.8, 3.7 | μ and σ |
| Exponential | 0.7, 1.2, 0.9, 2.1 | λ = 1 / mean |
| Gamma | 1.2, 2.7, 3.1, 1.8 | Shape and scale |
Frequently Asked Questions
What is maximum likelihood estimation?
MLE chooses parameter values making observed data most plausible. It evaluates a model’s likelihood function. The highest value determines the final estimate.
Why use log-likelihood?
Products of small probabilities can underflow numerically. Logarithms convert products into stable sums. They preserve the same maximizing parameter values.
What is the difference between probability and likelihood?
Probability treats parameters as fixed and data as variable. Likelihood treats observed data as fixed. Candidate parameter values are then compared.
When should numerical optimization be used?
Use it when no closed-form estimate exists. It also supports custom likelihood expressions. Starting values and bounds improve reliable convergence.
What do AIC and BIC measure?
Both compare fitted models using likelihood and complexity. Lower values generally indicate better tradeoffs. BIC penalizes extra parameters more strongly.
Why can standard errors be unavailable?
Standard errors require stable local curvature near the maximum. Flat or boundary solutions weaken that calculation. More data may improve numerical stability.
What is a bootstrap confidence interval?
Bootstrap intervals repeatedly resample the observed dataset. Each resample produces another parameter estimate. Percentiles summarize the resulting uncertainty distribution.
Can this calculator fit multiple distributions?
Yes, compatible continuous models can be compared automatically. Results are ranked using AIC values. Support restrictions may exclude unsuitable distributions.
Is the custom expression evaluated safely?
The calculator uses a restricted mathematical expression parser. Arbitrary PHP code is never executed. Only documented variables and functions are accepted.