Optimize the whole parameter space

Try

Three positive observations are 1,3,6 from an iid Uniform(0,θ) model. Let M=6 be their maximum. Using the density convention that includes the upper endpoint, the likelihood is θ⁻³ for θ≥6 and zero otherwise. Increasing θ above six spreads the same density over a wider interval and decreases likelihood. Thus θ̂=M. There is no interior stationary point. A procedure that only solves “derivative equals zero” would miss the answer entirely.

Predict

Predict what changes if the parameter space is restricted to θ>6 rather than θ≥6 for this observed record. The same supremum is approached but never attained. A maximum likelihood estimate need not exist. The general statistical model usually specifies θ>0 before seeing data; the feasible region then comes from the likelihood. Keep a predeclared parameter restriction separate from an observed support constraint. Boundary behavior is part of optimization, not an optional final check.

Experiment

Inspect the support-sensitive likelihood table. Explain why the likelihood at five is zero rather than 1/125, and why a logarithm cannot erase that distinction.

Candidate θ	Contains every observation?	Likelihood
5	No	0
6	Yes	1/216
8	Yes	1/512
12	Yes	1/1728

For comparison, fit an exponential rate λ to the same data. Its log likelihood is 3 log λ−10λ on λ>0, with derivative 3/λ−10. The unique maximum is λ̂=3/10. Here support does not depend on λ, and an interior derivative calculation works after checking positive curvature sign correctly: the second derivative is negative, not positive.

Understand

A maximum likelihood estimator maximizes the likelihood over the stated parameter space. It may be nonunique, lie on a boundary, fail to exist, or have inconvenient finite-sample bias. For a one-to-one reparametrization η=g(θ), the transformed likelihood is L(g⁻¹(η)); therefore the transformed maximizer is g(θ̂). This invariance follows from the same set of candidate distributions, not from linearity of expectation. A nonlinear transformation of an unbiased estimator is generally not unbiased.

The uniform example has an additional subtlety: continuous densities are defined only up to null sets for each fixed parameter. Choosing an open versus closed endpoint version changes whether a likelihood maximizer is attained at an observed maximum. This unit states its conventional closed-endpoint density when using that estimator. Such version sensitivity is another warning that boundary-dependent models need more care than regular interior likelihood calculations.

Apply

Derive the normal likelihood estimate with unknown mean and variance. For positive variance v, the log likelihood is a constant minus(n/2)log v−Σ(xᵢ−μ)²/(2v). The mean minimizing the squared sum is X̄. Substituting it gives v̂=Σ(xᵢ−X̄)²/n when that sum is positive. If all observations coincide, the likelihood grows without bound as v approaches zero and no positive-variance MLE exists. State this exception instead of presenting a zero variance outside the parameter space as an ordinary solution.

For Bernoulli data containing no successes, the likelihood(1−p)ⁿ is maximized at p=0 on [0,1], but has only a supremum on (0,1). Repair any optimization explanation that ignores this distinction. The data do not make the parameter space irrelevant.

Continue

For each maximum, document the support, parameter space, boundary limits and uniqueness. Numerical optimization can supplement these checks but does not replace them. Continue to Sufficiency and factorization, which asks when a small statistic retains all parameter information available within a specified model. Both likelihood maximization and sufficiency depend on the full family of distributions, not just the observed sample mean.