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 Incoming links: Likelihood-ratio test for an exponential support endpoint

Write

a = x_{(1)} = min_{i} x_{i}

T = \sum_{i} x_{i}

and

\overline{x} = T / n

L (θ, β) = β^{- n} exp [- \frac{T - n θ}{β}] 1_{{0 \leq θ \leq a}} .

For fixed feasible

θ

, differentiating its log-likelihood with respect to

β

gives

- \frac{n}{β} + \frac{T - n θ}{β ^{2}} = 0, β (θ) = \overline{x} - θ .

The derivative changes from positive to negative there. Substituting gives the profile likelihood

e^{- n} (\overline{x} - θ)^{- n}

, which increases with

θ

. Therefore the unrestricted maximum-likelihood estimates are

θ = a, β = \overline{x} - a,

while under the null hypothesis

θ = 0

the estimate is

β = \overline{x}

R = \frac{x ^{- n} e ^{- n}}{( x - a ) ^{- n} e ^{- n}} = (1 - \frac{a}{x})^{n} .

Since the likelihood-ratio test rejects for small

R

, the requested form is obtained with

g (x_{1}, \dots, x_{n}) = \frac{x _{(1)}}{x} = \frac{n min _{i} x _{i}}{\sum _{i} x _{i}}, C = {x : (1 - g (x))^{n} \leq k} .

The formulas assume

\overline{x} > a

, which holds almost surely for a continuous sample with

n \geq 2

; coincident observations give an unbounded unrestricted likelihood and can be treated by its supremum.

An exact calibration is available and also explains why Wilks theorem should not be applied here. Under the null, the memoryless property makes

A = n X_{(1)}

and

D = \sum_{i} (X_{i} - X_{(1)})

independent gamma distributions with common scale

β

1

and

n - 1

1 - g = \frac{D}{A + D} \sim Beta (n - 1, 1) .

Consequently, for

0 \leq k \leq 1

Pr_{0} (R \leq k) = k^{(n - 1) / n}, k_{α} = α^{n / (n - 1)} .

This yields an exact test size

α

, independent of the unknown scale. The support endpoint and boundary null make this a likelihood-ratio test for an exponential support endpoint, with a different calibration from a regular one-parameter test.