Monotone Likelihood Ratio and Uniformly Most Powerful Tests

Intuition Beginner

In the previous step we found the best test when there are only two stories about the data: one fixed null, one fixed alternative. But real questions are rarely that tidy. You usually do not ask "is the drug effect exactly $0$ versus exactly $3$?" You ask "is the effect zero versus anything positive?" The alternative is now a whole range of values, not a single one. So the natural worry is: the best test against an effect of $3$ might be a different test from the best one against an effect of $1$. Which test should you run when you do not know the size of the effect, only its direction?

The good news is that for many standard models the answer does not depend on the size at all. There is one single test that is simultaneously the best against every alternative on one side. A test that beats every rival at every alternative value at once is called uniformly most powerful. When such a test exists, you are done — there is nothing to trade off.

What makes this possible is a structural feature called monotone likelihood ratio. Loosely, it says that bigger values of some summary of the data always count as stronger evidence for bigger values of the parameter, in a consistent way that never reverses. When that holds, the rejection rule "reject if the summary is large enough" is the right rule no matter which positive alternative you have in mind.

The catch comes with two-sided questions — "is the effect zero versus nonzero in either direction?" There, no single test can be best both against positive effects and against negative effects, because the two pull the cutoff opposite ways. That failure is what forces statisticians to add a fairness side condition and, later, to fall back on a general-purpose test.

Visual Beginner

Figure: a horizontal parameter axis with the null value ${\theta }_0$ marked. Three bell-shaped curves sit above it, one centered at ${\theta }_0$, one at a small positive alternative, one at a large positive alternative. A single vertical cutoff on the data axis serves all of them: as the curve slides right (larger true parameter), more of its area falls past the fixed cutoff, so the chance of rejecting climbs steadily. The one cutoff is best for every curve on the right side at once — that is the uniformly-most-powerful picture. A second small panel shows the two-sided case: a curve to the left of ${\theta }_0$ would need the cutoff on the opposite side, so no single cutoff can serve both directions.

 chance of rejecting (power)
  1 |                               ____------  large theta
    |                        __----/
    |                  __--/
    |             __--/                 small theta
    |        __--/        ____------
  a |....__-/___----/...................  <- level a, reached at theta_0
    |__--/   ----
  0 +-------|-----------------------------> theta
         theta_0      (one cutoff serves every theta > theta_0)

Worked example Beginner

Return to the factory line measuring rod lengths, but now average $n=16$ rods so the spread of the average $\bar{X}$ is $2/\sqrt{16}=0.5$. The current target is ${\theta }_0=10$. The real question is not "is the average $10$ or exactly $11$?" but "is the average $10$, or has the line drifted higher by any amount?" So the alternative is every $\theta >10$. We want a false-alarm cap of $5$ in $100$.

Step 1. Pick the rule shape. Drift to any higher value pushes $\bar{X}$ up, and a larger $\bar{X}$ is stronger evidence the higher the true average — the same direction for every alternative above $10$. So one rule fits all: reject when $\bar{X}$ is bigger than a single cutoff $c$.

Step 2. Set the cutoff from the cap, using the null. If the true average is $10$, then $\bar{X}$ is centered at $10$ with spread $0.5$. We want the chance $\bar{X}$ exceeds $c$ to be $5$ in $100$. The bell-curve value leaving $5$ in $100$ in the upper tail is about $1.645$ spreads up, so $c=10+1.645\times 0.5=10.82$.

Step 3. Check it works for two different alternatives. If the true average is $10.5$, the chance $\bar{X}>10.82$ is the area past $(10.82-10.5)/0.5=0.64$ spreads, about $0.26$. If instead the true average is $11$, the chance is the area past $(10.82-11)/0.5=-0.36$ spreads, about $0.64$.

What this tells us. The very same cutoff $10.82$ gave a sensible, best-possible test against an average of $10.5$ and against an average of $11$ — and against every other value above $10$. We never had to know the size of the drift, only its direction. That is the whole point of a uniformly most powerful test: one rule, optimal across the entire one-sided range.

Check your understanding Beginner

Formal definition Intermediate+

Fix the statistical model $\{P_{\theta }:\theta \in \Theta \}$ on $(\mathcal{X},\mathcal{B})$ with $\Theta \subseteq \mathbb{R}$ an interval, densities $p_{\theta }=d{P}_{\theta }/d\mu$ relative to a common $\sigma$-finite $\mu$ (the dominated setting of 45.01.02). A test is a measurable $\phi :\mathcal{X}\to [0,1]$ with power function ${\beta }_{\phi }(\theta )={\mathbb{E}}_{\theta }[\phi (X)]=\int \phi d{P}_{\theta }$, size ${\sup }_{\theta \in {\Theta }_0}{\beta }_{\phi }(\theta )$, and level $\alpha$ if its size is at most $\alpha$ — all as in 45.02.01.

Definition (uniformly most powerful test). For testing $H_0:\theta \in {\Theta }_0$ against $H_1:\theta \in {\Theta }_1$, a level-$\alpha$ test ${\phi }^{\ast }$ is uniformly most powerful (UMP) at level $\alpha$ if for every level-$\alpha$ test $\phi$ and every ${\theta }_1\in {\Theta }_1$,

$${\beta }_{{\phi }^{\ast }}({\theta }_1)\ge {\beta }_{\phi }({\theta }_1).$$

The quantifier over all of ${\Theta }_1$ is the difference from the Neyman-Pearson notion of most powerful at one point: a UMP test is most powerful against every alternative simultaneously.

Definition (monotone likelihood ratio). The family $\{p_{\theta }\}$ has monotone likelihood ratio (MLR) in a real statistic $T(x)$ if there is a function $T$ such that for every pair ${\theta }^{\prime }>\theta$ in $\Theta$, the ratio

$$\frac{p_{{\theta }^{\prime }}(x)}{p_{\theta }(x)}$$

is a nondecreasing function of $T(x)$ on the set $\{x:p_{\theta }(x)>0\text{ or }{p}_{{\theta }^{\prime }}(x)>0\}$ (with the value $+\infty$ assigned where $p_{\theta }=0<p_{{\theta }^{\prime }}$). When the ratio is nonincreasing in $T$ for all ${\theta }^{\prime }>\theta$ the family has MLR in $-T$; the convention here takes the increasing orientation.

A one-parameter exponential family $p_{\theta }(x)=h(x)\exp (\eta (\theta )T(x)-A(\theta ))$ with $\eta$ strictly increasing has MLR in $T$: for ${\theta }^{\prime }>\theta$,

$$\frac{p_{{\theta }^{\prime }}(x)}{p_{\theta }(x)}=\exp ((\eta ({\theta }^{\prime })-\eta (\theta ))T(x)-(A({\theta }^{\prime })-A(\theta ))),$$

and $\eta ({\theta }^{\prime })-\eta (\theta )>0$ makes this increasing in $T(x)$. This is the structural link to 45.01.02: the natural statistic that is sufficient is also the statistic in which the ratio is monotone.

Definition (unbiased test). A level-$\alpha$ test $\phi$ is unbiased if ${\beta }_{\phi }(\theta )\ge \alpha$ for every $\theta \in {\Theta }_1$ — the test is at least as likely to reject under any alternative as under the null. A uniformly most powerful unbiased (UMPU) test is one that is UMP within the class of unbiased level-$\alpha$ tests. Unbiasedness is a side condition that restores existence of an optimum where the unrestricted UMP test fails.

Definition (similar test; Neyman structure). A test $\phi$ is similar of size $\alpha$ on the boundary ${\Theta }_b=\overline{{\Theta }_0}\cap \overline{{\Theta }_1}$ if ${\beta }_{\phi }(\theta )=\alpha$ for every $\theta \in {\Theta }_b$. If $U$ is a sufficient statistic for the nuisance parameter on ${\Theta }_b$, the test has Neyman structure with respect to $U$ if ${\mathbb{E}}_{\theta }[\phi (X)\mid U=u]=\alpha$ for almost every $u$ — the conditional rejection probability given $U$ is exactly $\alpha$. Completeness of $U$ (in the sense of 45.01.03) is the property forcing every similar test to have Neyman structure.

Counterexamples to common slips

• UMP is a both-conditions claim, not just high power. A UMP test must be level $\alpha$ and dominate every level-$\alpha$ competitor at every alternative. The constant test $\phi \equiv 1$ has maximal power but size $1$, not $\alpha$, so it is disqualified; the optimization is constrained.

• MLR is a statement for all pairs ${\theta }^{\prime }>\theta$, not just against the null. Checking the ratio is monotone only at $({\theta }_0,{\theta }_1)$ is the Neyman-Pearson input; MLR demands monotonicity for every ordered pair, which is what makes the rejection region alternative-free across the whole family.

• The support must be fixed for the standard one-parameter exponential conclusion. A location family on a moving support such as $\mathrm{Unif}(0,\theta )$ still has MLR in ${\max }_i{x}_i$ and admits a one-sided UMP test, but it is not an exponential family and the smoothness-based power-function arguments below need separate handling; MLR is the genuinely operative hypothesis, not the exponential form.

• Two-sided alternatives generally have no UMP test. For $H_1:\theta \ne {\theta }_0$ in a normal-mean family, the test best against $\theta >{\theta }_0$ rejects for large $\bar{X}$ and the test best against $\theta <{\theta }_0$ rejects for small $\bar{X}$; no single test is most powerful in both directions, so UMP existence fails and unbiasedness must be imposed.

Key theorem with proof Intermediate+

The central result lifts the single-pair Neyman-Pearson optimum to the whole one-sided family. The mechanism is that under MLR the Neyman-Pearson rejection region for any pair ${\theta }^{\prime }>{\theta }_0$ is the same region $\{T>c\}$, independent of ${\theta }^{\prime }$, so one test serves every alternative at once.

Theorem (Karlin-Rubin). Let $\{p_{\theta }:\theta \in \Theta \}$, $\Theta \subseteq \mathbb{R}$ an interval, have monotone likelihood ratio in $T$. For testing $H_0:\theta \le {\theta }_0$ against $H_1:\theta >{\theta }_0$, fix $\alpha \in (0,1)$ and choose $c$ and $\gamma \in [0,1]$ so that the test

$${\phi }^{\ast }(x)=\{\begin{array}{ll}1& T(x)>c,\\ \gamma & T(x)=c,\\ 0& T(x)<c,\end{array}\text{satisfies}{\mathbb{E}}_{{\theta }_0}[{\phi }^{\ast }(X)]=\alpha .$$

Then ${\phi }^{\ast }$ is UMP at level $\alpha$ for $H_0$ against $H_1$. Moreover its power function ${\beta }_{{\phi }^{\ast }}(\theta )$ is nondecreasing in $\theta$, so its size over ${\Theta }_0=\{\theta \le {\theta }_0\}$ is attained at ${\theta }_0$ and equals $\alpha$ ^{[Casella & Berger §8.3.2]}.

Proof. Existence of $c,\gamma$. Let $F_{{\theta }_0}(t)=P_{{\theta }_0}(T>t)$, the survival function of $T$ under ${\theta }_0$; it is nonincreasing and right-continuous with limits $1$ and $0$. Set $c=\inf \{t:F_{{\theta }_0}(t)\le \alpha \}$ and, when $P_{{\theta }_0}(T=c)>0$, set $\gamma =(\alpha -P_{{\theta }_0}(T>c))/P_{{\theta }_0}(T=c)\in [0,1]$, else $\gamma =0$. Then ${\mathbb{E}}_{{\theta }_0}[{\phi }^{\ast }]=P_{{\theta }_0}(T>c)+\gamma P_{{\theta }_0}(T=c)=\alpha$, by the same boundary-randomization construction proved in 45.02.01.

Monotone power function. Fix ${\theta }^{\prime }>\theta$ and write the likelihood ratio $r(x)=p_{{\theta }^{\prime }}(x)/p_{\theta }(x)$, which by MLR is a nondecreasing function $\rho (T(x))$. The test ${\phi }^{\ast }$ rejects for large $T$, hence for large $r$. Define $k=\rho (c)$. Where $T>c$ one has $r\ge k$ and ${\phi }^{\ast }=1$; where $T<c$ one has $r\le k$ and ${\phi }^{\ast }=0$. So ${\phi }^{\ast }$ agrees with a Neyman-Pearson test for $\theta$ versus ${\theta }^{\prime }$ at critical value $k$, off the boundary $\{r=k\}$. Now compare ${\phi }^{\ast }$ with the constant test $\psi \equiv {\beta }_{{\phi }^{\ast }}(\theta )$, which has the same value ${\beta }_{{\phi }^{\ast }}(\theta )$ at $\theta$. By the Neyman-Pearson sufficiency inequality (the variational identity $({\phi }^{\ast }-\psi )(p_{{\theta }^{\prime }}-k{p}_{\theta })\ge 0$ of 45.02.01),

$${\beta }_{{\phi }^{\ast }}({\theta }^{\prime })-{\beta }_{\psi }({\theta }^{\prime })\ge k({\beta }_{{\phi }^{\ast }}(\theta )-{\beta }_{\psi }(\theta ))=0,$$

and ${\beta }_{\psi }({\theta }^{\prime })={\beta }_{{\phi }^{\ast }}(\theta )$ because $\psi$ is constant. Hence ${\beta }_{{\phi }^{\ast }}({\theta }^{\prime })\ge {\beta }_{{\phi }^{\ast }}(\theta )$: the power function is nondecreasing. In particular ${\sup }_{\theta \le {\theta }_0}{\beta }_{{\phi }^{\ast }}(\theta )={\beta }_{{\phi }^{\ast }}({\theta }_0)=\alpha$, so ${\phi }^{\ast }$ has level $\alpha$.

Uniform most-powerfulness. Fix any ${\theta }_1>{\theta }_0$ and let $\phi$ be any level-$\alpha$ test, so ${\beta }_{\phi }({\theta }_0)\le \alpha$. By the argument above, ${\phi }^{\ast }$ is a Neyman-Pearson test for the simple pair $({\theta }_0,{\theta }_1)$ at critical value $k_1={\rho }_1(c)$, where ${\rho }_1$ is the monotone ratio for $({\theta }_0,{\theta }_1)$, and it has size exactly $\alpha$ at ${\theta }_0$. The Neyman-Pearson lemma part (ii) 45.02.01 then says ${\phi }^{\ast }$ is most powerful at level $\alpha$ for ${\theta }_0$ versus ${\theta }_1$, so ${\beta }_{{\phi }^{\ast }}({\theta }_1)\ge {\beta }_{\phi }({\theta }_1)$. Because ${\theta }_1>{\theta }_0$ was arbitrary and the rejection region $\{T>c\}$ does not depend on ${\theta }_1$, the same ${\phi }^{\ast }$ dominates $\phi$ at every alternative: ${\phi }^{\ast }$ is UMP. $\square$

Bridge. This theorem builds toward the confidence-bound and interval theory of the chapter, where inverting the one-sided UMP test family yields a uniformly most accurate confidence bound, and it appears again in every one-sided exponential-family test — the one-sample $t$, the variance test, the binomial and Poisson tests — whose optimality is this single statement. This is exactly the Neyman-Pearson lemma of 45.02.01 with the alternative quantified away: MLR makes the critical region $\{T>c\}$ independent of which ${\theta }_1>{\theta }_0$ one targets, so the simple-vs-simple optimum becomes a uniform one, and the foundational reason is that under MLR a larger value of $T$ is monotone evidence for a larger $\theta$ uniformly, never reversing. The construction generalises the boundary randomization of the simple lemma to a composite null by the monotone-power-function argument, which shows the worst null point is the boundary ${\theta }_0$. Putting these together, the Karlin-Rubin region is the Neyman-Pearson region read through a sufficient statistic of 45.01.02; the bridge is the factorization that makes $T$ both sufficient and the carrier of the monotone ratio, so optimality and data reduction coincide.

Exercises Intermediate+

Advanced results Master

Why two-sided alternatives admit no UMP test, and the unbiasedness remedy

For a one-parameter family with MLR, the one-sided problem $H_0:\theta \le {\theta }_0$ against $H_1:\theta >{\theta }_0$ has the Karlin-Rubin UMP test, but the two-sided problem $H_0:\theta ={\theta }_0$ against $H_1:\theta \ne {\theta }_0$ does not. The obstruction is structural: by Neyman-Pearson uniqueness the test maximizing power against a single ${\theta }_1>{\theta }_0$ rejects for large $T$, while the test maximizing power against a single ${\theta }_1<{\theta }_0$ rejects for small $T$, and these regions disagree off null sets. A UMP test would have to be both. The resolution is to restrict the competition to unbiased tests, those with ${\beta }_{\phi }(\theta )\ge \alpha ={\beta }_{\phi }({\theta }_0)$ for all $\theta \ne {\theta }_0$. In a smooth family unbiasedness forces ${\beta }_{\phi }^{\prime }({\theta }_0)=0$, a second linear side condition, and the generalized Neyman-Pearson lemma ^{[Lehmann & Romano §4.2]} with the two constraints ${\mathbb{E}}_{{\theta }_0}\phi =\alpha$, ${\mathbb{E}}_{{\theta }_0}[T\phi ]=\alpha {\mathbb{E}}_{{\theta }_0}T$ produces a unique optimum. For a one-parameter exponential family the UMPU test is the equal-tailed two-sided test $\{T<c_1\}\cup \{T>c_2\}$ with the two multipliers fixing $(c_1,c_2)$; the resulting region is alternative-free, so the unbiased optimum is uniform. Unbiasedness is the minimal extra requirement that rescues a uniform optimum from the two-sided collapse.

UMPU tests for exponential families with nuisance parameters

The decisive extension is to multiparameter exponential families with a nuisance parameter. Let $p_{\theta ,\vartheta }(x)=h(x)\exp (\theta U(x)+⟨\vartheta ,V(x)⟩-A(\theta ,\vartheta ))$, with $\theta$ the parameter of interest (testing $H_0:\theta \le {\theta }_0$ or $\theta ={\theta }_0$) and $\vartheta$ the nuisance vector. The statistic $V$ is complete and sufficient for $\vartheta$ on the boundary $\{\theta ={\theta }_0\}$ (the completeness of 45.01.03). The optimal one-sided test rejects for large $U$ conditionally on $V$: it uses the conditional distribution of $U$ given $V=v$, which is itself a one-parameter exponential family in $\theta$ free of $\vartheta$, and applies Karlin-Rubin within each $v$-slice. This conditional construction yields the UMPU test, and it is the theoretical origin of the one-sample and two-sample $t$-tests, the $F$-test for variance ratios, and Fisher's exact test for a $2\times 2$ table — each is the UMPU test of an exponential-family interest parameter after conditioning on the complete sufficient statistic for the nuisance ^{[Lehmann & Romano §4.4]}. The conditioning eliminates the nuisance exactly, with no asymptotic approximation.

Similar tests, Neyman structure, and the conditioning principle

A test is similar of size $\alpha$ on the boundary ${\Theta }_b$ if ${\beta }_{\phi }(\theta )=\alpha$ for all $\theta \in {\Theta }_b$. When $V$ is a sufficient statistic for the family restricted to ${\Theta }_b$, a test has Neyman structure if $\mathbb{E}[\phi \mid V]=\alpha$ almost surely. Any test of Neyman structure is similar — integrate the conditional identity. The converse is exactly where completeness enters: if $V$ is complete sufficient on ${\Theta }_b$, then every similar test has Neyman structure. The proof is one line of completeness: similarity says ${\mathbb{E}}_{\vartheta }[\phi -\alpha ]=0$ for all $\vartheta$, and since $\mathbb{E}[\phi -\alpha \mid V]=g(V)$ is a $\vartheta$-free statistic with ${\mathbb{E}}_{\vartheta }[g(V)]=0$ for all $\vartheta$, completeness forces $g(V)=0$ a.s., i.e. $\mathbb{E}[\phi \mid V]=\alpha$ ^{[Lehmann & Scheffé 1950]}. This reduces the search for an optimal similar test to a family of conditional problems given $V$, each a clean one-parameter testing problem solved by Karlin-Rubin — the technical heart of the UMPU constructions above and the precise sense in which completeness is the bridge from a composite nuisance-laden null to a tractable conditional one.

Confidence bounds by inversion of one-sided UMP tests

Inverting the one-sided UMP test family produces an optimal interval estimate. For each ${\theta }_0$, let $A({\theta }_0)$ be the acceptance region of the UMP level-$\alpha$ test of $H_0:\theta \le {\theta }_0$; the random set $\{{\theta }_0:X\in A({\theta }_0)\}$ is a $1-\alpha$ lower confidence bound for $\theta$. Because the underlying tests are UMP, the bound is uniformly most accurate: it minimizes the probability of covering every false value ${\theta }^{\prime }<\theta$, the confidence-set image of most-powerfulness. This is the duality that organizes the move from testing to interval estimation in the rest of the chapter, with MLR supplying the monotone, nested acceptance regions that make the inverted set an interval rather than a scattered collection.

Synthesis. The central insight is that monotone likelihood ratio quarantines the alternative out of the optimal rejection region: under MLR the Neyman-Pearson region for any pair ${\theta }^{\prime }>{\theta }_0$ is the single set $\{T>c\}$, so the simple-vs-simple optimum of 45.02.01 becomes uniform, and this is exactly the Karlin-Rubin theorem. The foundational reason the two-sided problem breaks is that the alternative now pulls in two directions whose Neyman-Pearson regions are disjoint, so most-powerfulness is dual to a constraint that ties the directions together — unbiasedness, whose smoothness condition ${\beta }^{\prime }({\theta }_0)=0$ is the second multiplier in the generalized lemma, restoring a uniform optimum within the unbiased class. Putting these together, the multiparameter exponential family carries a complete sufficient statistic $V$ for its nuisance, and completeness generalises the boundary-randomization device into a conditioning device: every similar test has Neyman structure, the nuisance is eliminated exactly by conditioning on $V$, and Karlin-Rubin runs inside each conditional slice to give the UMPU test — the structure underlying the $t$-, $F$-, and Fisher-exact tests. The bridge is the test-inversion duality: the one-sided UMP family inverts to a uniformly most accurate confidence bound, so the optimality proved here appears again in interval estimation, and where even unbiasedness fails to single out a uniform optimum — genuinely multiparameter or nonmonotone problems — the likelihood-ratio test of 45.02.03 is the general-purpose replacement, its very necessity certified by the nonexistence results established above.

Full proof set Master

Proposition 1 (MLR implies a monotone power function for threshold tests). If $\{p_{\theta }\}$ has MLR in $T$ and $\phi (x)=1\{T(x)>c\}+\gamma 1\{T(x)=c\}$, then ${\beta }_{\phi }(\theta )={\mathbb{E}}_{\theta }[\phi ]$ is nondecreasing in $\theta$.

Proof. Fix ${\theta }^{\prime }>\theta$. The ratio $r=p_{{\theta }^{\prime }}/p_{\theta }=\rho (T)$ with $\rho$ nondecreasing; set $k=\rho (c)$. Where $T>c$, $r\ge k$ and $\phi =1$; where $T<c$, $r\le k$ and $\phi =0$. Hence $\phi$ is the Neyman-Pearson test for $(\theta ,{\theta }^{\prime })$ at level $k$ off $\{r=k\}$, and the pointwise inequality $(\phi -\psi )(p_{{\theta }^{\prime }}-k{p}_{\theta })\ge 0$ holds for the constant test $\psi \equiv {\beta }_{\phi }(\theta )$. Integrating, ${\beta }_{\phi }({\theta }^{\prime })-{\beta }_{\psi }({\theta }^{\prime })\ge k({\beta }_{\phi }(\theta )-{\beta }_{\psi }(\theta ))=0$ since ${\beta }_{\psi }(\theta )={\beta }_{\phi }(\theta )$. As $\psi$ is constant, ${\beta }_{\psi }({\theta }^{\prime })={\beta }_{\phi }(\theta )$, so ${\beta }_{\phi }({\theta }^{\prime })\ge {\beta }_{\phi }(\theta )$. $\square$

Proposition 2 (Karlin-Rubin, full statement). Under MLR in $T$, the size-$\alpha$ threshold test ${\phi }^{\ast }$ rejecting for large $T$ is UMP level $\alpha$ for $H_0:\theta \le {\theta }_0$ against $H_1:\theta >{\theta }_0$.

Proof. By Proposition 1 the power function of ${\phi }^{\ast }$ is nondecreasing, so ${\sup }_{\theta \le {\theta }_0}{\beta }_{{\phi }^{\ast }}(\theta )={\beta }_{{\phi }^{\ast }}({\theta }_0)=\alpha$, giving level $\alpha$. Fix ${\theta }_1>{\theta }_0$. The same Neyman-Pearson identification shows ${\phi }^{\ast }$ is the level-$\alpha$ Neyman-Pearson test for the simple pair $({\theta }_0,{\theta }_1)$, so by the fundamental lemma's sufficiency part 45.02.01 it is most powerful at ${\theta }_1$ among tests of level $\alpha$ at ${\theta }_0$; a fortiori among tests of level $\alpha$ over all of $\{\theta \le {\theta }_0\}$, since the latter is a subclass. The rejection region $\{T>c\}$ is independent of ${\theta }_1$, so ${\beta }_{{\phi }^{\ast }}({\theta }_1)\ge {\beta }_{\phi }({\theta }_1)$ for every level-$\alpha$ $\phi$ and every ${\theta }_1>{\theta }_0$: UMP. $\square$

Proposition 3 (nonexistence of a two-sided UMP test). In a one-parameter family with strictly MLR in $T$ and continuous power functions, for $H_0:\theta ={\theta }_0$ against $H_1:\theta \ne {\theta }_0$ and $\alpha \in (0,1)$ there is no UMP level-$\alpha$ test.

Proof. Suppose ${\phi }^{\ast }$ were UMP. For each ${\theta }_1>{\theta }_0$, ${\phi }^{\ast }$ must attain the maximal power ${\beta }^{+}({\theta }_1)$, achieved by the upper-tail Karlin-Rubin test ${\phi }_{+}=1\{T>c_{+}\}$; by Neyman-Pearson necessity 45.02.01 ${\phi }^{\ast }={\phi }_{+}$ off the boundary. For each ${\theta }_1<{\theta }_0$, ${\phi }^{\ast }$ must attain ${\beta }^{-}({\theta }_1)$, achieved by the lower-tail test ${\phi }_{-}=1\{T<c_{-}\}$, so ${\phi }^{\ast }={\phi }_{-}$ off the boundary. With strict MLR and continuous laws the sets $\{T>c_{+}\}$ and $\{T<c_{-}\}$ have intersection of probability strictly less than each (they are essentially disjoint when $c_{-}\le c_{+}$, which the size-$\alpha$ constraint forces for $\alpha <1$). Then ${\phi }^{\ast }$ equals two essentially different tests off a null set — a contradiction unless $\alpha \in \{0,1\}$. Hence no UMP test exists. $\square$

Proposition 4 (completeness forces Neyman structure). Let $V$ be a complete sufficient statistic for the family on the boundary ${\Theta }_b$. Then a test $\phi$ is similar of size $\alpha$ on ${\Theta }_b$ if and only if it has Neyman structure with respect to $V$.

Proof. ($\Leftarrow$) If $\mathbb{E}[\phi \mid V]=\alpha$ a.s., then ${\beta }_{\phi }(\theta )={\mathbb{E}}_{\theta }[\mathbb{E}[\phi \mid V]]=\alpha$ for every $\theta \in {\Theta }_b$, so $\phi$ is similar. ($\Rightarrow$) Suppose ${\beta }_{\phi }(\theta )=\alpha$ for all $\theta \in {\Theta }_b$. Since $V$ is sufficient, $g(V):=\mathbb{E}[\phi \mid V]-\alpha$ is a statistic free of the boundary parameter, and ${\mathbb{E}}_{\theta }[g(V)]={\beta }_{\phi }(\theta )-\alpha =0$ for all $\theta \in {\Theta }_b$. Completeness of $V$ (the property of 45.01.03 that ${\mathbb{E}}_{\theta }[g(V)]=0$ for all $\theta$ implies $g(V)=0$ a.s.) forces $g(V)=0$, i.e. $\mathbb{E}[\phi \mid V]=\alpha$ a.s.: Neyman structure. $\square$

Proposition 5 (UMPU one-sided test in the presence of a nuisance parameter). For $p_{\theta ,\vartheta }(x)=h(x)\exp (\theta U+⟨\vartheta ,V⟩-A(\theta ,\vartheta ))$, the UMPU level-$\alpha$ test of $H_0:\theta \le {\theta }_0$ against $H_1:\theta >{\theta }_0$ rejects, conditionally on $V=v$, when $U>c(v)$ with $\gamma (v)$ on the boundary, where $c(v),\gamma (v)$ are determined by ${\mathbb{E}}_{{\theta }_0}[\phi \mid V=v]=\alpha$.

Proof. Any unbiased test is similar on $\{\theta ={\theta }_0\}$ by continuity of the power function; $V$ is complete sufficient for $\vartheta$ there, so by Proposition 4 the test has Neyman structure: ${\mathbb{E}}_{{\theta }_0}[\phi \mid V=v]=\alpha$ for a.e. $v$. The conditional density of $U$ given $V=v$ is the one-parameter exponential family $\propto \exp (\theta u)d{P}^{U\mid V=v}(u)$, free of $\vartheta$, with MLR in $U$. Within each slice, maximizing the conditional power against $\theta >{\theta }_0$ subject to the conditional size constraint $\alpha$ is the Karlin-Rubin problem (Proposition 2), whose solution rejects for $U>c(v)$ with boundary randomization $\gamma (v)$. A test maximizing conditional power in every slice maximizes the unconditional power among all tests of Neyman structure, hence among all similar — in particular all unbiased — level-$\alpha$ tests. So the conditional Karlin-Rubin test is UMPU. $\square$

Connections Master

• The Neyman-Pearson lemma 45.02.01 is the engine of this unit: every optimality claim here reduces to the simple-vs-simple lemma applied pair-by-pair, with monotone likelihood ratio guaranteeing that the resulting rejection region $\{T>c\}$ is the same for all alternatives on one side, so the most-powerful test of that unit becomes the uniformly-most-powerful test of this one.

• Sufficiency and exponential families 45.01.02 supply the structural hypothesis under which the theory bites: a one-parameter exponential family has MLR in its natural sufficient statistic, so the same $T$ that reduces the data optimally also carries the monotone ratio, and the multiparameter exponential form is what makes the nuisance-parameter conditioning of the UMPU construction exact.

• Ancillarity, completeness, and Basu's theorem 45.01.03 provide the completeness that converts similarity into Neyman structure: a complete sufficient statistic for the nuisance parameter forces every similar test to have constant conditional size $\alpha$, which is precisely what reduces the composite nuisance-laden problem to a conditional one solvable by Karlin-Rubin, the technical core of the $t$-, $F$-, and Fisher-exact tests.

• The likelihood-ratio test 45.02.03 is the successor that takes over exactly where this unit's optimality fails: when no UMP test exists (two-sided alternatives) and no UMPU test is available (genuinely multiparameter or non-exponential problems), the likelihood-ratio statistic is the general-purpose substitute, and the nonexistence results proved here are the motivation for needing it.

Historical & philosophical context Master

The uniformly-most-powerful concept and the monotone-likelihood-ratio condition were isolated by Samuel Karlin and Herman Rubin in The theory of decision procedures for distributions with monotone likelihood ratio (Ann. Math. Statist. 27, 1956, 272-299) ^{[Karlin & Rubin 1956]}, who showed that MLR is the exact structural property making the one-sided Neyman-Pearson region alternative-free, and developed the corresponding theory of monotone decision procedures beyond testing. The earlier Neyman-Pearson program of 1933 had already recognized that one-sided normal and exponential tests retained their form across alternatives, but the general MLR formulation that explains why was the Karlin-Rubin contribution.

The unbiasedness side condition and the theory of similar tests trace to Jerzy Neyman and Egon Pearson's work on similar regions in the 1930s and to Erich Lehmann and Henry Scheffé's Completeness, similar regions, and unbiased estimation (Sankhyā 10, 1950, 305-340) ^{[Lehmann & Scheffé 1950]}, which introduced completeness precisely as the property securing that every similar test has Neyman structure, tying the testing theory to the estimation theory through one algebraic condition. Lehmann's Testing Statistical Hypotheses (1959; third edition with Joseph Romano, 2005) organized the UMP, UMPU, and similar-test theory into the canonical form, deriving the standard parametric tests as conditional UMPU tests of exponential-family interest parameters and establishing the test-confidence-set duality that carries one-sided optimality into uniformly most accurate confidence bounds.

Bibliography Master

@article{karlinrubin1956monotone,
  author  = {Karlin, Samuel and Rubin, Herman},
  title   = {The theory of decision procedures for distributions with monotone likelihood ratio},
  journal = {Annals of Mathematical Statistics},
  volume  = {27},
  number  = {2},
  pages   = {272--299},
  year    = {1956}
}

@book{lehmannromano2005testing,
  author    = {Lehmann, Erich L. and Romano, Joseph P.},
  title     = {Testing Statistical Hypotheses},
  edition   = {3rd},
  series    = {Springer Texts in Statistics},
  publisher = {Springer},
  year      = {2005}
}

@book{casellaberger2002statistical,
  author    = {Casella, George and Berger, Roger L.},
  title     = {Statistical Inference},
  edition   = {2nd},
  publisher = {Duxbury},
  year      = {2002}
}

@article{lehmannscheffe1950completeness,
  author  = {Lehmann, Erich L. and Scheff\'e, Henry},
  title   = {Completeness, similar regions, and unbiased estimation. Part I},
  journal = {Sankhy\={a}},
  volume  = {10},
  number  = {4},
  pages   = {305--340},
  year    = {1950}
}

@article{neymanpearson1936similar,
  author  = {Neyman, Jerzy and Pearson, Egon S.},
  title   = {Contributions to the theory of testing statistical hypotheses},
  journal = {Statistical Research Memoirs},
  volume  = {1},
  pages   = {1--37},
  year    = {1936}
}