GATE Statistics Testing of Hypotheses Syllabus

Show that the one-parameter exponential family $f_\theta(x)=h(x)\exp\{\eta(\theta)T(x)-A(\theta)\}$ has MLR. In what statistic, and under what condition?

It has MLR in $T(x)$ provided the natural parameter $\eta(\theta)$ is strictly increasing in $\theta$, since $\dfrac{f_{\theta_2}}{f_{\theta_1}}=\exp\{[\eta(\theta_2)-\eta(\theta_1)]T(x)-[A(\theta_2)-A(\theta_1)]\}$ is increasing in $T(x)$.

Give the UMP test for $H_0:\mu\leq\mu_0$ vs $H_1:\mu>\mu_0$ based on $X_1,\dots,X_n\sim N(\mu,\sigma^2)$ with $\sigma^2$ known.

Reject $H_0$ when $\bar{X}>\mu_0+z_\alpha\dfrac{\sigma}{\sqrt{n}}$, equivalently when $Z=\dfrac{\sqrt{n}(\bar{X}-\mu_0)}{\sigma}>z_\alpha$; it is UMP because the normal family has MLR in $\bar{X}$.

Does a UMP test exist for $H_0:\mu=\mu_0$ vs $H_1:\mu\neq\mu_0$ in $N(\mu,\sigma^2)$, $\sigma$ known? Why?

No. The MP test for $\mu>\mu_0$ rejects for large $\bar{X}$ while the MP test for $\mu<\mu_0$ rejects for small $\bar{X}$; no single test is UMP on both sides, so we use the UMP unbiased two-sided test instead.

Define an unbiased test of level $\alpha$.

A test $\phi$ is unbiased if its power is at least $\alpha$ at every alternative and at most $\alpha$ on the null: $\beta_\phi(\theta)\leq\alpha$ for $\theta\in\Theta_0$ and $\beta_\phi(\theta)\geq\alpha$ for $\theta\in\Theta_1$.

Define a uniformly most powerful unbiased (UMPU) test.

A test that is UMP within the restricted class of all unbiased level-$\alpha$ tests: it is unbiased of level $\alpha$ and has power at least as great as any other unbiased level-$\alpha$ test, simultaneously at every alternative.

What is a similar test (test similar on the boundary), and why is it relevant to UMPU tests?

A test is similar of size $\alpha$ on a set $\Theta'$ (typically the boundary between $\Theta_0$ and $\Theta_1$) if $\beta_\phi(\theta)=\alpha$ for all $\theta\in\Theta'$. Under continuity of the power function, any unbiased test is similar on the boundary, so UMPU tests are found among similar tests.

State the relationship: unbiased + continuous power function $\Rightarrow$ what property on the boundary?

If the power function $\beta_\phi(\theta)$ is continuous in $\theta$ and $\phi$ is unbiased, then $\phi$ is similar on the boundary $\omega=\overline{\Theta_0}\cap\overline{\Theta_1}$, i.e. $\beta_\phi(\theta)=\alpha$ for all $\theta\in\omega$.

What is the role of Neyman structure in constructing UMPU tests for exponential families?

If a sufficient statistic $U$ for the boundary is complete, every similar test has Neyman structure, meaning $E[\phi(X)\mid U]=\alpha$ a.s. Reducing the similarity requirement to conditional size-$\alpha$ given $U$ lets one apply the (generalized) Neyman-Pearson lemma conditionally to obtain the UMPU test.

Write the multiparameter exponential family form used for UMPU theory.

$f_{\theta,\vartheta}(x)=h(x)\exp\{\theta\,U(x)+\sum_{j=1}^k \vartheta_j T_j(x)-A(\theta,\vartheta)\}$, where $\theta$ is the parameter of interest and $T=(T_1,\dots,T_k)$ is sufficient for the nuisance parameters $\vartheta$.

For the exponential family $f_\theta(x)=C(\theta)h(x)e^{\theta U(x)}$, give the UMPU test of $H_0:\theta=\theta_0$ vs $H_1:\theta\neq\theta_0$.

$\phi(x)=1$ if $U(x)<c_1$ or $U(x)>c_2$, $\phi=\gamma_i$ if $U(x)=c_i$, $\phi=0$ if $c_1<U(x)<c_2$, where $c_1,c_2,\gamma_1,\gamma_2$ solve $E_{\theta_0}[\phi]=\alpha$ and $E_{\theta_0}[U\phi]=\alpha\,E_{\theta_0}[U]$ (the unbiasedness/derivative condition).

What two conditions determine the constants of the two-sided UMPU test for a one-parameter exponential family in $U$?

(1) Size: $E_{\theta_0}[\phi(U)]=\alpha$. (2) Unbiasedness (zero-derivative of power at $\theta_0$): $E_{\theta_0}[U\,\phi(U)]=\alpha\,E_{\theta_0}[U]$. Together they pin down the two critical values and randomization probabilities.

For UMP one-sided tests in exponential families, which constants and conditions are needed, in contrast to the two-sided case?

One-sided UMP needs a single critical value $c$ and randomization $\gamma$ determined by the single size condition $E_{\theta_0}[\phi]=\alpha$; the two-sided UMPU additionally requires the unbiasedness (derivative) condition $E_{\theta_0}[U\phi]=\alpha E_{\theta_0}[U]$.

Give the UMPU test of $H_0:\mu=\mu_0$ vs $H_1:\mu\neq\mu_0$ for $N(\mu,\sigma^2)$ with $\sigma$ known.

Reject when $|Z|>z_{\alpha/2}$ where $Z=\dfrac{\sqrt{n}(\bar{X}-\mu_0)}{\sigma}$; by symmetry the equal-tailed test $\bar{X}<\mu_0-z_{\alpha/2}\sigma/\sqrt{n}$ or $\bar{X}>\mu_0+z_{\alpha/2}\sigma/\sqrt{n}$ is UMPU.

Give the UMPU test of $H_0:\mu=\mu_0$ vs $H_1:\mu\neq\mu_0$ for $N(\mu,\sigma^2)$ with $\sigma^2$ unknown.

The two-sided $t$-test: reject when $|t|>t_{n-1,\alpha/2}$ where $t=\dfrac{\sqrt{n}(\bar{X}-\mu_0)}{S}$ and $S^2=\frac{1}{n-1}\sum(X_i-\bar{X})^2$. It is UMPU, with $S^2$ playing the role of the nuisance-eliminating statistic.

Give the UMP one-sided test for $H_0:\mu\leq\mu_0$ vs $H_1:\mu>\mu_0$ in $N(\mu,\sigma^2)$ with $\sigma$ unknown.

Strictly no UMP test exists when $\sigma$ is unknown; the one-sided $t$-test rejecting when $t=\dfrac{\sqrt{n}(\bar{X}-\mu_0)}{S}>t_{n-1,\alpha}$ is UMP unbiased (UMPU).

For testing the variance $H_0:\sigma^2\leq\sigma_0^2$ vs $H_1:\sigma^2>\sigma_0^2$ in $N(\mu,\sigma^2)$ with $\mu$ known, give the UMP test.

Reject when $\dfrac{\sum_{i=1}^n(X_i-\mu)^2}{\sigma_0^2}>\chi^2_{n,\alpha}$; the family has MLR in $T=\sum(X_i-\mu)^2$, so this one-sided $\chi^2$ test is UMP.

For $H_0:\sigma^2=\sigma_0^2$ vs $H_1:\sigma^2\neq\sigma_0^2$ in $N(\mu,\sigma^2)$, $\mu$ unknown, what test is UMPU and how do its critical values differ from the equal-tailed test?

The UMPU test rejects when $T=\sum(X_i-\bar{X})^2/\sigma_0^2$ falls outside $(c_1,c_2)$, where $c_1,c_2$ satisfy the size condition plus the unbiasedness condition. The cutoffs are NOT the equal-tailed $\chi^2$ quantiles; they are determined so that the power has zero derivative at $\sigma_0^2$.

What is the key reason a UMP (not just UMPU) test exists for one-sided hypotheses in MLR families?

By Karlin-Rubin, the MLR structure ensures the Neyman-Pearson MP test against any single alternative $\theta_1>\theta_0$ does not depend on the particular $\theta_1$ (it always rejects for large $T$), so one test is simultaneously most powerful for all alternatives.

Compare: when does one use UMP vs UMPU tests (one-sided vs two-sided)?

One-sided hypotheses with MLR structure admit genuine UMP tests. Two-sided hypotheses (and one-sided with nuisance parameters) generally have no UMP test, so one restricts to unbiased tests and seeks the UMPU test.

State the condition under which a one-sided UMP test exists in a one-parameter exponential family with natural parameter $\eta(\theta)$.

If $\eta(\theta)$ is strictly monotone (say increasing) in $\theta$, then for $H_0:\theta\leq\theta_0$ vs $H_1:\theta>\theta_0$ a UMP test exists rejecting for large $T(x)$ (the natural sufficient statistic).

Why is completeness of the boundary sufficient statistic important for UMPU tests?

Completeness guarantees that every similar test has Neyman structure ($E[\phi\mid T]=\alpha$), which is what reduces the search for the optimal unbiased test to a conditional Neyman-Pearson problem given $T$, eliminating the nuisance parameters.

What is the power function derivative condition for unbiasedness at an interior boundary point $\theta_0$ of a two-sided test?

If $\theta_0$ is interior to where $H_1$ holds on both sides, unbiasedness forces $\beta_\phi(\theta_0)=\alpha$ to be a minimum, hence $\beta_\phi'(\theta_0)=0$. For exponential families this becomes $E_{\theta_0}[U\phi]=\alpha E_{\theta_0}[U]$.

In Neyman-Pearson testing, distinguish Type I and Type II error and relate them to $\alpha$ and power.

Type I error: rejecting $H_0$ when true, probability $\leq\alpha$ (the size). Type II error: failing to reject $H_0$ when $H_1$ true, probability $\beta_{II}=1-\text{power}$. The MP test minimizes $\beta_{II}$ (maximizes power) for fixed $\alpha$.

How is the Neyman-Pearson lemma used to find the MP test for $N(0,1)$ vs $N(1,1)$ from a single observation $X$? Give the rejection region.

The likelihood ratio $\dfrac{f_1}{f_0}=\exp\{x-\tfrac12\}$ is increasing in $x$, so the MP test rejects for large $x$: reject when $X>z_\alpha$ (e.g. $X>1.645$ for $\alpha=0.05$).

For the Bernoulli/Binomial family $\binom{n}{x}p^x(1-p)^{n-x}$, in what statistic does it have MLR, and what is the UMP test of $H_0:p\leq p_0$ vs $H_1:p>p_0$?

MLR in $T=\sum X_i$ (the number of successes). The UMP test rejects for large $T$: $\phi=1$ if $T>c$, $\phi=\gamma$ if $T=c$, $\phi=0$ if $T<c$, with $c,\gamma$ giving size $\alpha$ under $p_0$.

For a one-parameter exponential family, is the family with MLR also guaranteed to give UMP tests for the two-sided alternative? Explain.

No. MLR yields UMP tests only for one-sided alternatives. For $H_0:\theta=\theta_0$ vs $H_1:\theta\neq\theta_0$ no UMP test exists; the best available is the UMPU test using two critical values.

What is the relationship between UMP and UMPU tests when a UMP test exists?

If a UMP level-$\alpha$ test exists, it is automatically unbiased (its power $\geq\alpha$ everywhere on $H_1$ since the trivial test $\phi\equiv\alpha$ is a competitor), and hence it is also the UMPU test. UMPU is a strictly weaker optimality used when no UMP test exists.

State the unbiasedness inequality showing the NP most powerful test is unbiased.

For the MP test of simple $H_0$ vs simple $H_1$, comparing against the trivial test $\phi\equiv\alpha$ (which has power $\alpha$ at $\theta_1$) shows the MP test has power $\beta\geq\alpha$, so it is unbiased.

Summarize the logical chain used to derive a UMPU two-sided test in a multiparameter exponential family.

(1) Unbiased + continuous power $\Rightarrow$ similar on boundary; (2) boundary sufficient statistic $T$ complete $\Rightarrow$ similar tests have Neyman structure $E[\phi\mid T]=\alpha$; (3) apply the generalized Neyman-Pearson lemma conditionally on $T$ to maximize conditional power subject to size and zero-derivative constraints; (4) the resulting conditional test, free of nuisance parameters, is UMPU.