GATE Statistics Testing of Hypotheses Flashcards

24 sample cards from the Testing of Hypotheses deck

Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.

1. 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.

2. 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 $\theta\in\omega$.

3. 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.

4. 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 $\vartheta$.

5. 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).

6. 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.

7. 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 $E_{\theta_0}[U\phi]=\alpha E_{\theta_0}[U]$.

8. 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.

9. 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.

10. 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).

11. 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.

12. 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 $\sigma_0^2$.

13. 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.

14. 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.

15. 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).

16. 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.

17. 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 $E_{\theta_0}[U\phi]=\alpha E_{\theta_0}[U]$.

18. 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 $\alpha$.

19. 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 $\alpha=0.05$).

20. 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 $p_0$.

21. 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.

22. 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.

23. 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.

24. 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.

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