GATE Statistics Testing of Hypotheses Syllabus

Testing of Hypotheses syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Testing of Hypotheses in GATE Statistics, not a summary of it.

1. 1

   Neyman-Pearson lemma

   1 topic

   • Most powerful tests
2. 2

   Monotone likelihood ratio (MLR) property

   overview

   Examined as a single unit within Testing of Hypotheses — no further topic split in the official outline.

3. 3

   Uniformly most powerful tests

   1 topic

   • Uniformly most powerful tests for families having MLR property
4. 4

   Uniformly most powerful unbiased tests

   1 topic

   • Uniformly most powerful unbiased tests for exponential families
5. 5

   Likelihood ratio tests

   overview

   Examined as a single unit within Testing of Hypotheses — no further topic split in the official outline.

6. 6

   Large sample tests

   overview

   Examined as a single unit within Testing of Hypotheses — no further topic split in the official outline.

Testing of Hypotheses flashcards for GATE Statistics

22 of 49 cards from the Testing of Hypotheses deck — real questions with worked answers.

1. Define a simple hypothesis.

   A hypothesis that completely specifies the probability distribution of the data, i.e. it corresponds to a single point $\theta_0$ in the parameter space (e.g. $H_0:\theta=\theta_0$).

2. Define a composite hypothesis.

   A hypothesis that does not completely specify the distribution; it corresponds to a set of more than one parameter value, e.g. $H_0:\theta\leq\theta_0$ or $H_1:\theta\neq\theta_0$.

3. What is the size (level) $\alpha$ of a test with critical function $\phi$ for testing $H_0:\theta\in\Theta_0$?

   $\alpha=\sup_{\theta\in\Theta_0}E_\theta[\phi(X)]$, the supremum over the null of the probability of rejecting $H_0$.

4. Define the power function $\beta_\phi(\theta)$ of a test with critical function $\phi$.

   $\beta_\phi(\theta)=E_\theta[\phi(X)]=P_\theta(\text{reject }H_0)$, the probability of rejection viewed as a function of the parameter $\theta$.

5. What is a randomized (critical) test function $\phi(x)$ and how is it interpreted?

   A measurable function $\phi:\mathcal{X}\to[0,1]$ where $\phi(x)$ is the probability of rejecting $H_0$ when $X=x$. Values $0$ and $1$ give a nonrandomized test; intermediate values randomize the decision.

6. Define a most powerful (MP) test of level $\alpha$ for testing simple $H_0$ against simple $H_1$.

   A test $\phi$ with $E_{\theta_0}[\phi]\leq\alpha$ such that for every other level-$\alpha$ test $\phi'$, $E_{\theta_1}[\phi]\geq E_{\theta_1}[\phi']$; i.e. it maximizes power at $\theta_1$ among all level-$\alpha$ tests.

7. State the Neyman-Pearson Lemma (sufficiency part) for testing $H_0:\theta=\theta_0$ vs $H_1:\theta=\theta_1$ with densities $f_0,f_1$.

   Any test of the form $\phi(x)=1$ if $f_1(x)>k\,f_0(x)$, $\phi(x)=\gamma$ if $f_1(x)=k\,f_0(x)$, and $\phi(x)=0$ if $f_1(x)<k\,f_0(x)$, with $k\geq0$ and $\gamma\in[0,1]$ chosen so $E_{\theta_0}[\phi]=\alpha$, is a most powerful level-$\alpha$ test.

8. State the existence part of the Neyman-Pearson Lemma.

   For every $\alpha\in[0,1]$ there exists a test of the Neyman-Pearson form (a likelihood-ratio test) with $k$ and $\gamma$ chosen so that $E_{\theta_0}[\phi]=\alpha$ exactly.

9. State the necessity (uniqueness) part of the Neyman-Pearson Lemma.

   If $\phi$ is a most powerful level-$\alpha$ test, then it must have the Neyman-Pearson likelihood-ratio form ($\phi=1$ when $f_1>kf_0$, $\phi=0$ when $f_1<kf_0$) almost everywhere, except possibly on the boundary set $\{f_1=kf_0\}$.

10. In the Neyman-Pearson test, what is the likelihood ratio statistic used?

    $\lambda(x)=\dfrac{f_1(x)}{f_0(x)}$ (the ratio of the alternative density to the null density); we reject for large values of $\lambda(x)$.

11. Why is randomization needed in the Neyman-Pearson test for discrete distributions?

    Because the distribution of the likelihood ratio is discrete, the rejection probability jumps; randomization on the boundary $\{f_1=kf_0\}$ via $\gamma$ lets the size equal exactly $\alpha$ rather than being only attainable at discrete values.

12. For the Neyman-Pearson MP test, what is the relationship between its power and $\alpha$ when $f_0\neq f_1$?

    The power is at least $\alpha$: $\beta\geq\alpha$, with strict inequality $\beta>\alpha$ whenever the two hypotheses are distinguishable (a result related to the test being unbiased).

13. What is the generalized Neyman-Pearson lemma about?

    It maximizes $\int\phi f\,d\mu$ subject to several side constraints $\int\phi f_i\,d\mu=c_i$, $i=1,\dots,m$; the optimal $\phi=1$ when $f>\sum_{i=1}^m k_i f_i$ and $\phi=0$ when $f<\sum k_i f_i$. It is used to derive UMPU and similar tests.

14. Define a uniformly most powerful (UMP) test of level $\alpha$ for $H_0:\theta\in\Theta_0$ vs $H_1:\theta\in\Theta_1$.

    A level-$\alpha$ test $\phi^*$ such that for every $\theta\in\Theta_1$ and every other level-$\alpha$ test $\phi$, $\beta_{\phi^*}(\theta)\geq\beta_\phi(\theta)$; i.e. it is most powerful simultaneously at every alternative.

15. Why do UMP tests generally fail to exist for two-sided alternatives $H_1:\theta\neq\theta_0$?

    The MP test maximizing power for $\theta>\theta_0$ differs from the one maximizing power for $\theta<\theta_0$; no single test is best on both sides simultaneously, so no UMP test exists (motivating UMP unbiased tests).

16. Define the monotone likelihood ratio (MLR) property of a family $\{f_\theta\}$ in a statistic $T(x)$.

    The family has MLR in $T$ if for any $\theta_1<\theta_2$ the ratio $\dfrac{f_{\theta_2}(x)}{f_{\theta_1}(x)}$ is a nondecreasing function of $T(x)$ (over the set where at least one density is positive).

17. State the Karlin-Rubin theorem for testing $H_0:\theta\leq\theta_0$ vs $H_1:\theta>\theta_0$ when the family has MLR in $T$.

    There exists a UMP level-$\alpha$ test given by $\phi(x)=1$ if $T(x)>c$, $\phi(x)=\gamma$ if $T(x)=c$, $\phi(x)=0$ if $T(x)<c$, where $c,\gamma$ are chosen so $E_{\theta_0}[\phi]=\alpha$.

18. For an MLR family in $T$, what is the UMP test of $H_0:\theta\geq\theta_0$ vs $H_1:\theta<\theta_0$?

    Reject for small values of $T$: $\phi(x)=1$ if $T(x)<c$, $\phi=\gamma$ if $T(x)=c$, $\phi=0$ if $T(x)>c$, with $c,\gamma$ chosen so $E_{\theta_0}[\phi]=\alpha$.

19. For an MLR family, what monotonicity property does the power function of the Karlin-Rubin UMP test have?

    Its power function $\beta(\theta)=E_\theta[\phi]$ is nondecreasing in $\theta$ (strictly increasing wherever the power is strictly between 0 and 1).

20. Why does the MLR property let a UMP test for the composite null $H_0:\theta\leq\theta_0$ have size exactly $\alpha$ at $\theta_0$?

    Because the power function is monotone nondecreasing, $\sup_{\theta\leq\theta_0}\beta(\theta)=\beta(\theta_0)=\alpha$, so controlling the rejection probability at the boundary $\theta_0$ controls the entire composite null.

21. 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 $T(x)$.

22. 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 $\bar{X}$.

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