GATE Statistics Estimation Syllabus

Estimation syllabus — full chapter and topic list

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

1. 1

   Sufficiency

   1 topic

   • Minimal Sufficiency
2. 2

   Factorization Theorem

   2 topics

   • Completeness
   • Completeness of Exponential Families
3. 3

   Ancillary Statistic

   overview

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

4. 4

   Basu’s Theorem and Its Applications

   overview

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

5. 5

   Unbiased Estimation

   1 topic

   • Uniformly Minimum Variance Unbiased Estimation
6. 6

   Rao-Blackwell Theorem

   overview

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

7. 7

   Lehmann-Scheffe Theorem

   overview

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

8. 8

   Cramer-Rao Inequality

   overview

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

9. 9

   Consistent Estimators

   overview

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

10. 10

    Method of Moments Estimators

    overview

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

11. 11

    Method of Maximum Likelihood Estimators and Their Properties

    overview

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

Estimation flashcards for GATE Statistics

20 of 50 cards from the Estimation deck — real questions with worked answers.

1. Define a sufficient statistic for a parameter $\theta$.

   A statistic $T(X)$ is sufficient for $\theta$ if the conditional distribution of the sample $X$ given $T(X)=t$ does not depend on $\theta$. Intuitively, $T$ captures all information in the data about $\theta$.

2. State the Fisher–Neyman Factorization Theorem.

   $T(X)$ is sufficient for $\theta$ if and only if the joint density/pmf factors as $$f(x;\theta) = g(T(x);\theta)\,h(x),$$ where $g$ depends on $x$ only through $T(x)$ and $h(x) \geq 0$ does not depend on $\theta$.

3. Define a minimal sufficient statistic.

   A sufficient statistic $T$ is minimal sufficient if, for any other sufficient statistic $S$, $T$ is a function of $S$. Equivalently, $T$ induces the coarsest partition of the sample space among all sufficient statistics.

4. State the Lehmann–Scheffé characterization used to find a minimal sufficient statistic.

   Suppose the likelihood ratio $\dfrac{f(x;\theta)}{f(y;\theta)}$ is constant in $\theta$ (independent of $\theta$) if and only if $T(x)=T(y)$. Then $T(X)$ is a minimal sufficient statistic for $\theta$.

5. What is the relationship between a minimal sufficient statistic and any other sufficient statistic in terms of dimension/information?

   A minimal sufficient statistic is a function of every sufficient statistic, so it achieves the maximal data reduction while retaining sufficiency. It is the most economical summary that still loses no information about $\theta$.

6. Is a minimal sufficient statistic unique?

   It is unique only up to one-to-one (bijective) transformations. Any one-to-one function of a minimal sufficient statistic is also minimal sufficient.

7. For an i.i.d. sample from $N(\mu,\sigma^{2})$ with both parameters unknown, give a minimal sufficient statistic.

   $T(X) = \left(\sum_{i=1}^{n} X_i,\ \sum_{i=1}^{n} X_i^{2}\right)$, equivalently $\left(\bar{X},\ S^{2}\right)$. It is two-dimensional even though there are two parameters.

8. For an i.i.d. sample from $\text{Uniform}(0,\theta)$, give a minimal sufficient statistic.

   $T(X) = X_{(n)} = \max_i X_i$, the largest order statistic.

9. For an i.i.d. sample from $\text{Uniform}(\theta_1,\theta_2)$, give a minimal sufficient statistic.

   $T(X) = \left(X_{(1)},\ X_{(n)}\right) = \left(\min_i X_i,\ \max_i X_i\right)$.

10. What is the relationship between minimal sufficiency and the maximum likelihood estimator (MLE)?

    The MLE is always a function of any minimal sufficient statistic. If the minimal sufficient statistic is also complete, the MLE (when unbiased) is connected to the UMVUE via Lehmann–Scheffé.

11. Define a complete statistic.

    A statistic $T$ with family of distributions $\{f_T(t;\theta):\theta\in\Theta\}$ is complete if for every measurable function $g$, $$E_\theta[g(T)] = 0 \ \text{for all } \theta \implies g(T)=0 \ \text{a.s. for all }\theta.$$

12. Define a bounded complete statistic and how it relates to completeness.

    $T$ is bounded complete if $E_\theta[g(T)]=0$ for all $\theta$ implies $g(T)=0$ a.s. for every bounded measurable $g$. Completeness implies bounded completeness, but not conversely.

13. What is the intuitive meaning of completeness?

    Completeness means the family of distributions of $T$ is 'rich enough' that no nonzero function of $T$ can have zero expectation for all $\theta$. There is essentially only one unbiased estimator (up to a.s. equivalence) of any estimable function based on $T$.

14. State Basu's Theorem.

    If $T$ is a complete sufficient statistic for $\theta$ and $V$ is an ancillary statistic (its distribution does not depend on $\theta$), then $T$ and $V$ are independent for every $\theta$.

15. Define an ancillary statistic.

    A statistic $V(X)$ is ancillary for $\theta$ if its distribution does not depend on $\theta$. It carries no information about $\theta$ by itself.

16. Use Basu's theorem: for $N(\mu,\sigma^2)$ with $\sigma^2$ known, are $\bar{X}$ and $S^2$ independent, and why?

    Yes. $\bar{X}$ is complete sufficient for $\mu$, and $S^2 = \frac{1}{n-1}\sum (X_i-\bar X)^2$ is ancillary for $\mu$ (its distribution depends only on $\sigma^2$). By Basu's theorem $\bar{X}$ and $S^2$ are independent.

17. For $\text{Uniform}(0,\theta)$ i.i.d. sample, is $X_{(n)}$ complete? Justify briefly.

    Yes, $X_{(n)}$ is complete (and sufficient) for $\theta$. The density of $X_{(n)}$ is $\frac{n t^{n-1}}{\theta^{n}}$ on $(0,\theta)$, and $E_\theta[g(X_{(n)})]=0$ for all $\theta>0$ forces $g\equiv 0$ a.e.

18. Give a classic example of a sufficient statistic that is NOT complete.

    For $\text{Uniform}(\theta,\theta+1)$, the pair $(X_{(1)},X_{(n)})$ is minimal sufficient but not complete: $E_\theta[X_{(n)}-X_{(1)}]$ is constant in $\theta$, so $g = (X_{(n)}-X_{(1)}) - \frac{n-1}{n+1}$ has expectation $0$ for all $\theta$ but is not zero.

19. What is the relationship: does completeness imply minimal sufficiency?

    Yes. (Bahadur's theorem) A statistic that is both complete and sufficient is minimal sufficient, provided a minimal sufficient statistic exists.

20. Does minimal sufficiency imply completeness?

    No. A minimal sufficient statistic need not be complete. Example: $(X_{(1)},X_{(n)})$ for $\text{Uniform}(\theta,\theta+1)$ is minimal sufficient but not complete.

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