Question bank — Properties of estimators — unbiasedness, consistency, efficiency

False. $\hat{\theta }=X_1$ (sirf pehla data point) $\mu$ ke liye unbiased hai lekin iska variance sab $n$ ke liye ${\sigma }^2$ rehta hai, toh ye kabhi home in nahi karta — unbiasedness ek fixed-$n$ property hai, consistency ek limit property hai.

False. Divide-by-$n$ sample variance har finite $n$ ke liye biased low hai, phir bhi iska bias $\propto 1/n\to 0$ hai, toh ye phir bhi ${\sigma }^2$ par converge karta hai aur consistent hai.

False. Ye sirf asymptotic unbiasedness hai (centre $\theta$ approach kar raha hai). Variance ka bhi vanish hona zaroori hai; ek distribution sahi centred ho sakti hai phir bhi hamesha spread out reh sakti hai.

True — lekin sirf isliye kyunki unbiased estimators ke liye $\text{MSE}=\text{Var}+0^2=\text{Var}$, toh comparison variance par reduce ho jaata hai. Dekho Bias–Variance Tradeoff.

False on two counts. Ye sirf unbiased estimators ko bound karta hai (ek biased constant $\hat{\theta }=0$ ki zero variance hoti hai), aur ye sirf regularity conditions ke under hold hoti hai — differentiable likelihood aur support jo $\theta$ par depend na kare. Dekho Fisher Information.

False. Divide-by-$n$ estimator biased hai lekin iska variance chhota hai, aur normal data ke liye iska MSE actually chhota hota hai — unbiasedness aur MSE-optimal ek cheez nahi hai.

False. Sirf linear $g$ unbiasedness preserve karti hain. Jensen's inequality se $\mathbb{E}[{\hat{\theta }}^2]=\text{Var}+{\theta }^2>{\theta }^2$, toh ${\hat{\theta }}^2$, ${\theta }^2$ ko over-estimate karta hai.

False. Bound ek floor hai; kabhi kabhi koi bhi unbiased estimator use tak pahunch nahi paata. Tab MVUE (minimum-variance unbiased estimator) best available hota hai, lekin phir bhi $1/I(\theta )$ se strictly upar baith sakta hai.

True. $\text{Var}(\bar{X})={\sigma }^2/n$, toh $n\to 2n$ se ${\sigma }^2/2n$ milti hai — exactly half. Ye shrinking-to-zero variance hi wajah hai ki $\bar{X}$ consistent hai (ek Law of Large Numbers statement).

False. Uniform$(0,\theta )$ ke liye MLE ${\max }_i{X}_i$ hai, jo biased low hai ($\mathbb{E}[\max ]=\frac{n}{n+1}\theta <\theta$). MLEs sirf asymptotically unbiased aur consistent hoti hain — dekho Maximum Likelihood Estimation.

Deviations $\bar{X}$ se measure hote hain, true $\mu$ se nahi, aur $\bar{X}$ data ke andar baitha hai, toh squared deviations systematically chhote hote hain. Tumne mean estimate karne mein ek degree of freedom kharch kiya; correct karne ke liye $n-1$ se divide karo.

Number $1.5>1$ sahi hai lekin reading ulti hai: $A$ ki $B$ se relative efficiency $\text{Var}(B)/\text{Var}(A)$ hai; yahan $A$ ki variance badi hai, toh $B$ zyada efficient hai.

Ek tiny fixed bias jo $n$ ke saath nahi shrinkta, consistency rok deta hai — estimator $\theta +0.02$ par converge karta hai, $\theta$ par nahi. Consistency ke liye bias ka $\to 0$ hona zaroori hai, sirf chhota hona nahi.

Consistency convergence in probability hai, jo $n\to \infty$ ke saath sampling distribution ke baare mein ek statement hai, na ki kuch runs ke baare mein. Kuch lucky samples limit ke baare mein kuch prove nahi karte.

Sign galat hai: $\text{MSE}=\text{Var}+{\text{Bias}}^2$. Dono terms non-negative hain aur add hote hain; bias kabhi mean squared error kam nahi karta.

Equal weighting sirf tab optimal hai jab observations ki equal variance ho. Agar kuch $X_i$ zyada noisy hain, toh variance-weighted average plain $\bar{X}$ ko beat karta hai.

Kyunki $\bar{X}$ data par fit hota hai aur use hug karta hai, deviations $(X_i-\bar{X})$ average par $(X_i-\mu )$ se chhote hote hain; algebra deta hai $\mathbb{E}[\sum (X_i-\bar{X}{)}^2]=(n-1){\sigma }^2$, toh $n-1$ se divide karna unbiasedness restore karta hai.

Har observation $\theta$ ke baare mein sirf finite information rakhta hai; Fisher information $I(\theta )$ measure karta hai ki likelihood kitni sharply peak karti hai, aur ek flatter peak matlab $\theta$ ko tightly pin nahi kar sakte — toh variance $1/I(\theta )$ se neeche nahi ja sakti.

Ek unbiased estimator phir bhi wildly variable ho sakta hai (jaise $X_1$), kisi bhi diye gaye sample mein $\theta$ se bahut dur bikh sakta hai. Sahi centred rehna bekar hai agar wobble kabhi nahi shrinkta — tumhe chhoti variance aur consistency bhi chahiye.

Sirf variance comparison biased estimators ko unfairly reward karta hai (ek constant ki zero variance hoti hai). MSE = Var + Bias² estimator se dono uski wobble aur off-centredness ke liye charge karta hai, toh ye unhe equal footing par compare karta hai.

Ek value ki taraf shrink karna ek chhote bias ke badle variance mein bada drop trade karta hai; jab variance saving added squared bias se zyada ho, total MSE girta hai — yahi Bias–Variance Tradeoff action mein hai.

Law of Large Numbers kehta hai $\bar{X}\stackrel{P}{\to }\mu$; yahi bilkul $\bar{X}$ ke $\mu$ ke liye consistent hone ki definition hai, jo $\text{Var}(\bar{X})={\sigma }^2/n\to 0$ se drive hoti hai.

Independent observations log-likelihood mein additively contribute karte hain, aur information expected squared score hai, toh information add hoti hai: $I_n(\theta )=n{I}_1(\theta )$ — zyada data, proportionally zyada information, isliye CRLB floor $1/n$ ki tarah girta hai.

Sirf ek degenerate case $\theta =7$ mein; har doosre $\theta$ ke liye iska expectation $7\ne \theta$ hai, toh ye biased hai. Iska zero variance hai, jo dikhata hai ki sirf "sabse chhoti variance" akela ek bura criterion kyun hai.

Har observed value $\le \theta$ hai, toh unka maximum $\theta$ se kabhi exceed nahi kar sakta aur almost surely short fall karta hai; isliye $\mathbb{E}[\max ]=\frac{n}{n+1}\theta <\theta$, ek strictly-below bias jo $n$ badhne par shrinkta hai.

${\sigma }^2/n$ ke zariye MSE argument collapse ho jaata hai kyunki ${\sigma }^2=\infty$. $\bar{X}$ phir bhi Law of Large Numbers ke under consistent ho sakta hai (jise sirf finite mean chahiye), lekin neat variance-based reasoning fail ho jaati hai — tumhe zyada strong tool chahiye.

Nahi — CRLB attain karne ke liye unbiased hona zaroori hai aur variance $1/(n{I}_1(\theta ))\to 0$ hona chahiye, toh iska MSE vanish hota hai aur ye automatically consistent hai. Is classical sense mein efficiency teeno properties mein sabse strong hai.

Ye $\frac{0}{0}$ hai — undefined. Ek single point ke saath $\bar{X}=X_1$, toh deviation zero hai aur zero degrees of freedom hain; ek observation se spread estimate nahi kar sakte.

Nahi. Unbiasedness ke liye sirf expectation ki linearity chahiye, aur consistency ke liye sirf $\text{Var}\to 0$ chahiye. CLT $\bar{X}$ ke sampling distribution ki shape (approximate normality) describe karta hai, jo ek alag aur finer question hai.

Koi nahi — ye by definition equally efficient hain, kyunki efficiency unbiased estimators mein variances compare karti hai aur yahan tie hai. Relative efficiency exactly $1$ hoti hai.