Foundations — Statistical estimation — MLE, method of moments

4.9.17 · D1 · Maths › Probability Theory & Statistics › Statistical estimation — MLE, method of moments

Yeh page assume karti hai ki aapne parent note ka ek bhi symbol pehle kabhi nahi dekha. Hum unhe ek ek karke build karte hain, har ek pichle ke upar, har ek ek picture se anchor karke. Ant mein aap parent note ko ek sentence ki tarah padh paayenge.

1. Data: raw numbers $x_1,x_2,\dots ,x_n$

Topic ko yeh kyun chahiye? Parent page ki har recipe "given data $x_1,\dots ,x_n$…" se shuru hoti hai. Agar numbers ka koi naam na ho, hum unke baare mein kuch bhi nahi bol sakte.

2. Badi vs chhoti letters: $X$ (random) vs $x$ (observed)

Topic ko yeh kyun chahiye: ek estimator abhi bhi ghoomte hue $X$'s se bana hota hai (isliye woh khud bhi random hai aur hum uske spread ke baare mein pooch sakte hain), jabki ek estimate woh frozen number hai jo woh ek baar $x$'s ke land hone par deta hai. "Estimator vs estimate" ka poora distinction isi capital-vs-lowercase choice mein rehta hai.

3. Summation $\sum$ aur average $\bar{x}$

Topic ko yeh kyun chahiye: parent page ke lagbhag har answer ($\hat{\lambda }=1/\bar{x}$, $\hat{p}=\bar{x}$, $\hat{\mu }=\bar{x}$) "average, possibly flipped ya reshaped" hai. Agar aapke paas $\bar{x}$ hai, toh aapke paas aadhe results hain.

4. Parameter $\theta$ aur uska guess $\hat{\theta }$

Topic ko yeh kyun chahiye: estimation precisely yahi hai ki "$x_i$ se $\hat{\theta }$ produce karo". Parent page ka har boxed answer kisi hat ke liye ek formula hai.

5. Probability distribution, density $f$, aur "$;\theta$"

Topic ko yeh kyun chahiye: MLE literally in hill-heights ko aapke observed points par read karta hai aur unhe multiply karta hai. MoM in hills ke averages read karta hai. Dono $f(x;\theta )$ se shuru hote hain.

6. Expectation $E[\cdot ]$ aur moments ${\mu }_k$

Topic ko yeh kyun chahiye: MoM = "${\mu }_k=m_k$ set karo aur solve karo". Aap woh sentence bhi nahi padh sakte dono moment ideas ke bina.

7. Variance ${\sigma }^2$ aur standard deviation $\sigma$

Topic ko yeh kyun chahiye: normal distribution $\mathcal{N}(\mu ,{\sigma }^2)$ mein do dials hain, $\mu$ aur ${\sigma }^2$. Parent page dono ko estimate karta hai, aur famous "divide by $n$ vs $n-1$" bias story puri tarah ${\hat{\sigma }}^2$ ke baare mein hai.

8. Product $\prod$ — kyun likelihoods multiply hoti hain

Topic ko yeh kyun chahiye: likelihood ek product ke roop mein paida hoti hai. "Independent ⇒ multiply" samajhna hi poori wajah hai ki $L$ mein $\prod$ hai.

9. Logarithm $\log$ / $\ln$ — $\prod$ ko $\sum$ mein badalna

Topic ko yeh kyun chahiye: har MLE derivation $L\to \ell =\log L$ jaati hai, taaki $\prod$ $\sum$ ban jaaye aur calculus bearable ho jaaye.

10. Derivative $\frac{d\ell }{d\theta }$ aur use zero set karna

Prerequisite map

Ise top-down padho: raw data do streams ko feed karta hai — moment stream (average, expectation) jo MoM ko power deta hai, aur likelihood stream (density, product, log, derivative) jo MLE ko power deta hai. Dono final guess $\hat{\theta }$ mein pour hote hain.

Equipment checklist

Har item ko ek question ki tarah padho; ::: ke baad ka answer woh readiness hai jo aapko already feel honi chahiye.