---
# System prepended metadata

title: Statystyka

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# Statystyka

## Previous exam from Statistics course
<a href = http://wmirepo.xyz/Statystyka_A/Statystyka_-_Egzamin_-_2021-2022_zimowy.pdf>Link </a>

## Relaks
https://krunker.io/?game=FRA:21rzj

## Lecture notes
https://github.com/TheFebrin/UNIVERSITY/tree/master/Statistics%20and%20Linear%20Models/Lectures

Regularity conditions:
- R0: The pdfs are distinct -> $\theta \not= \theta' \implies f(x_i, \theta) \not= f(x_i, \theta')$
- R1: The pdfs have common support for all $\theta$
- R2: $\theta_0 \in int(\Theta_0)$
- R3: The $pdf(x, \theta)$ is twice differentiable as a function of $\theta$
- R4: The integral $\int f(x,\theta) dx$ can be differentiated twice under the integral sign as a function of $\theta$ 
- R5:
- R6:

## Task 1 [Maurycy]
![](https://i.imgur.com/5xpYZz5.png)
* Likelihood function: $L:\Theta \rightarrow \mathbb{R}$
$$L(\theta)=L(\theta, \underline{x}) = \prod_{i=1}^nf(x_i, \theta)$$
* Log-likelihood function: $l:\Theta \rightarrow \mathbb{R}$
$$l(\theta) = log{L(\theta)} = \sum_{i=1}^n log{f(x_i,\theta)}$$
$\theta \in \Theta$, $\underline{x} = (x_1, x_2, \ldots, x_n)$, $f(x_i, \theta)$ - pdf with parameter $\theta$
*MLE - maximum likelihood estimator
$$MLE = \hat{\theta} \iff \hat{\theta} = \underset{\theta \in \Theta}{\operatorname{argmax}} L(\theta, \underline{x})$$
* Fisher information
$$I(\theta) = \mathbb{E}[(\frac{\partial\log f(x,\theta)}{\partial\theta})^2]$$
* Theorem 1
Under assumptions (R0) - (R1):
$$
\forall_{\theta \neq \theta_0 }\quad\lim_{n\rightarrow\infty} P_{\theta_0}[L(\theta_0, \underline{x}) > L(\theta, \underline{x})] = 1
$$
jus the statement

* Theorem 2
asymptotic varianca of the estimator
optimal variance
![](https://i.imgur.com/NsJ8b9e.jpg)
Basic idea that if thata hat is the estimator, then g(theta hat) is the MLE of g(theta).

* Collorary 1
![](https://i.imgur.com/9ZhuGHM.jpg)

## Task 2 [Piotrek]
![](https://i.imgur.com/bt4fzaR.png)
* Efficiency of an estimator:
An efficient estimator is an estimator that estimates the quantity of interest in some “best possible” manner. The notion of “best possible” relies upon the choice of a particular loss function — the function which quantifies the relative degree of undesirability of estimation errors of different magnitudes.
![](https://i.imgur.com/NAuL5vm.png)

* Theorem 1 (Cramér–Rao inequality)
![](https://i.imgur.com/kkPsKPM.jpg)

Q: What is a difference between these two formulation? (look at the arrow on the image below)
The change is based on the assumnption that we assume unbiasness
![](https://i.imgur.com/pwp2roK.jpg)


* Corollary 1
![](https://i.imgur.com/2LMwJhb.jpg)


* Theorem 2
provide 3 asymptotic properties of the MLE (0 mean)
normality, optimiality and assimptotic unbiasness
![](https://i.imgur.com/GPG23Wp.jpg)



## Task 3 [Piotrek]
![](https://i.imgur.com/0OUjSNB.png)

* asymptotic efficienct of an estimator
![](https://i.imgur.com/r8dtEZX.png)

* minimum variance unbiased estimator
![](https://i.imgur.com/mjsiaxm.png)

* sufficient statistic
no other statistic that can be calculated from the same sample provides any additional information as to the value of the parameter
![](https://i.imgur.com/yTfeEmB.png)
![](https://i.imgur.com/pbnFavT.png)


cond distribution of the sample, stat is not depend on the parameter


## Task 4 [Oskar]
![](https://i.imgur.com/egfrU67.png)

* Fisher–Neyman factorization theorem - (Q) provide a name of the theorem that allows one to find the suffitient statistic
![](https://i.imgur.com/FhCR0pf.png)
![](https://i.imgur.com/Iy47Jom.png)


* Rao-Blackwell theorem:
finding minimum variance unbiased estimator
what is recepit 
![](https://i.imgur.com/z0ObxGb.png)
![](https://i.imgur.com/bBg8ECN.png)


* Theorem 2
link between MLE and sufficient statistic
![](https://i.imgur.com/q6qOmAn.png)
![](https://i.imgur.com/Av5tvSl.png)

## Task 5 [Krzysztof]
![](https://i.imgur.com/3wyt7aT.png)

complete statistic:
![](https://i.imgur.com/ZGoJs3W.jpg)

regular exponential class (also positive and negative example of regular exponential class):
![](https://i.imgur.com/aDgSVUJ.jpg)


Lehmann-Scheffe theorem:
![](https://i.imgur.com/Acfat7m.jpg)
Extra assumptions to Rao-blackwell: completness and MVUE is unique.

Example question:
* What is the difference between Rao-Blackwell theorem and Lehmann-Scheffe
    * when we assume completness (in L-S) then the resulting MVUE is unique 


Theorem 2:
![](https://i.imgur.com/A01QpyI.png)



## Srask 6 🥶 []
![](https://i.imgur.com/KCBfiNX.png)

test is at significance level alfa <=

alfa- size =
power under the alternative of the UMP test is at least alpha

* Definition 1 Neyman-Pearson Lemma
![](https://i.imgur.com/FzTBC0O.png)

* Theorem 1
![](https://i.imgur.com/35u2xzX.png)

* Corollary 1
![](https://i.imgur.com/kdNMatB.png)



## Plask 7 & 8
![](https://i.imgur.com/KAl6p2I.png)

* family with monotone likelihood ratio
![](https://i.imgur.com/lFXJIFP.png)

* Karlin Rubin Theorem
![](https://i.imgur.com/w2n4b33.png)


* likelihood ratio test 
![](https://i.imgur.com/IvreUxc.png)
![](https://i.imgur.com/Ejzgg3N.png)

## Task 9 [Oskar]
![](https://i.imgur.com/HDOfNwp.png)

* asymptotic likelihood ratio test
![](https://i.imgur.com/ABjr3Bq.png)
![](https://i.imgur.com/9gedMjy.png)


* Rao score test
![](https://i.imgur.com/AUfn1Uo.png)

* Wald test 
![](https://i.imgur.com/NrrqelO.png)


* Theorem 1
![](https://i.imgur.com/Y3YhHkL.png)



## Task 10 (za zero punktów) [Dawid]
just definition (gradient)
![](https://i.imgur.com/cyz66Ez.png)

Just simple definition of the matrix is needed here. :)

https://www.youtube.com/watch?v=pneluWj-U-o&t=626s&ab_channel=MutualInformation

https://en.wikipedia.org/wiki/Fisher_information
![](https://i.imgur.com/mANjShT.png)



## Task 11
![](https://i.imgur.com/sCqwHI5.png)

* Remark 1
![](https://i.imgur.com/3YL8OUY.png)

* Remark 2
![](https://i.imgur.com/hkzeg0a.png)
![](https://i.imgur.com/UjvAVy1.png)

* Remark 3
![](https://i.imgur.com/KelKlWT.png)

![](https://i.imgur.com/O6AtPtu.png)

## Task 12 [Dawid]
![](https://i.imgur.com/gtlsEfF.png)
![](https://i.imgur.com/yx3aqhD.png)
![](https://i.imgur.com/r5byInT.png)
![](https://i.imgur.com/JvI0oOh.png)



## Q
* likelihood function
* log likelihood function
* 3 properties of log likelihood estimator
* asymptotic properties of MLE
* type 1 error
* type 2 error
* alpha size
* difference between rao-blackwell and lehmann-schaffe theorems
* significance level