Questions tagged [statistical-inference]
The area of statistics that focuses on taking information from samples of a population, in order to derive information on the entire population.
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Log predictive density asmptotically in predictive information criteria for Bayesian models
I am reading this paper, Andrew Gelman's Understanding predictive information criteria for Bayesian models, and I will give a screenshot as below:
Sorry for the long paragraph. The things that ...
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14 views
Let X1,…, Xn be i. i. d. with N(0,theta). Show that the summation from xi=1 until n from (Xi)^2 is a Sufficient statistics for theta.
Help me to solve this problem about sufficient statistics please..
Let X1,..., Xn be i. i. d. with N(0,theta). Show that the summation from xi=1 until n from (Xi)^2 is a Sufficient statistics for ...
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1answer
38 views
Let $X_1,\ldots,X_n$ be i.i.d from a distribution with pdf $f(x) = (1-\theta)^x\theta$; with $x=0,1,2,3,\ldots$ zero elsewhere.
Help me to solve this problem please. I am a new comer in math exchange.
I have a homework from my lecture about sufficient statistics.
Would you be so kind help me?
Let $X_1,\ldots,X_n$ be i.i.d ...
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14 views
Variance of a sum of random variables with overlapping domains
This problem has stumped me for a while, so I'd love to get some fresh eyes on it.
Problem
Suppose an online retail provider served a large number of users ($N$) in a given time-window. Each user ...
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23 views
how to write the condition $\Lambda \beta$, in the linear model?
i'm dealing with the model:
$$Y_{1j}=\mu+\tau_{1}+\varepsilon_{1j}, \hspace{2cm} j=1,2,...,n_{1}$$
$$Y_{2j}=\mu+\tau_{2}+\varepsilon_{2j}, \hspace{2cm} j=1,2,...,n_{2}$$
$$Y_{1j}=\mu+\tau_{3}+\...
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28 views
Likelihood function for continuos observations [on hold]
I need to know how to calculate the likelihood function for numeric range observations, for example:
Given two observations of normal distribuition:
$y_1 = a \leq y_1 \leq b$ and
$y_2 = c \leq y_2 ...
1
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0answers
19 views
Derived parameter instead of parameter estimation
In a statistical model
$$
(\mathcal X,\mathcal F, (P_{\theta})_{\theta\in\Theta})
$$
we call
$$
\varphi(\theta)\in\mathbb R^d
$$
a derived parameter (literal translation from German, Source: ...
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0answers
14 views
Let Xn be a sequence of random variables that converges in distribution to a random variable X. [on hold]
Let Yn be a sequence of random variables such that for any finite number c,
limn→∞ P(Yn > c) = 1.
Show that for any finite number c,
...
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0answers
12 views
Prove an equality consists of increasing events
If $A$ and $B$ are correlated increasing events with respect to $X$ as it shown below and $X$, $Y$ , $W$ and $Z$ be independent random variables, is it true that we say:
$$P \left(A \cap B \right) \...
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2answers
28 views
Is it possible that probability of occuring intersection of two events will be greater than probability of occuring each of them?
If $A$ and $B$ are correlated events, Is it possible that we have :
$$P\left(A \cap B \right) \geq P\left( A\right)$$
and
$$P\left(A \cap B \right) \geq P\left( B\right)$$
Is it possible?
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2answers
46 views
Most powerful test for discrete uniform
Let $X$ be a random sample from a discrete distribution with the probability mass function
$f(x, \theta) =\frac{1}{\theta} , x=1,2,...,\theta;= 0 \ \text{otherwise} $
where $\theta \ \text {is ...
1
vote
0answers
17 views
Show that E(Y | X) = E(E(Y | X, W) | X). The right hand side may sometimes also be written as E(E(Y | X, W) | X) = E(E(Y | W) | X).
We know that, E(E(X|Y)) = E(v(Y)) = E(E(X|Y=y)) = E(X), where v(Y) = E(X|Y=y). Hence, E(E(X|Y)) = E(X).
Note: Let X and Y be discrete or jointly continuous random variables. The conditional ...
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35 views
Inference Statistic - Likelihood Function [on hold]
Can anyone help me understand this?
Consider the four observations from de Normal Distribution with variance equal to one $y_1 < 10$$, y_2 > 10 $, $5 < y_3 < 10 $ and $ y_4 = 10$.
...
1
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1answer
46 views
Limiting distribution of $T_n=\frac{\sum_{i=1}^n(2X_i−n)}{n}−3$ where $f_k(x) =\frac{x^k}{k!}e^{−x}I_{(0,\infty)}(x)$
$X_1$, $X_2$,..., are independent random variables. $X_k$ has pdf
$$f_k(x) =\frac{x^k}{k!}e^{−x}I_{(0,\infty)}(x)$$
Let
$$T_n=\frac{\sum_{i=1}^n(2X_i−n)}{n}−3$$
Consider the ...
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0answers
14 views
Range of Pearson's moment coefficient of skewness
The Pearson's moment coefficient of skewness is defined as
$\gamma_1 = \operatorname{E}\left[\left(\frac{X-\mu}{\sigma}\right)^3 \right]
= \frac{\mu_3}{\sigma^3}
= \frac{\...
2
votes
2answers
31 views
how to find the distribution of the least square estimator $\beta$?
i'm solving a problem that involve a linear model, and i'm trying to get the distribution of the least square estimator $\beta$.
i found in a book that:
$\widehat{\beta}\sim N_{p}(\beta, (X^{\...
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0answers
12 views
find a confidence interval in a linear model problem
i'm trying to solve a problem that involve a linear model given its normal equations, and the errors have a normal distribution
but i'm a little lost. the problem is about construct a 95% ...
1
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0answers
21 views
Central limit theorem and confidence interval
I have some questions concerning the following problem especially about b).
My problem is the following:
Let $X$ be an observation from a $Bin(n,\frac{1}{2})$:
a) prove that the ML estimator ...
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0answers
36 views
Does it make sense to use “linear transformation” to analyze effectiveness of a new technique?
I am relative new in statistical analysis, and I have question when I'm trying to measure the performance of new techniques.
Imagine we have $m$ machines to improve with new technique, and $n$ tasks ...
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0answers
22 views
Finding umvue of normal pdf [closed]
Let $X_1,X_2,\ldots,X_n$ be iid $\mathcal N(\mu,1)$ RVs.Let $T(\mathbf X) =\sum_{i=1}^nX_i$.Show that $\phi(x;t/n,n-1/n)$ is UMVUE of $\phi(x;\mu,1)$ where $\phi(x;\mu,\sigma^2)$ is the PDF of a $\...
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11 views
What is an unconditional model for a time series variable?
If I am being asked to do an unconditional analysis of a time series variable, lets say GDP starting in 2000, what model am I supposed to estimate?
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0answers
13 views
Compare KS test and Wasserstein distance or Earth mover's distance.
Consider two sets of data points A and B. Both these data set are from mixture of unknown number of Gaussians. The mean of the Gaussians are little different for each set (there may have few overlap ...
0
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1answer
33 views
Reasoning behind the statement 'Consistency is a property of a sequence of estimators rather than one point estimator"
I just started learning statistical inference course and I am doing this topic called consistency. I am not able to understand this line 'Consistency is a property of a sequence of estimators rather ...
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0answers
20 views
Understanding p-value and its relation with the value $\alpha$ used in tests
I am having a little trouble understanding these concepts. Let’s take a real problem, for instance. I buy some machinery, and I know newer models break down with an exponential probability of mean 5 ...
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1answer
15 views
Which distribution to use
After many generations, it is known that 75% of students pass certain subject. if a random sample of 40 students is such that it could be consider they have the same characteristics as the ones ...
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1answer
35 views
Hypergeometric Distribution over an interval
In a village with 2000 people, 100 people suffer from Alzheimer's disease. On a certain day, 40 people are admitted to a hospital.
Calculate the probability that between $15$ and $25$ people (...
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1answer
46 views
Binomial Distribution on finding the potency of a drug
An experiment is designed to test the potency of a drug on 40 rats. Previous animal studies have shown that a $10$-mg dose is lethal $10$% of the times within the first $4$ hours.
What is the ...
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24 views
Linear Least square estimate of $x^3$ given $x$ and the moments.
I have been struggling to find a direction on how to proceed with the following problem. Given that $x$ is a zero mean (non-Gaussian) random variable with moments
E$(x^n)=\mu_n$.
I need to find the ...
0
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1answer
61 views
Statistical problem involving a coin
Two friends flip a coin $100$ times, and $58$ it is head. One says the coin is rigged, the other says it is not, and that it happened by chance. The problem asks to verify the hypothesis $p=1/2$ ...
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22 views
Student´s-T interpretation for problem
According to the history of certain shop department, the purchase exhibit in a ticket can be modeled by a random variable with mean \$1,025. For planning purposes, the new marketing director decides ...
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20 views
How do they simplify the autocorrelation function?
In my book they state that
...the autocorrelation function can be defined as
$$r_X(s,t)=\frac{\text{Cov}[X(s),X(t)]}{\sqrt{\text{var}[X(s)]\text{var}[X(t)]}},\tag
1$$
where $X(t)$ and $...
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30 views
Let Y(1), Y(2), Y(3), Y(4), Y(5) denote the order statistics of a random sample of size 5 from a distribution [duplicate]
Let Y(1), Y(2), Y(3), Y(4), Y(5) denote the order statistics of a random sample of size 5 from a distribution having p.d.f. f(y) = e^(-y), 0 < y < ∞, zero elsewhere. Show that Z1 = Y(2) and Z2 = ...
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1answer
35 views
Let Y(1), Y(2), Y(3), Y(4), Y(5) denote the order statistics of a random sample of size 5 from a distribution having p.d.f.
Help me to solve this problem please..
Let $Y_{(1)}, Y_{(2)}, Y_{(3)}, Y_{(4)}, Y_{(5)}$ denote the order statistics of a random sample of size 5 from a distribution having p.d.f. $f(y) = e^{(-y)}, 0 ...
1
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1answer
39 views
Sufficient Statistics: Proof of a lemma by Halmos and Savage
I am reading the paper "Application of the Radon-Nikodym Theorem to the Theory of Sufficient Statistics" by Halmos and Savage and have much trouble following the proof of Lemma 7. Below is (my ...
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23 views
Markov chain transition kernel inference
Say we consider a continuous markov chain $X$ with state space $S \subset \mathbb{N}^n$, transition kernel $P$, rate $\lambda$ and initial state uniformly chosen at random in $\mathbb{N}^n$.
We do ...
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1answer
49 views
Finding an unbiased estimator of$ e^{−2λ}$ for Poisson distribution
The random variable X has a Poisson distribution with unknown
mean λ, where 0 < λ < ∞.
Based on a single X , Is it possible to calculate an unbiased estimator of $ e^{−2λ}$ ?
I have taken an ...
1
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0answers
24 views
How big of a sample size do you need to ensure the mean is within $\delta$ of the sample mean?
Visitors to a website click on an add with fixed unknown frequency $p$. You collect a sample of $n$ visits and compute the clickrate $\hat{p}$. Find the minimum $n$ such that $$P(|\hat{p}-p|<\delta)...
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1answer
26 views
Calculating joint cdf from joint pdf
I have the following joint distribution function
$f_{xy} = (12/5)(x+y^2)$ on $0<y<x<1$ (zero everywhere else)
I know I must integrate along both $y$ and $x$ to find this:
$$F(x,y) = \int \...
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60 views
Finding MLE when there are different distributions for different values of parameter
Suppose a random sample of size $n$ is drawn from a population having different distributions, one for different values of the parameter $\theta$. Let's say I consider the case when there are two ...
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0answers
18 views
Estimations from Standard Deviation
A while ago I was watching a show that centered around algorithms. At some point in it, the host of this show/documentary was given data from a fisherman about his haul of fish for that day. From this ...
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0answers
27 views
Student's-T distribution for problem
A biologist want to measure the mean height of a certain specie. If is assume $σ=2.5 inches$ and 100 males are randomly selected:
a) Find the probability that the difference between the sample ...
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0answers
32 views
Bias of a function of the avarage
$\overline{x}$ is an unbiased estimator of the exact average.
Now, we let's imagine that we want to estimate some function of the average $f(\langle x \rangle)\equiv f(X)$.
My first guess was $\...
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1answer
35 views
Finding an unbiased estimator for the variance
While doing some statistics exercises I found a question that I don't know how to solve. The question is as follows:
Let $X_{1},...X_{n} \stackrel{iid}{\sim} N(0,\sigma^2)$, where $\sigma^2$ $\in R^{+...
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0answers
21 views
Testing simple Null vs simple Alternative.
Let $X$ be a random variable with pdf $f$. Suppose we want to test $$H_0:f=f_0\quad $$ against $$ H_1:f=f_1$$. For any $\alpha>0$, define
$$T_\alpha(X)= 1\qquad if\quad f_1(X)>c(\alpha)f_0(X)$$
...
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0answers
28 views
Estimate parameters of Wishart matrix.
Given a sequence of real Wishart matrices $W_1 , \cdots , W_k \sim \mathcal{W}_m(n,\Sigma)$ where $\Sigma$ is a singular matrix. Are there good estimates for the degrees of freedom?
The MLE for $\...
2
votes
1answer
40 views
Estimate degrees of freedom in sample variance.
Given a sequence of independent identically distributed random variables $X_1,\ldots,X_m \sim \chi^2_n / n$ is there literature on estimates for the degrees of freedom $n$?
In an attempt to find the ...
2
votes
1answer
58 views
How to calculate a Bernoulli Distribution problem
I have my statistics exam quite soon and i came upon this question :
At the last referendum, $40\%$ of the Italian population supported the
constitutional reform. If a random sample of size $n = ...
1
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0answers
46 views
Differentiation of a Complicated Integration
Let, $h$ is differentiable function. $\theta_1\in\mathbb{R}$, $\theta_2>0$ $$F(\theta_1,\theta_2) = \int_{\theta_1-\theta_2}^{\theta_1+\theta_2}\int_{\theta_1-\theta_2}^{y}n(n-1)\frac{(y-x)^{n-2}}{(...
0
votes
2answers
34 views
Proof for Simple Linear Regression: What am I doing wrong?
I am trying to prove the well known formula for simple linear regression $$SS_{TOTAL}=SS_{MODEL}+SS_{ERROR}$$
i.e
$$\sum_{i=1}^n (y_i - \bar{y})^2 =\sum_{i=1}^n (\hat{y}_i-\bar{y})^2+ \sum_{i=1}^n(\...
3
votes
2answers
155 views
Tricky coin probability
Can someone please help, I am getting the wrong answer.
Consider four coins labelled as 1, 2, 3 and 4. Suppose that the probability of obtaining a ‘head’ in a single toss of the 𝑖-𝑡h coin is $𝑖/4,\...
