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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The distribution of roll of a dice $12$ times

What's the distribution of a variable $X$ if $X$ represents the number of times you get outcome $k$ when you roll a dice $12$ times? I thought that the distribution was a binomial distribution with ...
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2answers
21 views

confidence intervals for 20 different parameters - distribution, probabilit and most probable value.

I need help with the subexercise (c) in the following exercise. A researcher is planning a study where she must calculate confidence intervals for 20 different parameters. The intervals are ...
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41 views

How do I put together a set of modified conditional distribution into a single joint distribution?

I am abstracting my original problem to a simple scenario. Consider a bivariate multi-modal mixture of gaussian distribution, $P(x,y)$. When we slice through $x$ or $y$ we get a univariate multimodal ...
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1answer
20 views

How to figure out the respective sufficient statistic for a given vector of parameters?

Let $Y$ be a random sample from $N(\mu,\sigma^2)$ where both $\mu$ and $\sigma^2$ are unknown. Let $\theta$ be the vector of parameters of interest $\theta=(\mu,\sigma^2)$. I need to find the ...
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35 views

Limiting Distributions and the Weak Law of Large Numbers

I have that $Y_1, Y_2, ..., Y_n$ are i.i.d. Poisson random variables with mean 1, and that $U_n = \sqrt{\frac{\sum_{i=1}^{n}{Y_i^2}}{n}}$. Given that I have a sequence $U_1, U_2, ..., U_n$, I'm ...
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16 views

$\def\Var{\operatorname{Var}}\Var \left[\frac{(n-1)S^2}{\sigma^2} \right] = 2(n-1) \Longrightarrow \Var(S^2) = \frac{2\sigma^4}{(n-1)}$

$\def\Var{\operatorname{Var}}$ $$\Var \left[\frac{(n-1)S^2}{\sigma^2} \right] = 2(n-1) \Longrightarrow \Var(S^2) = \frac{2\sigma^4}{(n-1)}$$ I know that when you take a constant out of the variance, ...
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14 views
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24 views

Why is the sample Mean a consistent Estimator for the Logistic Distribution?

I think is this a very trivial question, but non the less: How can I show that the $ \hat\theta_n = $ $ \bar x $ is a consistent estimator of $ \theta _0 $. Since $ \theta _o $ is $ \mu $ for the ...
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12 views

which estimation are most effective(MY ATTEMPT)

To start with we have 2 variables $X_1$~$Bin(n,p_1)$ and $X_2$~$Bin(n,p_2)$. For example , we assume that we have an estimation $$p^*=p_1p_2(1-p_1p_2)$$ and another estimation ...
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38 views

Why is $\frac{\sum_{i=1}^n \log(X_i)}{n} = \overline{log X}$ [closed]

Why is $$\frac{\sum_{i=1}^n \log(X_i)}{n} = \overline{log X}$$ ($X_i$ are i.i.d samples)
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13 views

variance of the estimation $p^*=(x_1*x_2)/(n*n)$

I have that $X_1$~$Bin(n,p_1)$ and $X_2$~$Bin(n,p_2)$ and want to calculate the variance of the estimation $p^*=(x_1*x_2)/(n*n)$, which is an observation of the estimator ...
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0answers
16 views

Affine hypothesis

I'm looking at a data set containing income and expenditure on food for 235 household. We are interested in whether the cost of food depends on household income. I have verified that a workable model ...
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1answer
34 views

Why does $\hat\sigma^2 = \frac{1}{n}\sum_{i=1}^{n}X_i^2-(\bar X)^2= \frac{1}{n}\sum_{i=1}^{n}(X_i^2-\bar X)^2 = \frac{n-1}{n}s^2$? [closed]

Why does $$\hat\sigma^2 = \frac{1}{n}\sum_{i=1}^{n}X_i^2-(\bar X)^2= \frac{1}{n}\sum_{i=1}^{n}(X_i-\bar X)^2 = \frac{n-1}{n}s^2$$?
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1answer
57 views

domino's pizza claim

I just got a dominos promotional flier through the post and one of the graphics advertising 'create your own pizza' lists the various toppings and claims there are 'more combinations than people in ...
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1answer
22 views

Calculating the current age pdf from the lifetime pdf

Let's say I know the form of the lifetime pdf for some object class. If I select an arbitrary object from the class which is still alive and for which I have no ancillary information on its current ...
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0answers
40 views

Uniform Boundedness: Am I right or my TA?

I am a student, and I disagree with the solutions our TA has prepared. I am seeking verification that I am correct or explanation as to why I am wrong. It seems to be a disagreement or ...
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1answer
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limiting variances of iid sample mean

In the book Statistical Inference (George Casella 2nd ed.), page 470, there is an example: $\bar{X}_n$ is the mean of $n$ iid observations, and E$X=\mu$, $\operatorname{Var}X=\sigma^2$. "If we take ...
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8 views

Do higher order sample moments converge to the distributional mean?

The Methods of moments estimation is based on the law of large numbers, which says that the sample means of i.i.d. random variables from any distribution converge to the distributional mean as the ...
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8 views

Confidence interval using weigthed mean.

I'm doing an experiment in which I take n measures and their respective weight. Mean and variance population are unknown. If I don't care about their weights, I can do confidence interval using a ...
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1answer
41 views

Show the consistency of an estimator?

Let $Y_1,Y_2,Y_3,...,Y_n$ be a random sample from the exponential distribution having PDF $f(y;\lambda)= \lambda e^{-y\lambda},$ $y>0.$ A) Show that $\hat\lambda_n = Y_1$ is not consistent for ...
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27 views

Hazard function for proportional odds model

The Cox proportional hazards model for survival data with covariate ${\bf z}$ is defined through the hazard function $h(t,{\bf z})$ by $$ h(t,{\bf z}) = h_0(t)~\cdot\theta~~,~~~\theta = \theta(\beta, ...
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1answer
18 views

What is the difference between $\sigma, \sigma_{\bar{x}}, S, s,$ and $s_{\bar{x}}$?

What is the difference between $\sigma, \sigma_{\bar{x}}, S, s,$ and $s_{\bar{x}}$? My textbook uses lots of different symbols, and it's not clear to me what the difference between all of them are. ...
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13 views

Variance of log-odds ratio

For a 2x2 table: I know that $\widehat{var}\left(log\left(\widehat{OR}\right)\right)=1/a + 1/b + 1/c + 1/d$. I'm trying to use a Taylor Series approximation to show this, but I'm getting a bit ...
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22 views

A strange derivative. Why it is equal to the expectation of ${\bf{r}}({\bf{x}},{\bf{z}})$?

Suppose probability distribution of $\bf{x}$,$\bf{z}$ are defined as $P({\bf{x}},{\bf{z}}|{\bf{\theta }}) = \frac{{\exp \{ {\bf{r}}({\bf{x}},{\bf{z}}) \cdot {\bf{\theta }} + {r_0}({\bf{x}},{\bf{z}})\} ...
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11 views

Probability of taking a random sample of 24 measurements and getting a mean of at least 103.6 of true population

A random sample of size n = 24 measurements is drawn from a normal population. The sample has a mean of 103.6 and a standard deviation of 12.5. If the true population is 100, find the probability of ...
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15 views

Different definitions of minimal sufficiency

Let $X$ be a random sample from the family of distribution whic is indexed by $\theta$ and $T$ be a sufficient statistic for $\theta$. I have two definition of minimal sufficiency. Definition 1 $T$ ...
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16 views

Questions on the theory of Lasso

The linear model ${\bf Y}={X\beta}+\epsilon$, where ${\bf Y}$ is a $n\times 1$ vector, and ${\bf X}$ is $n\times p$ matrix. $n\lt p$ and $rank({\bf X})=n$. $\epsilon\sim N(0, \sigma^2)$. How to prove ...
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35 views

Why isn' t high order polynomial a good fit?

Let's say I have a set of data points $(x_i,y_i), i=1,2,...,N$, and I want to approximate it using a polynomial $p(x)=\sum_{i=0}^n a_i x^i$ with a least squares fit (so $n<N$). I know that the ...
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22 views

inferential statistics methods for population?

Is that true to use inferential statistics methods when we study whole population? I mean for example is that true to use hypothesis test when whole population are under study? Suppose I am studying ...
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15 views

Conditional expectation to define causal effect [migrated]

I'm reading these notes which are discussing the NRCM approach to analyzing causal relationships, that is to say, treats the causal inference problem like a missing data problem (where the missing ...
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1answer
40 views

Using Lindeberg’s Condition together with the Central Limit Theorem

I have the following problem: Problem. Let $ (X_{n})_{n \in \mathbb{N}} $ be a sequence of independent random variables such that $$ \mathbf{Pr} \! \left( X_{n} = \sqrt{n} + 1 \right) = ...
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34 views

$Z$ is Normal$(\sigma,1)$, find UMVUE of $P(Z\leq 0)$.

Given i.i.d. samples $X_1,...,X_n$ from Normal$(\sigma,1)$, find the UMVUE of $g(\sigma)=P_\sigma(Z\leq 0)$. I tried to use Lehman-Scheffe theorem. We now that $\sum_1^nX_i$ is sufficient and ...
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2answers
25 views

Trouble with True/False Stats Question

Having trouble determining the truth value of the two above statements. Please let me know if the following reasoning is correct. I believe the first statement is true, because of this statement I ...
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1answer
40 views

The function given by the Rao-Blackwell theorem is a statistic

$(X_1,...,X_n)$ is a random sample, $V_n$ is an unbiased estimator of the population parameter $\theta$ and $T_n$ is a sufficient statistic for $\theta$. Then by Rao-Blackwell theorem the rv ...
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34 views

What is the appropriate statistical test to see if a quantity has been distributed differently into discrete bins?

Say I have $10^6$ balls, $3$ bins $A,B,C$, and $2$ machines $X$ and $Y$ that distribute the balls into the bins according to an internal set of rules (i.e. a probability distribution). If I run both ...
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1answer
18 views

How to find the minimal MSE?

I'm confused as in how to find $⍴$ in c) and why $σ^2$ gives a smaller MSE than $s^2$ I know $MSE(θ) = E(θ - θ_0)^2 = Var(θ) + Bias(θ)^2 $ and that $ Bias(θ) = E(θ) - θ_0$ But I don't get what θ is ...
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1answer
22 views

How to estimate mean and variance of a normal distribution given the numbers?

Given the numbers generated in a normal distribution: $5.3299, 4.2537, 3.1502, 3.7032, 1.6070, 6.3923, 3.1181, 6.5941, 3.5281, 4.7433, 0.1077, 1.5977, 5.4920, 1.7220, 4.1547, 2.2799$ How would I ...
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18 views

How can I tell whether sample size is inadequate or not ?

I am given sample size of 15322 students and our research topic is to find out a relationship between students academic performance and participation in sports team. The question asks " do you think ...
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1answer
52 views

finding the probablity of type 2 error in a normally distributed RV using a Z test

I am getting a $0.913$ answer as opposed to $0.903$ from a text, or are we both wrong on this? Find the probability of type II error in $H_0 : X \sim \mathcal{N}( 84, 100)$ $H_1$: ...
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44 views

Confusing on some concepts of sufficient principle

Im reading Chapter6 of Casella Berger's statistical inference that talks about sufficiency principle. I've been confused a lot by the definition of sufficient statistics, here it is: Basically, ...
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Why not use always a binomial exact test to compare two proportions instead of chi square?

I am trying to figure out what test I should use in the following scenario: I know that there is a lot of room for improvement in a specific area at work - being extremely critical, let's say that ...
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22 views

Determining the weights of known parameters in a formula

I have a formula of the following form: $a_1*w + a_2*x + a_3*y + a_4*z$ In the above formula, the $a_i$s can be thought of as weights to the corresponding parameters. The values of the ...
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9 views

Determining the actual number of observations in a dataset

I have two datasets one is a dataset with doctors in which I have the procedures they have performed at a given hospital where the actual number of procedures is not captured by this data since it is ...
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11 views

Approximation of objective based on statistical distance

I am a computer science researcher (mostly theoretical) currently in midst of statistics and not able to figure out how to proceed. At an abstract level, I have a hypothesis for an unknown ...
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12 views

testing variance composite vs composite hypothesis normal distribution

I have a random sample from $\mathrm N(\mu,\sigma^2)$ with $\mu,\sigma^2$ unknown and I'd like to test the following hypotesis $ \mathrm H_0: \sigma^2 \le0.6$ vs $ \mathrm H_1: \sigma^2 >0.6$ ...
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16 views

how to find relationships in dataset with multiple variables

I have a large project data set ,which includes numeric values like dollar amounts, and non numeric quantities like country codes, purpose codes etc I want to find relationships between the variables. ...
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23 views

interpreting the multivariate Kalman filter update equations

consider a multi-dimensional Kalman filter model with these state transition and measurement probabilities: $P(x_{t+1} | x_{t}) = Normal(Fx_{t}, \Sigma_{x})$ $P(z_{t} | x_{t}) = Normal(Hx_{t}, ...
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1answer
14 views

Do we need to check that maximum likelihood estimator is a maximum?

For maximum likelihood estimation, do we theoretically need to check that the critical point is a maximum (rather than a minimum or saddle point) or is this automatic? I believe that it is automatic ...
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12 views

Showing that moment estimates are asymptotically bi-variate normal.

Let $X_1,\dots,X_n$ be iid $\Gamma(p,1/\lambda)$ with density $g_\theta (x) = \frac{1}{\Gamma(p)} \lambda^p x^{p-1} e^{-\lambda x}$, $x>0$, $\theta = (p,\lambda)$, $p > 0$, $\lambda > 0$. ...
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32 views

Geometric distribution $G(p)$ for independent random variables $X$ and $Y$

Question: If the random variables $X$ and $Y$ are independent and each have the geometric distribution $G(p)$ - that is, $P(X=k)=P(Y=k)=pq^k$ for $k=0,1,2,\ldots$ (where $q=(1-p)$) show that: (I) ...