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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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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43 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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42 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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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
49 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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38 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
23 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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25 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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61 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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1answer
57 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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17 views

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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23 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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20 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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12 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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17 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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1answer
30 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
15 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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36 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) ...
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7 views

Transformation of the LR-test for normal data

Given an iid normal sample $X_1,...,X_n$ with unknown mean $\mu$ and unknown variance $\sigma^2$, we want to test $H_0:\sigma^2=\sigma_0^2$ vs. $H_1:\sigma^2\not=\sigma_0^2$ using the likelihood-ratio ...
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39 views

Function Looks Poisson-Like: But What's the Parameter $\lambda$?

(On pause) I have $$f\left(x\right)=-x\left( x\sqrt{4-x^2}-4\arccos\left(\frac{x}{2}\right) \right)\arccos\left(\frac{x^2+d^2-1}{2dx}\right)$$ which looks a bit like the continuous version of ...
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36 views

Bivariate density function

For a distribution function $F$ density function is defined as $\dfrac{\partial^2 F}{\partial x \,\partial y}$. Is it essential that $F$ is differentiable? Is it required that $\dfrac{\partial^2 ...
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159 views

Fallacy of denying the hypothesis

I need help with this question Im not sure what a fallacy of denying the hypothesis is. Use a truth table to show that $p\to q$ and $\neg p$, $\therefore \neg q$ is not a valid rule of inference. It ...
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1answer
25 views

What is the meaning ‘uniformly converge’?

Assuming that, we randomly sample $n$ data following a distribution, then if someone claims that the average of these $n$ data uniformly converge to its expectation with rate $O(\sqrt{1/n})$. Here, ...
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1answer
135 views

Techniques for proving asymptotic normality by Taylor expansion?

Suppose I have a sequence of densities $$ f_{X_n}(x) = \exp[\ell_n(x)], \qquad (x \in A). $$ My goal is to prove a statement like $\sqrt n (X_n - \mu) \to N(0, \sigma^2)$ in distribution, for an ...
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22 views

Uniformly minimum variance unbiased estimator

How to prove $ \overline{X}=\frac{1}{n}\sum_{i=1}^nX_i$ is the uniformly minimum variance unbiased estimator of $\mu$ when $X_i\sim N(\mu,\sigma^2),$ and $\sigma$ is known. Idea: Let ...
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Interaction term (second order term) always implie the linear term (first order term)? (GLM)

I am doing a GLM model with interaction terms, but the same question can be asked with an ANOVA model. Suppose I have two independent variables $X_1,X_2$ and $Y$ the dependent one. I notice that, ...
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1answer
57 views

Proof of Central Limit theorem - infinite points

In the following proof I understand that we have taken power n because for sum of variables we take product of characterstic functions. Intuitively I understand why $n \to \infty$ is important but ...
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21 views

Prediction Intervals for Discrete Data

I would like to generate a prediction interval as described [here]. I suspect that my data comes from a normal distribution, but it was discretized into bins. How would one construct prediction ...
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5 views

How to parameterize some emprical data

I would like to describe a bunch of data that I have collected as a function of two variables. The data is phytoplankton absorption in my local area that has changed in concentration. The data looks ...
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1answer
14 views

Estimating population variance

Why is population variance estimated to be $\frac{1}{N-1}\Sigma_{1 \leq i\leq N}(x_i-m)^2$ as opposed to sample variance which is $\frac{1}{N}\Sigma_{1 \leq i\leq N}(x_i-m)^2$, where m is the mean? I ...
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1answer
25 views

Interpreting high p value and low correlation value

I am trying to run regression on financial data in R. I am new to regression analysis so I am finding it to difficult to interpret certain scenarios. I have the code as follows: Regression analysis ...
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1answer
16 views

Asymptotic test statistic for exponentially distributed data

I need an idea for tackling the following problem: Let $X_1,...,X_n\sim\mathrm{Exp}(\lambda)$ be an iid sample. We want to test $H_0:\lambda=\lambda_0$ vs. $H_1:\lambda\not=\lambda_0$ for some ...
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1answer
21 views

Transformation of a random variable.

In this transformation question, I managed to show the g(w) as given, yet I do not quite understand why the domain of the w is within plus and minus infinity. Shouldn't it be -1< w <1 as we ...
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1answer
141 views

Example of a real-world situation where multivariate analysis is applicable.

I have searched a lot of site to understand the situation where multivariate analysis is applicable. But not got any easily understandable example. Would you please give me a real-world example where ...
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22 views

Application of Multivariate Analysis

The following situation is proven valuable where multivariate analysis can be applied. This example is taken from the book Applied Multivariate Statistical Analysis ...
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125 views

Bayes estimator from a geometric distribution with a uniform prior

X is a random variable with Ber(p), 0 Y is the number of trials until a success occurs. Assume the prior p is unif(0,1). I have trouble in figuring out the posterior density f(p|Y). With the ...
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1answer
39 views

calculate expectation of MLE

It's a question about whether $\hat{\theta _{MLE}}$ is an unbiased estimator of $\theta$. n independent pairs $(X_{1},Y_{1}), (X_{2},Y_{2}),....(X_{n},Y_{n}), n\geq 3$, where $Y_{i}=\theta ...
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1answer
37 views

how to create unbiased estimator in uniform distribution

$X_{1}, X_{2},...X_{n},n\geqslant 2, $ is a random sample from unif[$\theta -1, \theta +1$] Followed with the problem, I got T(X)=($X_{(1)}, X_{(n)} $) is sufficient but not complete, But I got ...
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11 views

Hypothesis testing, find $n$ to limit type II errors

I'm trying to solve the following (basic) problem, but I don't seem to get the last part 3). The average lifespan of a certain mechanical part is set to be at least 2 years by experts. The ...
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1answer
25 views

computing expectation of two arm bandit

assume you have a two arm bandit with one arm having a fixed, known probability of payoff $p = 0.6$ and another having an unknown payoff $q$, which is drawn uniformly from $[0,1]$. Each game the ...
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1answer
25 views

Calculate size and power of a given PMF

Let $X$ be a random variable having probability mass function $f(x) = \begin{cases} \dfrac{2+4a_1+a_2}{6}, & \text{if $x=1$} \\ \dfrac{2-2a_1+a_2}{6}, & \text{if $x=2$} \\ ...
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1answer
11 views

Expectation of an estimator

I want to find the expectation of $\hat\theta$. I have the cumulative distribution of $\hat\theta$: $$\Pr{(\hat\theta>t)} = e^{n(\theta-t)}\quad \text{for $t>\theta.$}$$ Now to find the ...
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40 views

Likelihood Theory and finding maximum likelihood

Let $X_1,\,\ldots,\,X_n$ be independent random variables, each with probability density function $$f(x;\,\theta)=\frac{2x}{\theta^2}\qquad \text{for }0<x<\theta.$$ I want to find the maximum ...
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1answer
48 views

Find joint likelihood function of observations $x_1, x_2, \ldots, x_n$ and $y_1, y_2, \ldots, y_m$

Let $x_1,\ldots,x_n$ be observations from a normal distribution with mean $0$ and s.d $s_1$. Similarly let $y_1,\ldots,y_m$ be observations from a normal distribution with mean $0$ and s.d ...
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1answer
12 views

Estimating unknown weights of various parameters in an equation

I have an equation which has some unknown weights attached to various parameters. None of the weights are known. However, I have a history of data available with me which can be used to predict the ...
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44 views

$E\bigl(\frac{2}{1+x}\bigr)$ for Beta(2,$\frac{1}{2}$) random variable

Let x ~ Beta (2,$\frac{1}{2}$). Then calculate $E\left(\frac{2}{1+x}\right)$. So, ${E}[g(X)] = \displaystyle \int_{-\infty}^\infty g(x) f(x)\, \mathrm{d}x$ . $\displaystyle f(x;\alpha,\beta) ...
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1answer
21 views

Checking the consistency and Bias of $\frac{\sum X_i +\sqrt{n}/2}{n+\sqrt{n}}$

Let $X_1,\ldots,X_n$ be i.i.d. $B(1,\theta)$ random variables, $0<\theta<1$. Then, as an estimator $\theta$, check if $T(X_1,\ldots,X_n)= \dfrac{\sum_{i=1}^n X_i +\sqrt{n}/2}{n+\sqrt{n}}$ is ...