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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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Logistic Regression Explanation

I have two questions regarding logistic regression. 1) I understand that the results of a logistic regression model yield a table stating coefficients together with a p-statistic for each variable . ...
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Poisson distribution test using index of dispersion

I have a data set which Im trying to check if it is poisson distributed. I read some posts here and online but they are a bit "heavy" on the statistics for a newbie in statistics like me, so I would ...
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What are mean and variance of $W_i$, given that $Z_n=\frac{\sum{W_i}}{\sqrt{n}\sigma}\sim N(0,1)$? [on hold]

Let $$Z_n=\frac{\sum{W_i}}{\sqrt{n}\sigma}\sim N(0,1),$$ where $W_i=X_i-\mu$. What are the mean and variance of $W_i$?
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Statistics and Confidence Intervals

Given the following set of values: 10,11,14,95,73,30,29,9,97,94,70 How do I calculate a 99% confidence interval for the sample mean? I am assuming that the variance is 10 Well, the idea I have is ...
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Existence of a joint distribution given the conditional and marginal distribution

Can anyone point me a book where it has a proof of Theorem 1.7 (ii) of Jun Shao's book - Mathematical Statistics? I need this to show that given a distribution on one space and a collection of ...
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Sufficient statistic for class of distributions

For the class $\{F_{\theta_1}, F_{\theta_2}\}$ of two DFs where $F_{\theta_1}$ is $N(0,1)$ and $F_{\theta_2}$ is $C(0,1)$, find a sufficient statistic. Let, $X_1, X_2, \dots, X_n$ is a random sample ...
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probability of decetion / failure in decision theory

Let $\mathcal{Y}$ be some alphabet and a vector $Y$ with elements in $\mathcal{Y}$ be the observed data. Assume the data comes from either of the two hypotheses $H_1$ or $H_0$ and define $P_D$ and $...
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$T(X) = (\sum X_i, \sum X_i^2)$ is not complete for $\sigma$ [duplicate]

Let $X = (X_1,X_2,\dots, X_n)$ be a sample from $N(\alpha\sigma, \sigma^2)$, where $\alpha$ is known real number. Show that the family of distributions of $T(X) = (\sum X_i, \sum X_i^2)$ is not ...
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Rao-Blackwell Theorem Proof

Rao-Blackwell Proof Based on the Rao-Blackwell Theorem Proof shown in the image above, I have a question regarding the portion boxed in red: Why is $Var(\hat\theta|T) = 0$ if $\hat\theta=E(\hat\...
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Bayesian Statistics exercise?

I am having issues trying to solve this exercise in Bayesian analysis. The waiting time in minutes until being serviced by a phone call center follows an Exponential(λ) model, with E[y|λ] = 1/λ. Out ...
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Finding more information after finding MLE with Indicator functions.

Ex: $X_1 , X_2 , ... , X_n$ ~ $U(-\theta, \theta); f(x; \theta) = \frac{1}{2\theta}; -\theta \leq X \leq \theta; \theta > 0$ I believe this is the correct approach to finding the MLE in this ...
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Prove the equality of the following

How do i prove this? Can anyone help please. I have no idea how to start. $$ \sum_{k=1}^N (x_k-μ)^2 = N(x̄-μ)^2+V $$ where $$ x̄=μ_0=\sum_{k=1}^N\frac{x_k}{N} $$ and $$ V=\sum_{k=1}^N (x_k-x̄)^2 $$
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Bregman Divergences and Strictly Proper Scoring Rules

Let $S$ be a strictly proper scoring rule for probability functions. Define $EXP_{S}(Q|P) = \sum \limits_{w} P(w)S(Q, w)$ and let $D_{S}(P, Q) = EXP_{S}(Q|P) - EXP_{S}(P|P)$. Is it true that $D_{...
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How to find a confidence interval of a binomial distribution using a simulated random sample?

I have a random sample of 1000 values of deviates from binomial distribution with n = 52 and p^ So I have 1000 values from the distribution. How can I find a 95% confidence interval for the true ...
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Binomial distribution confidence interval using a random sample?

I am working with the binomial distribution Bin(52, 0.82) I am looking to find the confidence interval for the observed test statistic of 44 successes. I have generated a random sample of length ...
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In the context of the Cramer-Rao Lower Bound Theorem, how can I prove that $-E[\frac{d^2L(\theta)}{d\theta^2}] = {E[(\frac{dL(\theta)}{d\theta})^2]}$

I think this is called Fisher's Information and it is the final piece I prove Cramer-Rao Lower Bound Theorem.
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wilcoxon signed rank test interpretation

I try to understand the Wilcoxon signed rank test, but I have not been profoundly successful. I have studied various articles and educational videos, but I have not succeeded yet. The concept I'm ...
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Finding out what data is useful to use…

I'm trying to determine the best method for narrowing down data to use in a program I'm writing. I could use some help. Here is a use case... Sometimes we have bad data in our system and it can lead ...
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Margin of error and average for a sample.

In a sample of Petri dishes the number of possible infectious microorganisms was counted, obtaining the following results after 11 counts. 3,6,7,2,4,7,8,9,10,2,5 I have to prove that. I) The ...
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If our goal is to minimize $L_{2}$, why do we evaluate the accuracy of multivariate linear regression models with residual standard error?

If we're using $L_{2}$ as our criterion for choosing $f(x)$, why not use that for assessing fitted model accuracy? Or is that equivalent to using RSE for the purposes of minimizing our regression ...
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Sufficient statistic equivalence

Let $\theta'$, $\theta \in \Theta$ such that $\theta' \neq \theta$. I want to prove that $T$ is a sufficient statistic if and only if $$\frac{f(x,\theta')}{f(x,\theta)}$$ is a function dependent only ...
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Implication of Law of Large Numbers

I'm reading through a proof given for the consistency of the maximum likelihood estimator (MLE) of some parameter $\theta$. The begins as follows, Consider maximising $$\frac{1}{n}l(\theta) = \...
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Exercise 2.13 from “Mathematical Statistics - Jun Shao”

I'm trying to solve this exercise, but I think some information is missing. It is very vague. Anyone have any tips? Let $f$ be a function from $\Omega$ to $\Delta$. Show that a) $f^{-1}(B^c) =...
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Why is it valid to not fully expand the inequality in the convergence in probability definition?

Let $X_1,\ldots,X_n$ be iid random variables with common pdf $$f(x) = e^{−(x−\theta)}\quad, x > \theta ,\, − \infty < \theta < \infty\,; \quad 0 \quad\text{elsewhere}$$ Let $Y_n = \min({...
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Minimum sufficient statistic for logistic regression model

For the question in the link below, I am seeking the minimal sufficient statistic for $\theta$={$\beta_1$,$\beta_2$} in the linear regression model given. I have taken the ratio of likelihoods $...
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2answers
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Poisson Conditional Expectation ( searching best estimator for h(λ) )

Suppose $X_1$,$X_2$,$X_3$,.....,$X_n$ are i.i.d. random variables with a common density poisson(λ) (I is an indicator function) (t = a value) E $[$$X_2$ - I{$x_1$=1}|$\sum_{i=1}^n X_i=t$$]$ =E $...
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Mathematical Statistics (Significance level)

Let $X_1$, $X_2$ be a random sample of size $n=2$ from the distribution having pdf $$f(x;\theta)=\left( \dfrac 1{\theta} \right)e^{-\frac x{\theta}}, 0 \lt x \lt \infty$$ We reject $H_0: \theta=...
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Mathematical Statistics Question (Power Function)

Can someone explain to me why we would want to maximize the power function (the probability our parameter is part of our alternative hypothesis) if that minimizes Type II Error when Type I Error is ...
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Find the Unbiased Estimator (Poisson)

Suppose $x_1$,$x_2$,$x_3$,.....,$x_n$ are i.i.d. random variables with a common density poisson(λ) (I is an indicator function) Find an unbiased estimator for $λ^2$ E $[$$\left(\frac{2 }{e^{-λ}}\...
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Most likely value of k

A while ago I got this question on my exam, anyone got an idea how to solve this? Smarties are a chocolate candy that come in k different colors. Suppose that we do not know k. a. We draw three ...
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Consistent estimator for the variance of a normal distribution

So I have to show that $\hat{\sigma}_n^2=\frac{1}{n}\cdot (\sum_{i=1}^n(X_i-\bar{X})^2)$ is a consistent estimator for the variance $\sigma^2$ when $X_1,X_2,...,X$ are i.i.d. from a normal ...
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Point estimator as a vector

I am given the following definition of a point estimator. Definition: $\hat{\theta}$ is point estimator of $\theta$ if $\hat{\theta} = g(X_1,...,X_n)$ where $X_1,...,X_n$ are iid distributed with ...
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Sensor and Random Variable

Please, let us imagine that I have got a sensor D that measures the temperature in my room. My issue and questions are: some papers claims to model it as a Random Variable (RV) X. What does it mean? ...
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What exactly are sample elements from a population?

If there is a sample with sequence $ x_1 , x_2 , ... , x_n$ for example, that is taken randomly from a population, what exactly are these elements $ x_1 , x_2 , ... , x_n$? What do they represent? ...
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Distribution of General Pivotal Quantity

Let $f(x; \theta) = g(\theta)h(x)$ for $ a(\theta) \leq x \leq b(\theta)$ where $ a(\theta)$ decreases and $b(\theta)$ increases with $\theta$. I'm trying to show that \begin{equation} P(S ;\theta) ...
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Optimal Number of Realizations for a Discrete Stochastic Process

I have a curiosity concerning discrete stochastic processes. Let us say we have a discrete stochastic process $X_{i} = \left(x_1,x_2,...x_i,...,x_N \right)$, hence we have N random variables with an ...
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Proportion Confidence Interval [duplicate]

I understand this: There is a 95% chance any sample from a binomial distribution of samples of size n will have a sample proportion of success between: $$p\pm 1.96*\sqrt{\frac{p(1-p)}{n}}$$ I don't ...
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Prove $XY^{1/r}$ ~Gamma(r,1)

Given X~Gamma(r+1,1), Y~Uniform(0,1) How to prove the distribution of $XY^{1/r}$ is Gamma(r,1) Thanks!
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The natural sufficient statistic is minimal sufficient

We say the distribution of a r.v $X$ belongs to the exponential family with parameter $\theta$ if there exists functions $c,h,q_i,T_i$ independent of $\theta$ such that $$P_\theta(X=x)= c(\theta)h(x)e^...
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Definition of Degree of freedom of chi square statistics

My question is what the definition of degree of freedom on statistical model. Let $f(y|\theta)$ be a model (likelihood) for nominal data $y$ from $m$ categories, i.e., $y \in \{ C_1,C_2,...,C_m\}$ ...
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Neyman-Pearson Lemma for two hypothesis pairs and three parameters

Let $\Theta = \{\theta_0, \theta_1, \theta_2\}$. I want to test the hypotheses $H_0$: $\theta = \theta_0$ vs. $H_1$: $\theta = \theta_1$. Put $\Lambda_0(x) = \frac{\mathcal L(\theta_0|x)}{\mathcal L(\...
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Find the MLE for a two-parameter piecewise density.

Let $X_1,\cdots,X_n$ be a sample from the density \begin{align*} f(x|\theta_1,\theta_1)=\frac{1}{\theta_1+\theta_2} \begin{cases} e^{-\frac{x}{\theta_1}}, x>0,\\ e^{\frac{x}{\...
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MVUE for Bernoulli Random Variable

Let $X_1, X_2$ be a random sample from a Bernoulli distribution with $P(X = 1) = p$ and $P(X = 0) = 1-p$. I want to find a MVUE for $p$. $E[X_1]=p$ and $(X_1,X_2)$ is complete and sufficient for $p$. ...
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How do we find the asymptotic variance for the maximum likelihood estimator from the Rao-Cramer lower bound?

How do we find the asymptotic variance for the maximum likelihood estimator from the Rao-Cramer lower bound? As a concrete example, I have found that the Rao-Cramer lower bound for $$f(x;\theta)=\...
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Is the minimum-variance unbiased estimator consistent?

Well, as it is said in the title, my question is if there is a proof or a counterexample of the minimum-variance unbiased estimator being consistent. I know that if the estimator is efficient under ...
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Proof of w obeys Chi-square distribution with r degrees of freedom

In my econometrics book, it has a footnote saying: Let x be an m dimensional random vector, if x $\sim \chi^2(\mu, \Sigma)$ where $\Sigma$ is nonsingular, then $ (\mathbf{x} - \mu)'\Sigma^{-1} (\...
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Can I infer correlation from two parallel time series

Consider the following time series of two variables unemployment and underemployment in different years in the US. Can I infer a positive correlation between unemployment and underemployment? For ...
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Finding sufficient statistics from the loglikelihood function

Let Y1, . . . , Yn be a random sample from a N(0, θ) distribution. Yi ’s are independent. I need to derive the log-likelihood function and find a sufficient statistic of dimension one for θ. I ...
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$X$ and $Y$ are independent Poisson distributed values with means $θ$ and $2θ$. Consider the combined estimator $\gamma = k_1\cdot X + k_2\cdot Y$ [duplicate]

$X$ and $Y$ are independent Poisson distributed values, means are $θ$ and $2θ$. Consider the combined estimator $$\gamma = k_1\cdot X + k_2\cdot Y$$ where $k_1$ and $k_2$ are arbitrary constants. (a)...
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Why does the parameter estimator not depend on the parameter?

In the proof for the Cramer Rao Inequality, my book writes: $$E[\hat{\theta}] = \int{\hat{\theta}(\textbf{x})L(\theta;\textbf{x})d\textbf{x}=\theta}$$ Then differentiating both sides of this ...