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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Real Application for Takagi-Sugeno?

my question is short: Can anyone give a concrete example where a Takagi Sugeno controller is used and what the rules would look like for this example? I'm not quite sure where a Takagi-Sugeno rule was ...
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Is there a continuous parametric distribution for which beta is a conjugate prior?

Is there a continuous parametric distribution for which beta is a conjugate prior? It is, is there a parametric family of continuous distributions $f(x;\theta)$ such that if $\Theta \thicksim Beta(a,...
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Conditions for Weak and Strong law of Large Numbers

I am a second-year undergraduate student and was reading up the topics of the law of large numbers from Casella and Berger's Statistical Inference. They state the laws as follows: The link to the ...
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Is this proof that relative entropy is never negative correct?

I wish to prove that relative entropy(Kullback-Liebler divergence) is always non-negative. I.e. that $$I^{KL}(F;G)=E_F\left[\log\frac{f(X)}{g(X)}\right]\geq0$$ where F,G are two different probability ...
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Is $p(Y|X) p(\omega) = p(Y|X,w)$ true?

Given a distribution $p(Y|X)$ and a distribution $p(\omega)$, which are independent from eachother. Is then $p(Y|X) p(\omega) = p(Y|X,w)$ true? If not, are their any rules to calculate $p(Y|X) p(\...
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Determine the probability that any person is against government decisions

In one study, out of $80$ respondents, $23$ were against going to a concert. Determine if a person is against decisions to go to the concert if the confidence interval is 95%. Help me please. Thanks ...
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What prior should I choose for a hypoexponential likelihood (sampling distribution) if I want an analytic posterior?

The hypoexponential distribution is the distribution resulting from multiple exponential distributions with different rate parameters. When they are all the same, it is an Erlang distribution. For ...
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What is the best option of a graduate multivariate statistics book?

Right now I am in the middle of a graduate multivariate statistics but I am feeling to easy at the moment. As a mathematician I study univariate statistics from "The theory of statistical inference : ...
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Why does the number of possible probability distributions have the cardinality of the continuum?

Wikipedia's article on parametric statistical models (https://en.wikipedia.org/wiki/Parametric_model) mentions that you could parameterize all probability distributions with a one-dimensional real ...
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Fisher Information using Variance score [closed]

The answer to a question to find fisher information I don't understand how to get the variance of score. The distribution here is binomial.
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NP Lemma and rejection region having the size.

quick question. I need to find the rejection region of H_0, meaning that I need to find the constant value for which if X=x belongs to the region H_0 is rejected. The pdf is c(1-x)^(c-1) with x ...
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A baksetball probability question using Neyman–Pearson lemma

It is known that the probability of a basketball player to make his first shot is $p=0.6$ A player argues that it does not matter if he made the previous shot or not his odds stays the same. We say if ...
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When and why do formulae involving sums over $x_i$ change to formulae involving $X$ in statistics? Specifically when dealing with likelihoods.

I've been reading up on stats recently and a question I'm working through involves calculating the log-likelihood of a distribution w.r.t a parameter $\beta$. From my understanding, for some ...
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Why are you less likely to roll at least 1/6 of the dice as 6 when the number of dice increases?

So, I recently watched a V-Sauce video discussing a collaboration between Sir Isaac Newton and Samuel Pepys on a probability problem regarding the probability of rolling at least one six on six six-...
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Parameter estimation truncated Laplace distribution

Hello to the community, I have a problem with the parameter estimation from a model. Let's guess we have a sample $X = (X_1,...,X_n)$, $\forall i=1,...,n$ $X_i$ follows a truncated Laplace ...
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A research department is investigating the duration of a tire.The department wants to show it exceeds a certain value. Formulate the hypothesis

The research department of a tire manufacturer is investigating the duration of a tire using a new rubber component. 16 tires were produced and the duration was tested. The average duration and ...
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Is it possible to combine two statistics which may be dependent on each other without access to the underlying data?

I'm trying to find the average salaries for college graduates of specific universities with specific majors. The issue I'm running into is that there isn't good data available for this issue. However, ...
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Can we use both “Inferential Statistics” and “Statistical Inference” terms in an academic paper, interchangebly?

I've came upon these two definitions: Inferential statistics are techniques that allow us to use these samples to make generalizations about the populations from which the samples were drawn. ...
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How to estimate the “innate speed” of a leaping frog?

The motion of a leaping frog is set by a "hidden parameter" $V_\infty$ that we want to estimate: it is the "innate average velocity" of such a frog. The frog jumps a distance $J_i \in \mathbb{R}^+$ at ...
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Are these samples enough for a retention analysis? (T-Test)

I think that this is a dumb question but I would like to know if these differences in these KPIs are reliable or I have to change the sample in order to have statistically significant conclusions. ...
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Example 7.2.19 from Casella & Berger (EM algorithm)

$X_1, ..., X_n \sim Poi(\tau_i), Y_1, ..., Y_n \sim Poi(\beta\tau_i).$ X and Y are mutually independent. However, $X_1$ is missing. I would like to use the EM algorithm to estimate $\beta$ and $\tau_i$...
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Prove that $n \cdot\min\{T_1,…,T_n\}$ isn't allowable as an estimator of $\mu$

Let's suppose we have some electronic device which duration follows an Exponential distribution of unknown mean $\mu$. Some research team wants to estimate $\mu$ and uses a sample of $n$ devices to do ...
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Which of this two estimators of $\mu$ is better (Exponential distribution)?

The problem goes like this: "Suppose we have some electronic device which duration follows an Exponential distribution of an unknown mean $\mu$. We want to estimate $\mu$ and two teams will take care ...
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UMVUE of $\frac1\theta$ when $X_i\sim f_\theta(x)=\frac{1}{24}x^4\theta^{-5}e^{-\frac{x}{\theta}}$

Let $$f_\theta(x_1,\dots,x_n)=\frac{1}{24^n}\prod_{i=1}^nx_i^4\theta^{-5n}e^{\sum\limits_{i=1}^n\frac{-x_i}{\theta}}\hspace{0.5cm}\underset{\forall i=1\dotsm n}{x_i\in\mathbb{R}^+}\;\theta\in\mathbb{R^...
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Expectation value of the reciprocal of a sum of geometrical

I was studying statistical inference when I had a problem with the following probability problem. Problem: Suppose I have that $X_i \sim Geom(p)$ where $ p \in [0,1]$. Let us define $Y= \frac{1}{\...
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how to calculate the Wald test statistic

I am struggling to understand the definition as I am seeing several different definitions on the internet. Lets say I have a question that goes like : Given $Y_1 ...Y_n$ is a sample taken from ...
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36 views

How to use EM algorithm to find the mode of a distribution $f(x)$

The joint density of two random variables 𝑋 and 𝑌 is given by $$ f(x,y) \propto x^2y^2exp(-xy-5x-4y) $$ Treating 𝑦 as missing information, give full details of how the EM algorithm can be ...
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Finding Gaussian integrals with non-infinite limits to justify z-scores

I'm new to studying z-scores and I've been told that for a gaussian statistic, around 95% of the values lie within the area two standard deviations above and below the mean, which (in accordance to my ...
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Let $T(X_1,\dots,X_n)=\sum\limits_{i=1}^n X_i\sim$ Poisson$(n\ln\theta)$ find $\text{Var}[T^2]$ [duplicate]

Let $T(X_1,\dots,X_n)=\sum\limits_{i=1}^n X_i\sim \operatorname{Poisson}(n\ln\theta)$ $\DeclareMathOperator{\Var}{Var}$ I want to find $\Var\left[\frac{T^2-T}{n^2}\right]$ So what{I did $$\Var\left[\...
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UMVUE of $(\ln\theta)^2$ where $p(x|\theta)=\frac{\left(\ln\theta\right)^x}{\theta x!}I_{\{0,1,2,\dots\}}(x)$

Let $$p(x|\theta)=\frac{\left(\ln\theta\right)^x}{\theta x!}I_{\{0,1,2,\dots\}}(x)\; $$ I have to find UMVUE for $h(\theta)=\ln\theta$ and for $h(\theta)=(\ln\theta)^2$ I know $T(X_1,\dots,X_n)=\...
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Compare the least square estimators and the residuals of two Linear Regression with an alternative regressor

So I have been given this as an assignment for my econometrics course and I seriously can't understand where to begin here: Consider the least square regression $ y ∈ R^n $ on $ X ∈ R^{n*k} $, and the ...
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Finding MVUE of Uniform Distribution

Let X1, . . . , Xn be a random sample from Uniform(0, θ). Find the MVUE of E(X1) and Var(X1)? I'm new in inference, so to find MVUE I need first to find the an unbiased estimator T(x) and improve it ...
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Find UMVUE for $\theta$ where PDF is $e^{-x+\theta}$. $x > 0$ [duplicate]

I know that $\sum X_{i}$ is sufficient but I do not know how to continue
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Most Powerful test

Using Neyman Pearson Lemma, find a Most Powerful test for $H_0 : \sigma = 2$ vs. $H_1 : \sigma = 1 \ $ at level $\alpha$ based on a random sample $X_1, \dots , X_n \ $ from $N(3, \sigma^2 )$. Also ...
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AP STATISTICS 2008 EXAM Question on standard error of estimated proportion

This question is taken from the 2008 AP Statistics free-response question 4) An experiment was conducted to study the effect of temperature on the reliability of an electronic device used in an ...
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Finding Sufficient Statistics

Let X1, . . . , Xn be a random sample from the following pmf. P(X = 0) = θ, P(X = 1) = 2θ, P(X = 2) = 1 − 3θ, 0 < θ < 1/3 Find a non-trivial sufficient statistic. I start like this: L(θ)=L(θ)=∏...
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How to compute confidence interval for variance with unknown mean from a normal $(a,\sigma ^2)$ sample?

When mean is known, we note that $\frac{\bar{(X-a)^2}n}{\sigma ^2}$ has a Chi-squared distribution with $n$ degrees of freedom. However, what to do when $a$ is unknown? I can't just substitute sample ...
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Weighted Least Squares with statistical distance

I am currently wondering, if there is a sound way to determine weights in a weighted least squares regressional problem based on some statistical distance measure. Assuming I have data points coming ...
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What can be inferred from low sample size binomial statistics?

Consider the binomial (or Bernoulli) process with probability $p$ of passing or not passing a certain test. Say you have a limited sample size $n$ (because the test is hard to run, or the population ...
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Statistical significance of a classifier's precision

Suppose I have a sample of $n$ data points (examples) that have to be classified into one of two classes (positive and negative). Let's say I have a method to generate a score for each example. The ...
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Sample variance formula vs. Population variance formula usage

Just for sake of not having to write this equation out a few times, I will denote the residual sum of squares for a set of data points $X$ to be $$ RSS= \sum_{x\in X} \left(x - \overline{x}\right)^2 $$...
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Sample correlation coefficient of i.i.d Cauchy variables

Let $X_1,X_2, \dots, X_n, \dots$ be i.i.d Cauchy random variables, and let $Y_1,\dots, Y_n,\dots$ be i.i.d Cauchy random variables independent of the $X_i$'s. Let $\rho_n$ be the sample correlation ...
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Unbiased estimator of variance

My question is why is the best and most commonly used estimator for the variance (in a Gaussian distribution) the sample variance with constant 1/n-1 when the sample variance with constant 1/n+1 ...
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Best way to approximate a changing state from an imperfect test

suppose there is a box with a coin in it, the coin may or may not flip (say every 2-4 secs). Every second I am testing for its orientation and the test has an accuracy of say 70%. what is the best way ...
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79 views

Finding non-trivial sufficient statistic

I have this question. To find a sufficient I have to find first the joint pmf of this pmf's? Let $X_{1}, . . . , X_{n} $be a random sample from the following pmf: $P(X = k_{1}) = \frac{1 − \theta}{...
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Fisher Information. Density function.

I'm reading about Fisher's information. However, a question arises. What is the difference between the density function $$f(x,\theta)$$ and the same density function when composing with the random ...
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difference between bias vs variance

I am confused about variance and bias of function.How one can tell if function is overfitting or underfitting?how can you write formulas that express that?In machine learning if approximator has ...
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How does the posterior density g(μ|x) change if we find out x could only be observed if it were greater than 0.

Given prior density g(μ) and observation X ∼ Poi(μ) , you compute g(μ|x), the posterior density of μ given x. Later you are told that x could only be observed if it were greater than 0. Does this ...
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Is this the correct interpretation of the regression output (panel data)?

I've got some sample data which looks like this: For example, firm 1 took 3 weeks to innovate following policy change X, firm 2 took 6 weeks to innovate following policy change X, etc. Since this is ...
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Expectation over an expected value in ergodic process

From ergodic theorem we can see that: $\frac{1}{N} \sum_{i}f(x_{i}) \to {E}[f(x)]$ as $n \to \infty$ Does this imply that in ergodic process we can similarly write: $\frac{1}{N} \sum_{i}E[f(x_{i}...

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