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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Expected Value of sample mean and single observation
I'm trying to calculate the expected value of the autocovariance estimator,
$$\mathbb{E}(\hat{\gamma}(h)) = \mathbb{E} \left(\frac 1n\sum_{t=1}^{n-|h|}(X_{t+|h|}-\bar{X}_n)(X_t-\bar{X}_n)\right), \...
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2-dimensional complete statistic
Suppose it is given that the statistics $U(\mathbf X)$ and $V(\mathbf X)$ are independent, and furthermore the family of distributions of $U(\mathbf X)$ and $V(\mathbf X)$ are complete. Does it follow ...
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Sufficiency for Pareto distribution
Let $X_j$ for $1 \leq j \leq n$ be i.i.d. Pareto (type 1) random variables with scale $\alpha > 0$ and shape $\beta > 2$. The density is given by
$$
f(x; \theta) = \beta \alpha^{\beta} x^{-(\...
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Computing entropy of inverse gamma distribution
I'm trying to compute the entropy of the inverse gamma distribution and it keeps coming out negative. Obviously this is wrong because entropy is always positive. Wikipedia has this derivation for the ...
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Connection between general loss functions and the MLE
first up a big disclaimer: This question might already be answered somewhere, so if I have missed it, please just redirect me.
Let $X,Y$ be two (real-valued) random variables. I will assume that their ...
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Doubt in Convergence in Distribution . [closed]
The Limit funtion in the definition of Covergence in Distribution
$$lim_{n\to\infty} F_{n}(x)=F(x)$$
It is said "F(x) or the limit funtion may or may not be Distibution function".
I need the ...
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Python implementation of type 2 error [closed]
When reading python demonstration of Data science from scratch, I encounter the following notes:
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Can you use the beta-binomial distribution instead of MCMC?
So, i have a project to test the hypothesis that a marketing campaign with a new art generates more purchases than the old one, i have 2 samples of data, one using the standart ad and one using the ...
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Sample variance of residual series of an AR($p$) model
Ruey Tsay writes on pages 49-50 of his book Analysis of Financial Time Series (Third Edition):
For a specified AR($p$) model in Eq. (2.9), the conditional least-squares method, which starts with the $...
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Are There Bayesian Models Where the Posteriors of $\mu$ and $\sigma^2$ Are Independent?
In Bayesian inference, the Normal-Inverse-Gamma (NIG) distribution is often used as a conjugate prior for jointly modeling the mean $\mu$ and variance $\sigma^2$ of a normal distribution. However, the ...
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Which proposal function should be used in this particular case of the Metropolis-Hastings algorithm?
As part of my research, I would like to apply the Metropolis-Hastings in order to sample from some posterior distribution. More precisely, the data comes from a multivariate normal distribution in the ...
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How to check the Variances between 2 estimators are same or not [migrated]
Let say I have 2 batches of electric bulb from some manufacturing processes
First batch was run from 10 am to 2 pm (just assume). In this batch total $N_1$ number of bulbs are produced and among them $...
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Computing expectation under change of random variable
I am following the work from Kingma and Welling, where they introduce Variational Autoencoders. To train such models they use the so called Evidence Lower Bound and maximize it. One way to solve this ...
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Using differentiation under the integral sign for computing the gradient of the expectation.
I am following the work from Kingma and Welling, where they introduce Variational Autoencoders. To train such models they use the so called Evidence Lower Bound and maximize it. One way to solve this ...
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Is it possible to design a random partition with good concentration performance?
Assume that I have n points that I wish to partition into two groups A, B. My requirement is that for each point $x_i$, the marginal distribution $P(w_i=\mathbb{1}\{x_i \in A\})=e_i$, for some pre-...
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Equality regarding score function
I'm reading Bogetoft and Otto (2011) page 249 and this equality regarding the score function stumped me:
$Var\Big[- \Big(\frac{\partial^2 l}{\partial \beta^2}\Big)^{-1} \frac{\partial l}{\partial\beta}...
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How can I estimate the population size from subpopulation statistics?
Say there is a some population of N samples that are sampled from an unknown Normal distribution, where N is unknown. The samples of the population are divided into k independent subpopulations, each ...
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Paired vs 2-sample t-test
I’m comfortable with the general decision and underlying assumptions between paired and 2-sample t-tests.
Suppose I wish to collect data on, for example, how long it takes my oven to heat up to 200 ...
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Mean of the interval of normal distribution
Given a normally distributed data.
And the population mean $\mu$
I want to find estimate the mean of any arbitrary interval $[a,b]$ using the population mean $\mu$, i know it can be done easily using ...
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Does regular exponential family have an MLR only when $w(\theta)$ is a non-decreasing function?
Just under Definition 8.3.16 of Statistical Inference by Casella & Berger, it is claimed that any regular exponential family with $ g(t|\theta) = h(t)c(\theta)e^{w(\theta)t} $ has an MLR if $w(\...
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Prediction intervals for simple linear regression
I want to understand how to construct the test statistic for the case of predictive inference for a simple linear regression model and would be grateful if someone might confirm if my derivation is ...
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Moments or Mean of transformed Weibull Distribution.
Given the PDF and CDF of the Weibull distribution, denoted as $f_X(x)$ and $F_X(x)$ respectively, the range is $x \geq 0$.
Also, clearly that the mean, $E(x)$ would be $\int^{\infty}_{0}xf_X(x) dx$.
...
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$E((\ln X)^m)$ for Weibull Distribution.
I want to find the $E\left[ \left( \ln\left(X\right)\right)^m \right]$ for Weibull Distribution, which is $f_X(x;\tau,\theta)=\frac{\tau}{x}\left( \frac{x}{\theta}\right)^{\tau} e^{ -\left( \frac{x}{\...
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Is this claim justified at the 99% confidence interval?
A sample of 200 bees from a colony is tested and 40 bees are found to
be infected. The colony of bees will collapse and not survive if 35%
or more are infected. Why it is possible at the 99% ...
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Experimental design: Formula for D-efficiency/D-error
I've been looking for a software package to create an optimal experimental design for an instance where it's not possible to implement a full factorial design. How optimal a design is can be measured ...
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Are $\mu_{\hat{p}}$ and $\sigma_{\hat{p}}$ considered parameters or statistics?
Is $\mu_{\hat{p}}$ (the mean of the sampling distribution of $\hat{p}$) and $\sigma_{\hat{p}}$ (the standard deviation of the sampling distribution of $\hat{p}$) considered parameters or statistics, ...
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Logic behind hypothesis testing and one-tailed tests
Suppose I have the following:
Let's consider a hypothetical experiment to determine whether James Bond can tell the difference between a shaken and a stirred martini. Suppose we gave James Bond a ...
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Measuring departure between the posterior predictive distribution and the true data generating distribution
Suppose, I am trying to measure the departure between the posterior predictive distribution and the data generating distribution.
So, in this case, assume that there is a single observation $$X \sim N\...
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Average Rank versus Ranked Average in Parameter Estimation
I have the following problem:
In a cricket tournament, the eleven batsmen of a team play 100 matches before the final. The runs scored by each are available. Determine the average rank of the batsmen ...
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Understanding equicontinuity in asymptotic normality with nonsmooth objective functions
I am working on the normality of extremum estimators with nonsmooth objective functions. Assume that my objective function is $Q_n(\theta)$, where $n$ is the sample size. I denote by $\hat \theta_n$ ...
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Help developing intuition behind sufficient statistics (Casella & Berger)
Migrated to Cross Validated
I am trying to understand the following intuition for sufficient statistics in Casella & Berger (2nd edition, pg. 272):
A sufficient statistic captures all of the ...
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Creating an Estimator for the Dimension of Bernoulli-distributed Vectors from Observed pairwise Dot Products
I have I individuals defined by vectors $P_i \sim \mathcal{B}(1,1/2)^d$ iid. We can note $\overline{P}_i = \langle P_i, \textbf{1} \rangle$ the proportion of 1's in individual i; $c_{ij} = \langle P_i,...
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Fisher's information for a function that consist in many indicator functions
I have the following pdf:
$$
f(x) = \theta I_{(-\frac{1}{2},0]}+ I_{(0,\frac{1}{2}]}+(1-\theta) I_{(\frac{1}{2},1]}
$$
I've tried the following
\begin{align}
I(\theta) &=-E[\frac{d^2}{d\...
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Intuition for bounds of Adaptive Conformal Inference
I have been reading the paper by E. Candès and Gibbs about Adaptive Conformal Inference (here is the original papel). The main idea is to update the miscoverage level $\alpha_t$ as
$
\begin{cases}
\...
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The sufficient statistic and unbiased estimator of normal variance
Suppose we have a normal distribution with mean $\theta_1$ and variance $\theta_2$.
I know that $\frac{1}{n-1}\sum_{i=1}^n (X_i-\bar{X})^2$ is an unbaised estimator of $\theta_2$ and has a variance $2\...
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The idea/intuition behind replacing elements in bootstrapping(Statistics)
I have read several posts on this, none of them directly deals with this.
I don't understand the idea/intuition behind replacing elements when one bootstraps(Statistics).
As in given a data set that ...
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Maximum Likelihood Estimation for Poisson Mean with Given Observations
You have a sample of $n$ i.i.d. realizations of the random variable $X$ distributed as a Poisson with parameter $\lambda$. It is known that:
$n_1$ values are greater than or equal to $2$;
$n_2$ ...
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How to calculate risk of choosing option0 vs option1
I have an expirement where every day a person has to pick between $image0$ and $image1$, where based on your selection you can win $1$, loose $1$, or $0$.
I am given the following probabilities:
$r0$: ...
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the sum of $O_p$ --$ O_p(s^2\frac{\log d}{n}+s\sqrt{\frac{\log d}{n}}) $
I read papers in the area of inference for high-dimensional graphical models and these papers always state the convergence rate of the estimator. Using $O_p$ is a good choice.
Maybe I made some ...
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How to express Variational Auto-Encoder, ELBO with Random Variables?
In the context of VAE, variational inference and Bayesian statistics more broadly the equations typically involve densities that are somewhat loosely defined. Often the notation refers to a ...
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Probability that one estimator is larger than another
Imagine having $n$ independent and identically distributed (i.i.d.) observations from a variable $X$, which in the population follows a Gaussian distribution $\mathcal{N}(\mu, \sigma^2)$. For $\sigma^...
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How to compare real-time physiological data between pre-defined time blocks
I have a set of physiological data for approx. 20 patients measured per second over approximately 8 to 10 minutes. The data is grouped into the following blocks of time: calibration, test, first ...
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P-value for testing a median to be $M \geq M_0$
I'm trying to solve a question from an introductory textbook on statistics. I am to use the Sign Test and determine if there is significant evidence that the median of a dataset $M$ is "at least&...
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What is the frequentist's Bayesian prior for a coin with unknown bias
A "coin" has a fixed unknown bias $0\le p\le1$ for heads, and out of $n\ge0$ tosses it yielded $0\le h\le n$ heads. Note that this occurs with probability $P(h\;|\;p,n)=\binom{n}{h}p^h(1-p)^{...
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Estimate the correlation coefficient of a two-dimentional normal distribution $(X,Y)$, given some samples of $(|X|, |Y|)$
I have two random variables $X,Y$, whose joint distribution is a two-dimensional normal distribution, and the expectations of both $X,Y$ are zero. Let $\rho={\rm cov}(X,Y)$ be their correlation ...
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When to use t-distribution?
Let the dataset $6, 12, 12, 9, 7, 16, 10$ be given. Each data point is considered to be an outcome of a random variable $X_i$, where the $X_i$'s are assumed to be independent and Poisson distributed ...
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Given the maximum likelihood function- estimate the value of the parameter
Lets say I have the pdf and maximum likelihood function:
$
f_X(x) = \begin{cases}
\frac{\alpha \beta^\alpha}{x^{\alpha+1}}, & x > \beta, \\
0, & x \leq \beta.
\end{cases}
$
$
\begin{...
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What exactly is P-value and what is its relation with the significance level?
The formal definitions I have seen differ but the one I thought I understood was: "the probability of an observed or more extreme result assuming that the null hypothesis is true".
Let's say ...
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Expected Prediction Error from The Elements of Statistical Learning
I am working self-studying the Elements of Statistical Learning and have a question regarding how equation $(2.28)$ is derived. Similar question here but without a satisfactory answer: How to ...
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Evaulating conditional pdf efficiently given marginal pdfs
Let $\mathbf{A}$ and $\mathbf{B}$ be two random variables with joint distribution $p(\mathbf{A},\mathbf{B})$. The joint prior is defined in relation to a data model, given by
$$\mathbf{y} = f(\mathbf{...
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