# Tagged Questions

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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### Existence of asymptotic variance for an estimator when it doesn't converge to normal.

The definition of an asymptotic variance says: For sequence of estimators $\mathbf{U}=(U_1, U_2,...)$, where: $U_i=U_i(X_1,...,X_i)$, if for a sequence of constants $\{k_n\}$: ...
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### Asymptotic variance of MLE of normal distribution.

I am trying to explicitly calculate (without using the theorem that the asymptotic variance of the MLE is equal to CRLB) the asymptotic variance of the MLE of variance of normal distribution, i.e.: ...
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### Comparing definitions of limiting and asymptotic variances - what is the intuition behind?

In Casella's inference, it says: Definition 10.1.7: For an estimator $T_n$, if $\lim_{n\to \infty} k_n Var T_n = \tau^2 < \infty$, where $\{k_n\}$ is a sequence of constants, then ...
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### Relationship between 0-1 Loss and Type I and II error in Neyman Pearson

In the context of hyphotesis test $$H_0:\theta\in \Theta_0$$ $$H_1:\theta\notin \Theta_0$$. Find the relationship between the 0-1 loss defined by $$L(\theta,\delta)=1-\delta \theta\in\Theta_0$$ ...
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### Clarification on tests for independence and homogeneity?

If the null hypothesis for a test of independence is true, what distribution does the test statistic have? Would it still follow a chi-square distribution, or a normal distribution? In a ...
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### Can't figure out the correct degrees of freedom for my goodness of fit

I am trying to solve the following question: The number of goals scored by a certain football team was recorded for each of 100 matches and the results are summarised in the following table. ...
So I generally understand the basis for the t-test: i.e. you take advantage of the fact that you can make $\bar{X}-\bar{Y}$ standard normal: $$Z = \frac{((\bar{X}-\bar{Y})-(\mu_X-\mu_Y))}{\sigma ... 1answer 41 views ### Desirable properties of statistical estimators? What are some of the properties that people will consider when designing a statistical estimators? For example, unbiasedness and sufficiency are some of the factors considered. Please give some ... 0answers 10 views ### Drawing conclusions from given data as percentiles For a problem I want to determine the number of errors. I have data from two observers, let's say observer A and B. This data is given as 5\%, 50\% and 95\% percentiles. Observer A: ... 1answer 67 views ### Difference beteween ANOVA and ANCOVA In the context of using only experiment data for ANOVA analysis, ANCOVA offers post hoc statistical control. Is this a valid conclusion and why? 1answer 16 views ### Understanding the shape of T distributions I'm trying to understand why a T distribution with a small sample size has fatter tails and what this means. My textbook says "...t distributions have more probability in the tails and less in the ... 0answers 40 views ### Calculate calculate E(\frac{1}{\bar X}), Var(\frac{1}{\bar X}) Consider n i.i.d. observations from a Poisson (λ) distribution. Suppose \bar X =\frac1n\sum_{i=1}^n X_i. How do I calculate Var(\frac{1}{\bar X}) or even E(\frac{1}{\bar X}) for that matter. ... 0answers 14 views ### Multilinear loss in Uniform-Exponential model Let a prior \pi(\theta)=\frac{1}{3}(\mathbb{I}_{[0,1]}(\theta)+\mathbb{I}_{[2,3]}(\theta)+\mathbb{I}_{[4,5]}(\theta)) and f(x\mid\theta)=\theta e^{-\theta x}. Taking the multilinear loss ... 1answer 14 views ### using the Cramer-Rao bound to find the approximate variance of a bernoulli trial Given you have an independent random sample X_1, X_2,..., X_n of a Bernoulli random variable with parameter p, estimate the variance of the maximum likelihood estimator of p using the ... 1answer 22 views ### What is the standard deviation of the sample distribution for the sample standard deviation? I know that for the sample distribution for the sample mean given a large sample or a normal underlying distribution, the mean of the sample distribution is the population mean of the underlying ... 1answer 36 views ### rigorous statistics book recommendations I am learning statistical inference by myself, I have skim through a few books like Casella Hoggs and I find it omitted lots of details, for example, they didn't introduce the conditional expectation, ... 0answers 16 views ### What statistical tests should I perform to determine significance in differences between control, treatment, and sham groups? Say I have 3 groups of scores: one control, one treatment, and one sham. I want to determine whether there is a significant increase (or decrease) in the mean scores with the treatment compared to the ... 0answers 18 views ### Hypothesis testing using proportions Suppose that you interview 100 exiting voters about who they voted for governor. Of the 100 voters, 55 reported that they voted for the Democratic candidate. Is there sufficient evidence to suggest ... 0answers 20 views ### Hypothesis testing and the size of critical region I'm studying stats in order to obtain a certificate and I'm struggling with this question. I've searched a lot and I really didn't understand how to procede in this question. I even don't know how to ... 2answers 28 views ### How can I determine the best relationship for 3 variables, given several data points? What is the best way to determine the relationship for three apparently related variables? The relationship does not appear to be linear, and may follow a combination of non-linear functions. I have ... 0answers 8 views ### Bayesian Inferences: Finding Posterior HPD Interval I am currently working with Beta-Bernoulli and Beta-Binomial models. I have been searching around for the specific steps in obtaining the Posterior HPD intervals for both. Does anyone know how to find ... 0answers 42 views ### Finding the Cramer-Rao Lower Bound Given the probability density function$$f(x; \theta) = \frac{ \left(\ln(\theta)\right)^{x}}{\theta x!}, \quad x = 0,1,\ldots ; \theta > 1and 0 otherwise, find the Cramer-Rao Lower Bound ... 0answers 20 views ### Transformation of continuous random variables Let's say that Y = \log T = \alpha + \sigma W. I know that If W has logsitic distribution, the T will have the log-logistic distribution. Also, if W has the standard normal distribution then ... 0answers 23 views ### If the conditional expectation of the random variable does follow a linear function, can we show the probability a particular data set happens? Suppose that \mathbb{E}[Y\mid X=x]=\beta_0+\beta_1x where X, Y are random varibles. Given a set of observations consisting pairs of X,Y, is it possible to attach it as probabiltiy density ... 1answer 16 views ### Moment generating function of quadruple-form of Gaussian RVs Let X \sim N(0, I_d) be a d-dimensional Gaussian random variable. Let \beta_1 and \beta_2 be two d-dimensional vectors. I would like to compute expectation \begin{align} \mathbb{E}\Bigl \{ ... 1answer 19 views ### Using the normal approximation, what is the z-value of a sample difference of \hat p_1−\hat p_2=−0.18? [closed] A few concepts from my textbook that I do not understand: If the Random, Normal, and Independent conditions are met, then is it true that \hat p_1+2\hat p_2 is approximately normally ... 1answer 18 views ### how to determine the expected value of the first element of an ordered statistic let X be a random variable with densityf_x(x) = e^{\alpha-x} \: x>a$$(a) what is the maximum likelihood estimator for alpha (b) make the estimator in (a) unbiased, and evaluate its ... 1answer 28 views ### Estimate of Proportion An airline is interested in determining the proportion of its customers who are flying for reasons of business. If they want to be 90 percent certain that their estimate will be correct to within two ... 1answer 52 views ### Obtain a (1-\alpha) 100% confidence interval for \theta using the moment estimator Suppose x_1,..x_n is a random sample from a distribution with probability density f(x|\theta )=\theta x^{\theta -1} 0<x<1 \ and \ \theta >0. Find the moment estimator of \theta and ... 1answer 15 views ### One-sample t tests for sample mean with outliers In carrying out a one-sample t test for a sample mean, how should outliers be dealt with? For example, for the data {110, 110, 110, 118, 122, 150} in a sample size of 6, evidently 150 is an ... 0answers 9 views ### How to find the effects of the parameters with the Latin Hypercube Simulation? We are working with an epidemic model where we have 21 parameters which affect the R_{0}-value. We want to study the effect of those parameters on the R_{0}-value with the Latin Hypercube ... 0answers 30 views ### Why this equation holds, using Bayes formula? My mysterious equation is:$$p(x|\chi)=\int_{\theta\in\Theta}p(x|\theta)p(\theta|\chi)d\theta$$where \chi is some samples drawn from sample space parameterized by \theta\in\Theta. Follows the ... 1answer 35 views ### Correctly calculating the bias of an estimator I'm currently learning about method of moments and maximum-likelihood estimators and have confused myself with this issue: First, let me estimate the parameter \lambda from the exponential ... 2answers 18 views ### Confidence intervals and significance tests A sample is used both to construct a 95% confidence interval for a population proportion p and to run a significance test with null hypothesis H_0:P=0.07 and significance level \alpha=0.05. Is it ... 0answers 9 views ### Additive model ANOVA with interaction. How to obtain gammas from the graph? I try to complete this exercise of statistic that said that I need to compute the values of alpha, beta and gammas from the graph. Graph: I allready compute the values algebricaly, and this are: ... 0answers 9 views ### How to prove the ancillary statistic problem? Let X_1, ...,X_n be a random sample from the following pdf: f(x) = e^{-(x-\theta)} \exp(e^{-(x-\theta)}), -\infty<\theta<\infty a. Define W=\log(\sum_{i=1}^{n}e^{-X_i}).Shoe that ... 0answers 13 views ### Distribution of Subvector Sums Suppose X_1,\dots,X_N \sim_{iid} \mathcal{N}(0,1) are iid normal, and let K=N/2. Let S denote the collection of all subsets of \left\{1,\dots,N\right\} with K elements. For any s\in S ... 1answer 18 views ### estimating \mu of a poisson random variable using method of moments or maximum likelihood A used car salesman is willing to assume the number of sales he makes, per day, is a Poisson random variable with parameter \mu. Over the past 30 days, he made 0 sales on 20 days and one or ... 1answer 23 views ### Simple random sample of a Bernoulli and probability function of a statistic. Let x_1, \ldots, x_n an simple random sample of a Bernoulli distribution of parameter p, 0<p<1. Let \overline{\textbf{x}} be the mean sample and S^2=\frac{1}{n-1}\sum_{i_1}^n ... 1answer 19 views ### For X = (X_1,X_2,…,X_n) i.i.d. N(0,\theta), show  T = \bar X  is not complete for \theta. For X = (X_1,X_2,...,X_n) i.i.d. N(0,\theta), show  T = \bar X  is not complete for \theta. Definition: Let f (t, θ), θ ∈ Θ be a family of distributions for a statistic T (X). The family is ... 1answer 64 views ### Non-Linear Model Transformation I want to transform this Non-Linear Model y= 8-ae^{bx} to Linear.And my issue is in this step lny=ln(8-ae^{bx}) how can simplify it to reach in a linear model which is like this y*=b0+b1x ... 1answer 30 views ### expected value of random variable times sample mean Assume a population of M numbers, where M_i have values y_i, i = 1,2,...,k; thus \sum_{i=1}^k{M_i} = M. Thus the population mean is \mu = \sum_{i=1}^k{\frac{M_iy_i}{M}} and the ... 0answers 20 views ### Can I bound P[R > x + \epsilon] independently of R? I have this probability distribution: P[\Theta < \varphi] = \frac{\varphi}{\pi} for \phi \in [0,\pi]. Now I have n samples of D = R\Theta i.i.d. (R>0) and I want to estimate R as ... 1answer 28 views ### Definition of Power and relationship with Type II error I've seen two definitions of Power: P(\text{Rej. } H_0|\theta \in \Theta_1), from Wiki. P_{\theta}(\text{Rej. } H_0), from Casella and Berger 'Statistical Inference'. Which one is true? If we ... 1answer 17 views ### Statistics for Sports League Qualification How can one quantify and predict the needed points for qualify in a league given an up-to-date results registry? For instance, regarding Basketball Euroleague, there's 8 teams in a league with direct ... 1answer 18 views ### confidence about range of standard deviation Suppose X_1, X_2 is a random sample of size 2 from a normal population known to have mean 0 and variance \sigma^2; further assume x_1 = -0.75, x_2=0.16. How sure (or confident) would you ... 0answers 20 views ### How do I show that the two methods of permutation test are both the same? My main objective is to show the methods described below are really the same. However, I am having difficult both formulating the idea clearly and proving my assertion. Below is my attempt. Suppose ... 1answer 27 views ### Why can we simply pool the realized observations in a permutation test? Let a vector of i.i.d random varibles (X_1,X_2,X_3,\cdots, X_m) and another vector of i.i.d rvs (Y_1,Y_2,Y_3,\cdots, Y_n) be given. Suppose X_i stands for the recovery time using a new treatment ... 0answers 12 views ### Singular matrix in GMM based on variational Bayes computation I have implemented some codes according to the variational inference here. In the code W_k is computed according to$$ W_k^{-1}=W_0^{-1}+N_kS_k+\frac{\beta_0N_k}{\beta_0+N_k}(\bar x_k-\mu_0)(\bar ...
We've got some data containing two variables, where $x$ is the predictor and $y$ is the response variable. We make a model of the form of: $$y=\alpha+\beta \cdot x + \epsilon$$ Then we see that in the ...