Tagged Questions

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Does consistent estimators have in-variance property?

If $(T_n)$ is a sequence of consistent estimators of a parameter $\theta$ ( i.e. for every $\epsilon >0$ , $\lim_{n \to \infty} P [ \space |T_n -\theta|< \epsilon ]=1$ ) , then is it true that ...
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Maximum likelihood estimate vs likelihood ratio tests?

Can someone explain to me the intuition behind why we need likelihood ratio tests. From my understanding, they make use of maximum likelihood estimators over different parameters space and they are a ...
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Working with the sum of two independent random variables, and estimating a parameter

A network source sends a sequence of zeros and ones, $X_1, X_2, ...$ with $X_i$(iid) Bernoulli with $p = P(X_i = 1), 0 < p < 1$. Due to disturbances the received sequence is $Y_1, Y_2, ...$ ...
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Estimation of real estate

I'm working on a project on estimating the price value of real estate. First I have collected a lot of data (500 000 instances) with details such as postal code, number of bedrooms, build year, ...
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Biased MLE estimate of mean (expectation)

Please give an example of p.m.f. or p.d.f. , the maximum likely-hood estimate of whose mean (expectation) is a biased estimator . Thanks
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What is the problem with this model parameter estimation algorithm?

In a statistical model with parameters $\theta$ and unobserved laten variables $Z$, the model likelihood is $$L(\theta;X)=Pr(X|\theta)=\sum_ZPr(X,Z|\theta)$$ The standard way to estimate $\theta$ ...
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estimation problem for two-parameter weibull distribution

Suppose the two-parameter Weibull distribution is given by the pdf $$f(x;a,b) = \left(\frac{x}{a}\right)^b\frac{b}{a}\exp\left\{-\left(\frac{x}{a}\right)^b\right\},$$ where $x,a,b>0$. I am ...
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Shapiro-Wilk test

I am trying to determine if a given sample comes from a Normal distribution. For that purpose I want to perform a Shapiro-Wilk test in the way stated on wikipedia. My concern comes with the vector ...
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Finding an efficient estimator for $\theta$ in $U[0, \theta]$ in terms of the sample maximum

This question appeared in a past exam paper, in the form: Let $X = (X_1\dotsc X_n)\in\mathbb{R}^n$ be an i.i.d. sample from $U[0, \theta], \theta>0$ Apply Rao-Blackwell's theorem to the unbiased ...
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paramter estimation (maximum likelihood) of a mixture density

I have this mixture distribution $f(x) =w \cdot \mathcal{LN}(\mu_1,\sigma) + (1-w)\cdot \mathcal{LN}(\mu_2,\sigma)$ where $\mathcal{LN}(\mu,\sigma)$ is a lognormal distribution. I now have random ...
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questions on bias of estimator

a) Let $X_{1},...,X_{n}$ be i.i.d Uniform$[0,\theta]$. Show that estimator $\beta(X)=max(X_{1},..,X_{n})$ is a biased estimator for $\theta$.Find an unbiased estimator, based on $\theta$. My attempt: ...
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Predicting future outcomes from samples when sample sizes and distributions are not controlled and vary

I'm very stale in my statistics and am trying to calculate my confidence around a certain mean outcome from an investment firm (I'll use lay person terms so that I am not assuming any particular type ...
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statistics inequality

Let $\theta$ be a discrete pararmeter and $\gamma_{n}$ be an estimator. Prove that for any $c>0$ we have that $$\text{E}[(\gamma_n-\theta)^2] \ge\Pr[|\gamma_n-\theta|>c]\cdot c^2$$
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log likelihood function of a cauchy distribution

What is the log likelihood function of a random varible x with cauchy distribution (0,1)? I've tried to work it out. I think its $\log (1+x)^2$. Is that correct?
Let $X_n$ be a Poisson process with mean $\lambda^*$. The following sequence estimates the parameter of the Poisson process: $X_{n+1} = \hat{\lambda}_{n+1} + ... 2answers 81 views Maximum likelihood estimator? I am looking at some questions from Mods 2010 and I can't figure this one out. I think my problem is technical... We have a sample (L1,R1), ...,(Ln,Rn) with Lj and Rj normally distributed independent ... 1answer 26 views Given 50 IID normals, find the exact SE for the estimate of$\sigma^2$? Given 50 I.I.D Normal distributions random variables$X_i$, the Maximum Likelihood estimator for$\sigma^2$is$\hat{\sigma}^2$, as proven in my lecture notes. Find the EXACT SE. My Attempt: ... 0answers 26 views Convergence rate of an estimator Say we are interested in estimating some unknown real scalar parameter$\alpha$using data. Suppose the estimator$\widehat \alpha_N$of$\alpha$using the data is consistent. I want to know what it ... 1answer 56 views Finding the MLE of pareto dist., and trouble interpreting$\prod$notation properly. I am generally having trouble understanding how to use product notation when calculating Maximum Likelihood Estimators. The example bellow is from a random sample$X_1,...,X_n$. Find the MLE of ... 1answer 27 views Mean Square Estimate problem I have to find$\textbf{s}_{MS}$given$\textbf{r} = h\textbf{s}+\textbf{n}$where$h$is a Bernoulli random variable with$Pr(h=1)=Pr(h=0) = 1/2$and$\textbf{s}$and$\textbf{n}$are independent ... 0answers 10 views Estimating variance from the sequence Suppose that we have$\{X_n\}\to X\sim N(0,\Omega)$where$X_n$can be obtained from observations. My problem is to estimate$\Omega$consistently. If$var X_n$converges to a "finite" matrix, then ... 1answer 110 views Find the Method Moment Estimator of parameter$\theta$Find the MME of parameter$\theta$in the distribution with the density$f(x,\theta)=(\theta +1)x^{-(\theta+2)}$, for$x>1$and$\theta >0$. So far I think I have a basic understanding of the ... 1answer 59 views Is Sample Covariance Tied to a Specific Distribution In many sources on data analysis, the author(s) talk about calculating covariance of the data, and the formula is given as such $$\Sigma = cov(X) = E[(X-E[X])(X-E[X])^T]$$ This formulation is given ... 1answer 66 views How is the “cooking” done in surveys In my country there's an official center undertaking surveys of voting intention every 4 months. However, they provide only "direct" voting intention, and the statistics obtained are usually pretty ... 2answers 69 views Calculating likelihood of event based on retrospective analysis I have a simple dataset consisting of the dates/times at which certain medications were taken by a patient. By looking retrospectively I'd like to make a best guess estimate as to which medication ... 0answers 23 views Prove that an estimator is UMVU under the usual “assumptions of regularity” I'm asked to prove that some estimator is UMVU under the usual assumptions of regularity. I'm not sure what is meant with 'usual assumptions of regularity'. Do they mean with this that I can just ... 1answer 101 views Maximum likelihood estimator of$P(X < y)$for fixed$y$I'm having a problem understanding the following question. Given the following density function$f_X(x; \theta) = (\theta +1)x^\theta$on$0<x<1$, find the maximum likelihood estimator for ... 0answers 21 views Estimate the population mean when random selection is not possible Consider I have a jar with marbles labeled 0 and 1 in it. They're not well mixed so the possibility of obtaining a sample sized 1000 with mean 0.6 and another sample sized 1000 with mean 0.4 is not so ... 1answer 42 views proving unbiasedness of an estimator Question given independent random variable$X_{1},X_{2},...,X_{n}$from a geometric distribution with parameter$p$. we have an estimator for$p$, mainly$T=Y/n$where Y is number of$i$that ... 1answer 54 views calculating mean squared error for the Mean. Exam Question There are two independent random variables$X_{1}\&X_{2}$that are having normal distribution with mean$\mu$. Further Var$(X_{1})=1$and Var$(X_{2})=2$.an unbiased estimator ... 0answers 41 views Variance of a difference in estimated proportions with trivariate discrete distributions Let a multivariate distribution be given by$P(Y,S_1,S_2)$, where all three variables are discrete,$Y$is multivalued,$S_1=(0,1)$and$S_2=(0,1)$, respectively, and all may be dependent. Define the ... 0answers 46 views Unbiased estimator with conditional expectation. Suppose that$X$has a binomial distribution with parameter$N=1$and$p=1/2$. Y, which is independent of$X$, has a normal distribution with mean$\mu$and variance 1. Consider the estimator$\mu$of ... 1answer 40 views Variance of unbiased estimator Let$Y_1,Y_2,...,Y_N$be a random sample from a distribution with probability density function$f_Y(y,\theta) = 2y/\theta^2$if$0<y<\theta$and$0$otherwise. (a) Show that$W = 3\bar{Y}/2$... 2answers 90 views Finding the MLE of a function when L'($\theta$) doesn't depend on$\theta$Here's the problem: Find the MLE of of$\theta$when$f(x\mid\theta)=(1+x\theta)/2$for$-1<x<1$,$=0$otherwise.$0<\theta<1$Find the maximum likelihood of$\theta$and find its ... 1answer 41 views Statistics: why is this probablility smaller? a shipment of goods contains two containers, one container has 300 units and the other container has 700 units. A supervisor checks 30 units in the first container and he finds$X_1$broken units and ... 2answers 58 views Why we always put log() before the joint pdf when we use MLE(Maximum likelihood Estimation)? Maybe this question is simple, but I really need some help. When we use the Maximum Likelihood Estimation(MLE) to estimate the parameters, why we always put the log() before the joint density? To use ... 1answer 47 views Suitability of skew normal for rating task and calculation in an experiment, I ask participants to rate qualities on a continuous scale. I expect the results to be normal distributed and I am confident that assuming a normal works fairly well for most values. ... 1answer 205 views CRLB to find UMVUE In what situation can one obtain an estimator that reaches the Cramer-Rao lower bound, i.e. an efficient estimator? I know the rules for finding UMVUEs, and I know they are efficient if they reach ... 1answer 158 views Does an UMVUE's variance match the Cramer-Rao lower bound? I know it can match the CRLB, but does it have to, if it is an UMVUE? 2answers 152 views Does convergence in probability not imply convergence in distribution for Least Squares estimators? I have a question relating to convergence in probability and distribution for least squares estimators. Frequently, I see in textbooks that$\hat{\beta} \rightarrow^p b$. Where$b$is the population ... 1answer 203 views Find maximum likelihood estimator, trick? Let$Y_1, Y_2, \ldots, Y_n$iid random variables with density$f(y)=\theta\cdot y^{\theta-1}$,$0<y<1$,$\theta >0$. I need to show that the maximum likelihood estimator of$\theta$is ... 0answers 23 views ML Estimation for number of animals in a park. Hypothesis Testing. A park of area$S=10 000 km^2$was surveyed for bears, and out of$n$disjoint regions of equal area$s=1km^2$, there were$n_k$regions with$k=0,1,....,N\$ bears. On each of these regions, the amount ...
Consider a Bernoulli random variable: $$X_i= \begin{cases} 1, & \text{with probability }p \\ 0, & \text{with probability }1-p \end{cases}$$ You observe the outcomes of two Bernoulli trials ...