0
votes
1answer
28 views

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 ...
1
vote
1answer
35 views

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 ...
1
vote
0answers
25 views

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 $Y_1, Y_2, ...$ ...
0
votes
0answers
20 views

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, ...
0
votes
2answers
40 views

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
0
votes
0answers
23 views

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 $\theta$ ...
0
votes
1answer
23 views

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 ...
1
vote
1answer
29 views

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 ...
3
votes
1answer
46 views

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 ...
2
votes
0answers
32 views

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 ...
1
vote
0answers
60 views

Expectation of $\cos(\|X\|)$ where $X \sim \mathcal{N}(\mu,\Sigma)$

Do: $$ \int_{-\infty}^\infty \int_{-\infty}^\infty \cos\left(\sqrt{x^2+y^2}\right) e^{-\frac{1}{2}\left[\frac{(x-\mu_x)^2}{\sigma_x^2} + ...
1
vote
1answer
75 views

Order related to Empirical distribution function and Normal distribution

Let $X_1,\dots,X_n$ are i.i.d with distribution function $F$. Let $\hat F_n$ be its empirical distribution function, i.e., $$ \hat F_n(x)=\frac1n\sum_{i=1}^n1_{\{X_\le x\}}(x) $$ where $1_A(x)$ is the ...
0
votes
0answers
21 views

When does Cramer-Rao Lower Bound fail to hold?

When does Cramer-Rao Lower Bound fail to hold? I have computed the bound, and still one of my estimators has less variance below the bound. Why does this happen? Should not this bound hold always?
0
votes
0answers
26 views

compute maximum likelihood estimator and confidence interval

Let $X$ have a poisson distribution with parameter ${\lambda}$. Find the maximum likelihood estimator of ${\alpha}=P(X=0)$. In a sample of size 100 from a poisson distribution, it is found that the ...
2
votes
1answer
59 views

How to compute the unique MLE from an Exponential Family of Distributions?

Let $$ f(x;\theta)=\frac{1}{\pi} \frac{e^{\theta x}\cos(\theta \pi/2)}{\cosh(x)}, x\in{\mathbb{R}} $$ be a family of densities and which is clearly exponential family. Then what is the Maximum ...
1
vote
1answer
59 views

Conditional expectation and Rao Blackwell

Consider a family of densitites $f(x,\theta)=\frac{\exp(-\sqrt{x})}{\theta}$. Let $X_1$ be a single observation from this family. I have shown that $\sqrt{X_1}/2$ is an unbiased estimator. Now ...
2
votes
0answers
40 views

Alternatives to Fisher information

The Fisher information matrix is defined as the following: $$\mathcal{I}(\theta)=E[(\frac{\partial \log f(x;\theta)}{\partial \theta})^2]=-E[\frac{\partial^2 \log f(x;\theta)}{\partial \theta ...
0
votes
1answer
141 views

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: ...
0
votes
0answers
29 views

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 ...
1
vote
1answer
94 views

statistics inequality

Let $\theta$ be a discrete pararmeter and $\gamma_{n}$ be an estimator. Prove that for any $c>0$ we have that $$\text{E}[(\gamma_n-\theta)^2] \ge\Pr[|\gamma_n-\theta|>c]\cdot c^2$$
0
votes
1answer
73 views

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?
0
votes
0answers
15 views

Looking for hints on how to prove the convergence of this iterative estimator! [duplicate]

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} + ...
1
vote
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 ...
0
votes
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: ...
1
vote
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 ...
1
vote
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 ...
0
votes
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 ...
0
votes
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 ...
1
vote
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 ...
1
vote
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 ...
3
votes
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 ...
3
votes
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 ...
0
votes
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 ...
0
votes
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 ...
0
votes
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 ...
0
votes
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 ...
0
votes
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 ...
0
votes
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 ...
2
votes
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 ...
0
votes
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 $W = 3\bar{Y}/2$ ...
1
vote
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 ...
1
vote
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 ...
2
votes
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 ...
0
votes
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. ...
-1
votes
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 efficient estimator? I know the rules for finding UMVUEs, and I know they are efficient if they reach ...
0
votes
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?
1
vote
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 ...
1
vote
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 ...
1
vote
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 ...
1
vote
1answer
80 views

Using Neyman pearson lemma when ratio comes out to be zero.

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 ...