0
votes
0answers
80 views

Second derivative wrt complex parameter

I'm facing an estimation problem and I need to calculate the Cramer-Rao Lower Bound of an estimator. So I have 2 unknown parameters: the amplitude of the signal $A$ and its direction of arrival $u$. ...
2
votes
0answers
38 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
30 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 $W = 3\bar{Y}/2$ ...
0
votes
1answer
99 views

Maximum Likelihood

Find maximum likelihood estimator $\hat\theta$ of $f(x;\theta) = (1/2)\exp(-|x-\theta|)$, for $-\infty \leq x < \infty$ and $-\infty \leq x < \infty$. I am confused of how to deal with the ...
1
vote
1answer
41 views

Overestimate of $|\oint_{|z|=R} f(z) \mathrm{d}z|$ with $f(z)=\frac{z^a}{z^2+1}$, $0<|a|<1\mathrm{with} \;a \in \mathbb R$

How can I overestimate, $|\oint_{|z|=R} f(z) \mathrm{d}z |$ with $f(z)=\frac{z^a}{z^2+1}$, $0<|a|<1 \; \mathrm{with} \;a \in \mathbb R$ ? I tried this: $|\oint_{|z|=R} f(z) \mathrm{d}z | <= ...
0
votes
0answers
25 views

ML Estimation, graphical method?

I have a problem in statistics that I don't quite know how to do: "Generate a 1000-element data sample from the Rayleigh distribution. Graph the log-likelihood function $\ln L(\alpha; \vec{x})$ as a ...
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 ...
3
votes
1answer
349 views

CRLB/UMVUE estimation of $\theta$

We have a random sample $X_1,X_2,\ldots,X_n$ from a probabilitiy distribution with density $f(x;\theta) = \theta x^{-\theta-1} $ given that $x > 1$, and $0$ else. where $\theta >1 $ is an ...
0
votes
1answer
12 views

Estimating $\hat{p}$

let $X\sim Bin(n,p)$ and $\hat{p} =\frac{X}{n}$ a) Find a constant c such that $E[c\hat{p}(1-\hat{p})]=p(1-p)$ My work: $$ \begin{align} cE[\hat{p}(1-\hat{p})] ...
2
votes
1answer
57 views

Finding the MLE of $\theta$ where $\theta \leq x$

consider the following PDF $$ \begin{eqnarray} f(x;\theta) &=& \left\{\begin{array}{ll} 2\frac{\theta^2}{x^3} & \theta \leqslant x\\ 0 & x< \theta; 0 < \theta ...
1
vote
1answer
262 views

Maximum likelihood estimators, hypergeometric and binomial

I'm trying to solve a two part problem. The set up is as follows: consider a bag with $\theta$ red marbles and $7-\theta$ blue marbles, with $\theta$ being unknown. Let $x$ denote the number of red ...
1
vote
1answer
443 views

Rao-Blackwell unbiased estimator geometric distribution

I'm looking at review questions and having trouble with this one! Let $X_1,\ldots,X_n$ be i.i.d. geometric R.V.s with the pmf: $(1-p)^{x-1}p$, for $x=1,2,\ldots$ and $0<p<1$. I need to use ...
0
votes
1answer
90 views

How to estimate parameters of a normal distribution?

Suppose one knew that 105 workers were evaluated by their boss. Such evaluation is distributed according to a normal distribution with mean $\mu$ and std. deviation $\sigma$. We also know that 20 ...
-1
votes
1answer
234 views

How to estimate parameters of a uniform distribution?

I have information of the order in which students were classified in regard to their scores in a SAT test. I know the distribution of scores for each student is uniform with support [a,b]. I also know ...
0
votes
1answer
54 views

Several Unbiased Estimators

If I have some data set $ D={X_1,...X_N} $ and have an esitmator be "pick the first point" $X_1$, how can I show that this estimator is unbiased? I also have to show why its highly undesirable, and I ...
1
vote
0answers
66 views

Showing that statistic is unbiased

Let $X $ be observed data. Let $\hat{\theta}(X)$ be an unbiased estimate of $\theta$ and let T be a sucient statistic for $\theta$. Define the new estimator $\hat\theta^{*}$ of $\theta$, $$ ...
1
vote
1answer
131 views

Sufficient Estimators and Generalized Likelihood Ratios

If you can make the assumption that a sufficient statistic exists for some parameter - let's call it $\theta$. How would you show that the critical region of a likelihood ratio test will depend on ...
0
votes
1answer
149 views

Estimation Theory - Maximum Likelihood Estimation

The below homework question comes from Larsen and Marx, 4th edition. Is the maximum likelihood estimator for $\sigma^{2}$ in a normal pdf, where both $\mu$ and >$\sigma^{2}$ are unknown, ...
2
votes
1answer
3k views

Prove the sample variance is an unbiased estimator

I'm trying to prove that the sample variance is an unbiased estimator. I know that I need to find the expected value of the sample variance estimator $$\sum_i\frac{(M_i - \bar{M})^2}{n-1}$$ but I get ...
7
votes
1answer
2k views

Minimum variance unbiased estimator for scale parameter of a certain gamma distribution

Let $X_1, X_2, ..., X_n$ be a random sample from a distribution with p.d.f., $$f(x;\theta)=\theta^2xe^{-x\theta} ; 0<x<\infty, \theta>0$$ Obtain minimum variance unbiased estimator of ...