Estimation theory is a branch of statistics and signal processing that deals with estimating the values of parameters based on measured/empirical data that has a random component.

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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
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1answer
31 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 $W = 3\bar{Y}/2$ ...
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1answer
45 views

is any upper bound for mean square error of an unbiased estimator?

There is always a lower bound for an unbiased estimator called Cramer-Rao Lower Bound. Does any one remember any upper bound for unbiased estimator? The upper bound is used for worst-case analysis of ...
1
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2answers
85 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 ...
0
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0answers
35 views

maximum likelihood estimate

Two random variable X and Y have the following joint pdf: $P_{XY}(x, y) = \begin{cases} 10y,& \text{for } 0\leq y \leq x^2 \text{and} 0 \leq x \leq 1,\\ 0, & ...
4
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2answers
54 views

estimation of a parameter

The question is: $x_i = \alpha + \omega_i, $ for $i = 1, \ldots, n.$ where $\alpha$ is a non-zero constant, but unknown, parameter to be estimated, and $\omega_i$ are uncorrelated, zero_mean, ...
1
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1answer
26 views

Conjugate Bayesian analysis

Suppose that conditional on $\tau$, the random variable $X$ has normal distribution with mean zero and variance $1/ \tau$. The prior distribution for $\tau$ is Gamma with parameter $\alpha$ and ...
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0answers
34 views

Maximization of The Likelihood Function of Vector Entries and Its Norm

I'd be happy for assistance with the maximization of the likelihood function of the following model. The Parameters Vector $ \mathbf{\Theta} = [{x}_{1}, {x}_{2}] $. The measurement vector is $ ...
2
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1answer
21 views

How to show this estimation?

i have this polynom $$p(x) = \sum_{i=0}^{m}a_ix^i$$ I want to show, that if $\tilde{z}$ is the approximation to the simple zero digit $z \neq 0$ in first approximation, the following estimation ...
0
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1answer
24 views

How to show, that a relative mistake of a special function can be estimated in a given way

how to show, that if you have a function like this $$ y = f(x_1,...,x_m) := c \frac{x_1 *...*x_r}{x_{r+1},...,x_m}, \quad 1 < r \leq m,$$ the relative mistake in first order can be estimated ...
0
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1answer
66 views

A quick chanllenge: height and weight probability problem

The average height and weight of a group of people is 175cm and 70kg; Find the upper bound of the portion of the people who are over 200cm and over 100kg. I thought about Markov inequality, but I ...
0
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1answer
101 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
108 views

Optimal combination of two estimates

I have a set of random variables, $X_1,\dots,X_N$. They are i.i.d. Gaussian with zero mean and $w$ variance. I observe $Y_1,\dots,Y_N$ where $Y_i=\sum_{j=1}^N a_{ij} X_j+N_i$ where all $a_{ij}$s are ...
1
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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 ...
1
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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
1answer
153 views

expectation of Gamma distribution help

If x∼Gamma(1,λ) how would i find the expected value E(e^bx) where b=aλ I'm kinda stuck as to how to approach the question. Some help will be greatly appreciated Thank you in advance
2
votes
2answers
47 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
43 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. ...
0
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0answers
38 views

Assigning prior to $\gamma$ in composite power function $P(t) = max[\lambda t^{-\beta}, \gamma]$

I want to estimate the parameters $\lambda, \beta$ and $\gamma$ using a bayesian approach and an MCMC sampler. With the exception of $t$ all variables are random variables between $0$ and $1$. $t$ is ...
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0answers
16 views

A question on Stochastic Approximation

I have just started learning stochastic approximation methods, so the question I'm going to ask may be a trivial one in this field, but I need to know this seriousely. I know, that if $g(x,\xi)$ is a ...
-1
votes
1answer
164 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
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1answer
132 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?
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2answers
134 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 ...
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0answers
18 views

Difference between two model fitting schemes

We have some experimental data, $x \mapsto \hat{f}$ and we're trying to fit a known model of the form $$f(x\ \left|\right.\ a_1, a_2, a_3, b_1, b_2, b_3) = a_1 F(b_1, x) + a_2 F(b_2, x) + a_3 F(b_3, ...
2
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0answers
35 views

What are the properties of median-unbiased estimators?

On Wikipedia it says that " A median-unbiased estimator minimizes the risk with respect to the absolute-deviation loss function, as observed by Laplace." How to prove this? Note that I asked on Cross ...
1
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1answer
57 views

Is it possible to “customize” the multinomial distribution to your specifications?

So according to the multinomial distribution, the probability function $\Pr(X_1 = x_1, X_2 = x_2, \dots, X_k = x_k)$ is equal to $\dfrac{n!}{x_1! x_2! \cdots x_k!} \cdot p_1^{x_1}\cdot p_2^{x_2} ...
1
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1answer
139 views

How to calculate probability using multinomial distribution?

So according to the multinomial distribution, the probability function $\Pr(X_1 = x_1, X_2 = x_2, \dots, X_k = x_k)$ is equal to $\dfrac{n!}{x_1! x_2! \cdots x_k!} \cdot p_1^{x_1}\cdot p_2^{x_2} ...
0
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0answers
49 views

What is a zero order estimate?

In a paper I am examining (Multiband signal processing by using nonuniform sampling and iterative updating of autocorrelation matrix by Modris Greitans), the author uses the term zero order estimate. ...
1
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1answer
166 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 ...
3
votes
2answers
77 views

Estimating time in harmonic signal

I hope someone can help me with the following problem: Assume a periodic signal of the form $$\begin{align} s(t) &= \sum\limits_{p=1}^P \sin(p\Omega_0t)\\ &= \sum\limits_{p=1}^P ...
0
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0answers
76 views

Markov Chain : Montonicity of Sample Mean

Let $\{X_n\}_{n\geq1}$ be an irreducible, ergodic Markov chain with discrete state-space $S$, transition probability matrix $P$ and steady state distribution $\pi = \{\pi_j\}_{j\in S}$. Let $f$ be a ...
0
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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 ...
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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 ...
0
votes
1answer
68 views

Bounds on least squares and weighted least squares estimator

I was wondering if I can get some help in getting bounds on the parameters estimated by least squares (LS) and weighted least squares (WLS) methods. Suppose our observation model is: $\mathbf{y} = ...
1
vote
1answer
73 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 ...
2
votes
1answer
46 views

Monotonicity of Sample Mean

$X_1,X_2,\ldots$ are drawn i.i.d. from a distribution with mean $\mu$. Define $\bar{X}_n = \frac{1}{n}\sum_{i=1}^n X_i$ Prove that $\forall t \quad E[|\bar{X}_t - \mu|] \geq E[|\bar{X}_{t+1} - \mu|]$ ...
0
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1answer
45 views

Estimator in a one dimensional normal setting with only one observation

Let $X$ have the distribution $N(\theta,1)$ where $\theta \ge 0$. Is $T=X$ an admissible estimator with respect to the mean squared error? Construct an estimator that respects the assumption $\theta ...
3
votes
1answer
2k views

Maximum Likelihood Estimator for Multinomial.

Suppose that 50 measuring scales made by a machine are selected at random from the production of the machine and their lengths and widths are measured. It was found that 45 had both measurements ...
0
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1answer
181 views

Finding UMVUE from Lehmann-Sheffe Directly

I am having some trouble with an example from the book I am following. Let $X_1,X_2,...,X_n$ for $n>2$ be an iid set of $N(\mu,\sigma^2)$ random variables with, $\mu\in\mathbb{R}$ and ...
0
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0answers
42 views

How to calculate $Q_n$estimator

How to calculate $Q_n$estimator from this formula for sample $1\;3\; 6\;2\;7\;5$: $$Q_n=d\left\{|x_i-x_j|;\;\;i<j\right\}_{(k)}$$ where d is constant factor and $k={h \choose 2}\thickapprox ...
3
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2answers
110 views

Estimation of the number of prime numbers in a $b^x$ to $b^{x + 1}$ interval

This is a question I have put to myself a long time ago, although only now am I posting it. The thing is, though there is an infinity of prime numbers, they become more and more scarce the further you ...
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0answers
25 views

Proportional-Integral Estimator Question with Agents and Graphs

My question concerns a collection of $n$ agents, with interconnections described by a graph $G = (V,E)$, with Laplacian $L$ and adjacency matrix $A$. The point behind the following dynamical-system is ...
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0answers
35 views

How to obtain confidence intervals for a sample of independent normally distributed random variables

Let $X_1$, ..., $X_n$ be independent observations with distribution $N(\mu,\sigma^2)$ where $\mu \in \mathbb{R}$ and $\sigma^2>0$ are both unkown. What is an easy way to derive confidence ...
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2answers
74 views

What exactly is simple consistency?

Now yes, I know the definition: A sequence of estimators $\{T_n\}$of $\tau(\theta)$ are consistent if for every $\epsilon > 0$ $$ \lim_{n\to\infty}P[|T_n-\tau(\theta)|\leq\epsilon]=1 \\\text{ ...
4
votes
1answer
146 views

Estimate the scale of $e^{-(m+1) t} \sum _{k=0}^{\infty } \frac{t^k}{k!}\left(\sum _{r=0}^k \frac{t^r}{r!}\right)^{m}$

I would like to estimate the scale of the following series, $$S(m,t)=e^{-(m+1) t} \sum _{k=0}^{\infty } \frac{t^k}{k!}\left(\sum _{r=0}^k \frac{t^r}{r!}\right)^{m},$$ where $e$ is the base of ...
3
votes
1answer
351 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
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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
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1answer
65 views

find $\theta_{MLE}$ for a function

For $$ f(x;\theta)=(\theta+1)x^{-\theta-2} $$ find the maxmimum likelihood estimators (MLEs) for $\theta$ based on a random sample of size $n$. My work so far: $$ \begin{align} \prod_{i=1}^n ...
0
votes
1answer
42 views

Random Poisson Sample, Probability in terms of $\vartheta$

If $X_1, X_2, \ldots, X_n$ are a random sample from a Poisson Distribution with mean $\vartheta>0$, how do you find $P(X\le 1)$ in terms of $\vartheta$? I've proven that summing $X_i$ for ...