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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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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52 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
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37 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 $\theta$ ...
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1answer
106 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 ...
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1answer
45 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 $(...
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93 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 ...
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840 views

how to calculate vehicle speed using mathematics and Image processing?

i am doing my project in image processing.using segmentation i have detected the moving part(i.e the car) in the video successfully. But now i want to calculate speed of vehicle. in the above figure ...
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58 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 ...
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1answer
42 views

Bayes - dual estimation of parameter value and parameter growth

I am trying to find an bayesian approach to the following problem: Image a bucket with 100 white balls and an unknown number of red balls During each year, one can take a sample with replacement of ...
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102 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} + \frac{(y-\mu_y)^2}{\sigma_y^2}\right]}\:\mathrm{d}x\:\mathrm{d}...
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92 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 ...
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1answer
91 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 ...
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2answers
94 views

MAP Estimator with Laplacian Noise

I need to calculate the MAP estimator of $ x $ in the following case: $$ \left [ \begin{matrix} {y}_{1}\\ {y}_{2} \end{matrix} \right ] = \left [ \begin{matrix} x\\ x \end{matrix} \right ] + \...
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1answer
162 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 ...
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1answer
312 views

Is a probability density function necessarily a $L^2$ function?

If a nonnegative continuous real valued function $f$ is integrable over $\mathbb{R}$ with $$\int_\mathbb{R} f\,\mathrm{d}x = 1,$$ does it hold true $$\int_\mathbb{R} f^2 \,\mathrm{d}x<\infty?$$ ...
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94 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 \...
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2answers
61 views

how to estimate the phase parameter of a complex function

There is a complex serie: $f(t_n)=\alpha_n+\beta_n i$, for $n = 1,...,N$,$t_n,\alpha_n$ and $\beta_n$ are known.When we have know that $f(t)$ has the following form: $$f(t)=Ae^{-iBt}$$ with unknown ...
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1answer
31 views

How to compute MAP estimate of y?

Suppose that a scalar random variable y is of the form $y=z+v$, where the pdf of $v$ is $p_{v}(t)=\frac{t}{2}$ on the interval $[0,2]$, and the pdf of $z$ is $p_{z}(t)=2t$on the interval $[0,1]$. Both ...
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164 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: $...
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1answer
96 views

statistics inequality

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

ML estimate of sum of guassian variables?

consider the sum $z=x_{1}+...+x_{k}$, where the scalar variables $x_{i}$ are statistically independent and Gaussian, each having the same mean $0$ and variance $\sigma^2_{x}.$ how can I construct the ...
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1answer
679 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?
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184 views

When is a minimum distance decoder also a maximum likelihood decoder?

It is well known that if we have a binary symmetric channel with crossover probability $\epsilon<0.5$ and we send a word $x$ through it, the most likely word is the one with minimum hamming ...
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1answer
34 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: ...
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56 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 ...
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85 views

Identification of real functions

this my second question, so I'm still new... thanks in advance for any help! Basically, I'm looking for some references and tools to study the following problem. Consider the following function $f(\...
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1answer
85 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 $\...
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1answer
40 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 ...
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62 views

Literature study for Optimal Estimation Theory

It seems Optimal Estimation/Control Theory requires a lot more than undergraduate maths. Any good book that would help me get started? I have so far referred the following books but found them quite ...
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109 views

Estimation of a Ito's semi-martingale linear functional

Could someone check my solution for the following problem please? Or maybe propose a smarter/shorter solution. Consider a stochastic process $X=(X_t)_{t \in [0,1]}$ defined in a filtred ...
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276 views

Maximum Likelihood Estimator of the exponential function parameter based on Order Statistics

Let $X_1, \ldots, X_n$ be a random sample from the exponential distribution $\exp(\lambda)$. Let $$M_n=\max\{X_1, \ldots, X_n\}$$ with probability density function $$g_{M_n}(x)=n\lambda e^{-\lambda x}(...
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1answer
524 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 ...
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1answer
128 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 ...
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1answer
22 views

Likelihood function for only one trial

I have a trial $\mathbb X = (X_1,X_2,....X_n)$. $X_i$ has specified distribution with unknow parameter $\theta$. I want to find an estimator of this parameter. So I can use methods like Likelihood ...
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1answer
78 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 ...
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1answer
34 views

Estimator of a Random Variable

Given a random varable $Y$ where $$ f_Y(y) = \begin{cases}e^{-(y-k)} \quad x>k\\0\quad \text{otherwise}\end{cases} $$ Given $n$ observations of $Y$. Is the sample mean $\bar{Y}$ an unbiased ...
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130 views

Estimation theory: Finding MSE and Variance of an Estimator

Let $\hat{A}$ be an estimator of $A$ where $a<A<b$ and $\tilde{A}$ be another estimator such that$$\tilde{A} = \cases{a \text{ if } \hat A \leq a\\\hat A \text{ if } a< \hat A <b\\{b \text{...
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116 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 ...
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3answers
107 views

More accurate estimation of mathematical constant $e$

Very often in books and also on Wikipedia we can find that: $$e \approx \left(1+\frac{1}{n}\right)^n$$ but I want more accurate estimation, it means instead using $\approx$ I wonder if I can use $\leq$...
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1answer
122 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 $P(X&...
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26 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 ...
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144 views

If $(X_i)$ are i.i.d. exponential $\lambda$, then $\hat\lambda=n/\sum{X_i}$ is a biased estimator of $\lambda$

the main problem is that i have no clue on calculating $E(\frac{1}{x})$ let $U = \frac{1}{\frac{1}{n}\sum{X_i}}$ then, $E(U) = n*E(\frac{1}{\sum{X_i}})$. I think that i'm supposed to calculate: $\...
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1answer
61 views

How to estimate a vocabulary size?

I have a list of the 1 million most common English words ordered by number of times they appear on all books in Google Books. I want for the user to select from a list of 100 words (small sample) ...
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1answer
71 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 $X_{i}=...
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95 views

Time series (stochastic process) estimating parameters using characteristic function

I have a time series of assets ${A_1, A_2, ..., A_n}$, which is described by a sophisticated distribution having the following characteristic function: $\phi(u; t;\theta)$, where $\theta$ is a vector ...
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1answer
196 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 ...
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20 views

Estimating Attendence

I am trying to estimate attendance at events. What I have is daily/weekly ticket sales numbers. The goal is to analyze the ticket sales each day and estimate where we might be at by the day of the ...
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46 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 ...
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98 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 ...
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1answer
95 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 $W = 3\bar{Y}/2$ ...