For questions about estimation and how and when to estimate correctly

learn more… | top users | synonyms

0
votes
0answers
24 views

Lp estimates from Elliptic Equation

Using the theorem: Let $f \in L^{p}(\Omega)$, $1<p<\infty$, and let $w$ be the Newtonian potential of $f$, $w(x)=\int_{\Omega}\Gamma(x-y)f(y)dy$. Then $w\in W^{2,p}(\Omega), \Delta w=f$ a.e and ...
0
votes
0answers
17 views

Optimal combination of multiple estimates of a random variable

For the following estimation problem: y = hx + n, x is the sent data, y is the observation (received data), h is a scaling factor (known), n is an AWGN random variable with zero mean ...
2
votes
1answer
20 views

Probability that a sample comes from one of two distributions

Let's say I have two normal distributions with means $\mu_1$, $\mu_2$ and standard deviations $\sigma_1$, $\sigma_2$ (which I know). I am handed a random variate from one of the distributions (I don't ...
0
votes
0answers
31 views

Approximation of optimum for two linear programs

Suppose you got two linear programs. They are the same except that one has a shifted objective by a positive constant (1) $$\min c^Tx$$ (2) $$\min c^Tx + d$$ For (2) there exists a ...
2
votes
0answers
98 views

Is it compulsory to make transformation to the econometric model in order to have only diagonal elements on variance-covariance matrix of errors?

I need some sharped and advanced advices for the following issue ... Model and its assumptions I'm working on the methodology of a two-way error component model. Here is the model: $y_{jis} = ...
0
votes
1answer
110 views

Test if estimator is unbiased

I'm having problems with the following question for my econometrics homework. Is $\ \ \hat \beta_2 = (y_n - y_1)/(n - 1)\ $ an unbiased estimator of $\beta_2$ for $\ \ y_t\ =\ \beta_1\ +\ \beta_2 \ ...
1
vote
1answer
288 views

Tricks to solve inequalities

I am wondering if there are some tricks to solve inequalities which are not manageable analytically. For example consider the inequality (say we restrict on positive $x$): $\displaystyle \frac {\text ...
0
votes
0answers
37 views

Estimating “task duration” variance through questions in a questionnaire

I am working to estimate the average task duration for complex processes - a total of 19 tasks / processes (each task made of several subtasks of varying duration). I will be organising a ...
0
votes
1answer
26 views

Show that $\hat{\delta}_1=\hat{\beta}_1+(X_1^T X_1)^{-1} X_1^TX_2\hat{\beta}_2$

Let $\hat{\beta}=(\hat{\beta}_1,\hat{\beta}_2)^T$ be the least squares estimator in the regression model $Y=X_1\beta_1+X_2\beta_2+u$. Let $\hat{\delta}_1$ be the least squares estimator of the ...
0
votes
0answers
42 views

Linear model: Show that $\hat{\theta}$ and $\hat{e}$ are independent

Show that under the assumptions $Y\sim N(X\theta,\sigma^2I_n)$and $\text{rang}(X)=\text{rang}(\theta)$ the residual vector $\hat{e}$ and the least squares estimator $\hat{\theta}$ are ...
0
votes
1answer
36 views

Estimating the Growth Rate of Worms

I would like to know how much worms I'd have on hand given an initial amount and period of time. Here are some metrics regarding the growth rate of these worms. ...
5
votes
1answer
117 views

Estimating a gaussian distribution from a GMM

Suppose that we have a Gaussian mixture model (GMM) in n-dimensional space: $$P_1(x) = \sum_{i=1}^{C}\pi(c_i)\mathcal{N}(\mu_i,\Sigma_i)$$ We want to estimate a single Gaussian distribution from ...
2
votes
1answer
37 views

Estimating sum of n elements by throwing away half of elements

I've got a task where i need to proove the asymptotic big-Theta equation: $$ \log n! = \Theta(n \, \log n) $$ $ \ $ Since $f(\mathit{n}) \in \Theta(g(\mathit{n}))$ means that $g(n)\cdot k{_1} \leq ...
0
votes
0answers
16 views

Estimating the accuracy of precipitation odds.

Modern weather forecasts often include a percentage chance of rain. If this estimate was a constant it would be a simple matter of calculating the ratio. But the reality is quite another. This is the ...
2
votes
1answer
51 views

Estimate for integral of sine to the power of $-(1+a)$ where $a>0$

I'm trying to solve or estimate this integral $$ I=\int\limits_{\arcsin{k}}^{\pi/2}\dfrac{1}{(\sin{x})^{1+a}}\mathrm{d}x, $$ where $0<k<1/2$ and $a>0$. The estimate should depend on $k$. I ...
1
vote
3answers
278 views

Using loop to approximate pi (Monte Carlo, MATLAB)

I've written the following code, based on a for loop to approximate the number pi using the Monte-Carlo-method for 100, 1000, 10000 and 100000 random points. ...
0
votes
1answer
32 views

Maximum-Likelihood Estimator: What problems occur if data is not i.i.d.?

This is a question from an exam: You want to estimate the parameters for a gaussian distribution using the Maximum-Likelihood Method for an i.i.d. set of data. What role does the property ...
1
vote
1answer
30 views

show that the sample median of normal distribution is median unbiased

$X_i \sim N(\mu,\sigma^2)$, $i = 1,2,\ldots,n$. Show that the sample median is unbiased in median for $\mu$. I have obtained the pdf of sample median for $n=2m+1$ as: $$f(x_{(m+1)}) = \frac ...
0
votes
0answers
16 views

OLS estimators in stationary process

Given a stationary process xt=a+b*t+et with et a white noise, how can I find the OLS estimators for a and b? Cheers
1
vote
0answers
75 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} + ...
0
votes
1answer
26 views

Confidence/Tolerance interval for a percentage of a population

I have a problem I'm not sure how to solve. It goes something like: ...
2
votes
1answer
29 views

How to estimate the lower bound of a given Toeplitz matrix's eignvalue?

A given Toeplitz matrix is $$\left( \begin{matrix} 1 & a & a^2 & \cdots & a^n \\ a &1 &a & \cdots & a^{n-1} \\ a^2&a & 1 & \cdots& \cdots ...
1
vote
1answer
73 views

Estimation solving for binomial k?

Hello all trying to do an estimation problem at work and wondering if I'm on the right track! I'm running a study and its on the internet. I'm trying to determine how many people I need to show an ...
3
votes
1answer
579 views

What initial guess is used for finding n-th root using Newton-Raphson method?

I would like to know what is an optimal initial guess for use with Newton-Raphson method when finding n-th root. I develop some program which uses GMP C++ library. GMP manual says: The initial ...
1
vote
2answers
53 views

Estimate of L2 norm of a product of functions

Assume that $f,g, f\cdot g\in L^2(\mathbb{R}^n)$. Is the estimate $\|fg\|_{L^2(\mathbb{R}^n)}\leq \|f\|_{L^2(\mathbb{R}^n)}\|g\|_{L^2(\mathbb{R}^n)} $ correct? Thank you!
1
vote
0answers
36 views

Derive Maximum Likelihood Estimator of a Generalised Linear Regression Model

I understand how to find the MLE estimator for $b$ if it is a simple linear regression model. However, when $u\sim N(0,\sigma^2\Omega)$ where $\Omega\ne I$. I am getting confused. The model is: ...
0
votes
0answers
21 views

Is it possible to calculate the width of this table

Is it possible to calculate the length of x (width of the table) using the given values and any information that can be inferred from the image. If not, what is the best estimate that can be found.
0
votes
0answers
17 views

Parameter estimation of matrix-valued processes

I have a hard question for the probability theorist under you. Suppose we have the following process as defined by $dX_t = [-M(\bar{X} - X_t) - (\bar{X} - X_t)M^T] dt + \sqrt{X_t}dB_tQ + ...
2
votes
0answers
45 views

Lower bound of $\sum_{k = 1}^{N}1/(x + k)$

Let $f(x) := \sum_{k = 1}^{N}1/|x + k|$ for $x \in [0, N]$. Why is $f(x) \geq C\log N$ for all $x \in [0, N]$ where $C$ is an absolute constant. My work is: Since $x \in [0, N]$, we can remove the ...
1
vote
0answers
61 views

How to find the MLE of the mean of Gamma distribution

If I parameterize Gamma distribution in the way as $\Gamma(\alpha,\frac{\mu}{\alpha})$, am I able to find the maximum likelihood estimator of $\mu$. Here, $\alpha$ is the shape parameter, ...
0
votes
0answers
21 views

Find the covariance of estimation error

Define the linear operator $O_T: \mathbb{R}^n \to L_2[0,T] $ \begin{equation} O_Tx = Ce^{At} x, \end{equation} where $t \in [0, T]$, $ C$ and $A$ are matrices with compatible dimentions, and $x \in ...
2
votes
2answers
95 views

What's the best strategy to count the eggs in the jar?

It's Easter time, and in my workplace we have a "Count the eggs in the jar!" kind of game. What would be the best mathematical strategy to get as close as possible to the correct count? Update: ...
1
vote
2answers
86 views

Proving that the line integral $\int_{\gamma_{2}} e^{ix^2}\:\mathrm{d}x$ tends to zero

Let $f(z) = e^{iz^2}$ and $\gamma_2 = \{ z : z = Re^{i\theta}, 0 \leq \theta \leq \frac{\pi}{4} \} $. All the sources I have found online, says that the line integral $$ \left| \int_{\gamma_2} ...
0
votes
0answers
67 views

Can I estimate Variance of Gamma from Negative Binomial distributed data, given NB is Poisson-Gamma mixture

I believe the data I have follows Negative Binomial distribution (over-dispersed Poisson). We know Negative Binomial is a mixture of Poisson and Gamma. The variance of this Gamma distribution is ...
0
votes
1answer
215 views

Uniform distribution unbiased estimator

Let xi be iid observations in a sample from a uniform distribution over [0,θ]. Now I need to estimate θ based on N observations and I want the estimator to be unbiased. I thought about simple ...
1
vote
2answers
52 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 ...
1
vote
1answer
105 views

How to calculate the initial approximation in Newton - Raphson division algorithm.

I would like to know how to calculate first approximation in N-R division algorithm. I want to find the inverse of R. Here is the formula: $x_{i+1} = x_{i}(2-R*x_{i})$ I'm trying to implement it in ...
1
vote
1answer
66 views

Find an unbiased estimator

Let $X$ be an r.v defined by $P(X=0)=p$ and $P(X=1)=1-p$. Find an unbiased estimator for $2p$. My solution: $E(X)=1-p$ so $2-2E(X)$ is unbiased. Is this correct?
0
votes
1answer
26 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 ...
0
votes
0answers
55 views

Probabilities and Estimation of average and standard deviation

I've done a good bit of this number but I have trouble with part 2. I'll show you my work and the questions I can't figure in bold. A guy has a machine that scans his apples. The machine rules are : ...
0
votes
1answer
24 views

How to call $(E\hat{x} - x)^2$?

Let $\hat{x}$ be an estimation of $x$. Quantity $E(\hat{x} - x)^2$ is called Mean Squared Error. How one would call $(E\hat{x} -x )^2$?
1
vote
0answers
44 views

What are some good general estimates?

For example, the triangle inequality for complex numbers and summations is a good one. Also, the ML-Estimate (Estimation Lemma), Cauchy Estimates $|zw|=|z||w|$. As you can probably notice, I really ...
2
votes
1answer
58 views

Expectation of the MLE $e^{-\frac{1}{\overline{X}}}$

I am having a bit of a problem with examining the properties of a maximum likelihood estimator. I feel like I am missing something simple, but I have been unable to find someone doing an example quite ...
3
votes
1answer
37 views

Why is the MLE a special case of the minimum contrast estimator?

In my statistics lecture, we had two definitions, namely Let $X_1,\ldots.X_n$ be iid random variables, each with density $p_{\Theta_0}(x)$. Furthermore, let $\varrho$ be a real function such that ...
0
votes
0answers
76 views

MMSE estimate for scalar gaussian & uniform prior

I am trying to analyze the behavior of an MMSE estimator given Guassian measurement with scalar variability on an underlying uniform prior distribution. The measurement is generated according to the ...
0
votes
1answer
45 views

Bounds on coefficients of close polynomials

I've got two polynomials $p, \hat{p}:\mathbb{R}^2\rightarrow \mathbb{R}$ of degree $2\times2\ $ which are close together around $0$: $$|p(\mathbf{x})-\hat{p}(\mathbf{x})|<\varepsilon \quad \forall ...
1
vote
0answers
49 views

Parametric transition matrix in Markov Chains

I am trying to model a discrete-time MC with transition probabilities that depend on some function of parameters i.e $p_{ij} = f(X_0,X_1)$. Suppose we take a log-linear model where $p_{ij} = ...
1
vote
0answers
26 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 ...
0
votes
0answers
11 views

Maximum Likelihood parameters

I have a generative model with class conditional probability distribution $\Bbb P(x | C_k)$ and class priors $\Bbb P(C_k)$. I am having trouble with deriving the Likelihood function and hence the ...
2
votes
0answers
24 views

Estimation kernel

I just wonder if someone could just give me the proof for the following estimation: $$ \| \nabla (e^{t\Delta} f) \|^{2}_{L^{2}} \leq \|{f}\|_{L^{1}} \ \|e^{t\Delta} f\|_{L^{\infty}} $$ where ...