For questions about estimation and how and when to estimate correctly

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1answer
40 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
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1answer
145 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
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1answer
41 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 ...
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0answers
18 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
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1answer
54 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 ...
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3answers
986 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
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1answer
41 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 ...
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1answer
46 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 ...
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0answers
96 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} + ...
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1answer
43 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: ...
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1answer
34 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 ...
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1answer
79 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
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1answer
1k 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 ...
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2answers
84 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!
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0answers
44 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: ...
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0answers
24 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.
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0answers
46 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 ...
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0answers
83 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, ...
2
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2answers
146 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: ...
2
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2answers
120 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} ...
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1answer
696 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 ...
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2answers
53 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
191 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
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1answer
71 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?
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1answer
28 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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1answer
25 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 $(E\hat{x} -x )^2$?
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0answers
46 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
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1answer
67 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
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1answer
48 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 ...
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1answer
46 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 ...
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0answers
59 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} = ...
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0answers
27 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 ...
2
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0answers
25 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 ...
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0answers
46 views

Performance estimation of shellSort

I'm trying to make a performance estimation for shell-sort algorithm. And I fail in it. My formula: equals to where dz is outer while-loop, dy is middle for-loop, and dx is inner for-loop ...
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1answer
1k views

Unbiased estimators in an exponential distribution

We have $Y_{1}, Y_{2}, Y_{3}$ a random sample from an exponential distribution with the density function $ f(y) = \left\{ \begin{array}{ll} (1/\theta)\mathrm{e}^{-y/\theta} & y \gt 0 \\ 0 ...
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0answers
19 views

Determining the liklihood in Baye's rule for parameter estimation

I have used Bayesian statistics in classes but what I am trying to do now is different than anything I have done in class. Previously, I was given information and certain numbers adn I could ...
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0answers
60 views

Convergence in Probability of an estimator

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} + ...
0
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1answer
23 views

Generate function from discrete data (time-series)

How to transform discrete data into continous function ? I am working extensively with time series data and I would like to reduce amount of data in our frontend application. It would be cool to ...
0
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1answer
874 views

Determine the Asymptotic Distribution of the Method of Moments Estimator of $\theta$, $\tilde{\theta}$

I am having difficulty understanding what it means to find the asymptotic distribution of a statistic. I have the correct answer (as far as I know), but I am unconvinced that I understand the process ...
0
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1answer
32 views

Analytical solution to fitting two functions

I have two oscillatory functions $f(x)$ and $(k x)^2 g(x)$ where $f$ and $g$ are known and it is also known that the two functions are approximately similar. How can I analytically find the best ...
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0answers
88 views

MME of $\theta^2x^2e^{-{\theta}x}$

I need to find the Method Moment Estimator of parameter $\theta$ based on a random sample $X_1…X_n$ with the following pdf: $f(x;\theta)=\theta^2x^2e^{-{\theta}x}$; $0<x$, zero otherwise; ...
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0answers
87 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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4answers
454 views

Why does maximum likelihood estimation for uniform distribution give maximum of data?

I am looking at parameters estimation for the uniform distribution in the context of MLEs. Now, I know the likelihood function of the Uniform distribution $U(0,\theta)$ which is $1/\theta^n$ cannot ...
3
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0answers
73 views

Upper bound for area of polygons

is there a formula for an upper bound for the area of a polygon, knowing the length of its edges? In the ideal situation, the answer would be a function $f_n(l_1,l_2,\dots,l_n)$ of the edges lengths ...
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0answers
66 views

Calculating a metric to compare multiple posterior probability distributions

I am beginner in mathematics/statistics and apologise in advance for my faulty use of language. Especially because I assume this to be a simple problem. I am working on a problem in statistical ...
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1answer
86 views

Rolling a die 100 times and adding results

Simple problem. We role a die 100 times and we add the results. What is the probability of getting sum between 330 and 380 ? I got this: $P(330 \le X \le 380) = P\left( \frac{330 - n * ...
0
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1answer
90 views

Commulative degree distribution of nodes in a scale-free network

In a Barabasi-Albert model, which is a special kind of scale-free graphs, the degree distribution of each node is $$P(k) \sim k^{-3}$$ Given $\| V \|$ (number of nodes), how can I compute "number of ...
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1answer
50 views

Some mean value limiting result

Let $\phi$ be continuous in a neighborhood of $0\in\mathbf{R}^3$ (you may assume it to be uniformly continuous, if you like). Do we have that $$\lim_{\epsilon\rightarrow ...
2
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1answer
73 views

Help show that a second derivative is always negative

How do I show that the second derivative is always negative? I've computed the second derivative to be: $\displaystyle\frac{n}{2\sigma^4}-\frac{1}{\sigma^6}\sum\limits_{i=1}^n(x_i-\mu)^2$ Then I ...
2
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3answers
101 views

Why does finding the $x$ that maximizes $\ln(f(x))$ is the same as finding the $x$ that maximizes $f(x)$?

I'm reading about maximum likelihood here. In the last paragraph of the first page, it says: Why does the value of $p$ that maximizes $\log L(p;3)$ is the same $p$ that maximizes $L(p;3)$. The ...