For questions about estimation and how and when to estimate correctly

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1answer
77 views

When to we accept a hypothesis when using Wald test statistic? [closed]

Hello I had to test two hypothesis, one hypothesis gave a wald test statistic with value 0.00015 and the other a value of approximately 40. Is it true when I then say that we accept the hypothesis ...
0
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0answers
4 views

How to computing Wald test in R based on what I have so far?

I have to simulate an AR(1) process with $\rho = 0.5$ and then estimate $\rho$ based on the first 100 $X_T$ values I did this the following way: ...
1
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2answers
23 views

When does it make sense to build a confidence interval for the mean with known standard deviation?

While estimating with confidence interval the mean value for a population, there are two options: If the standard deviation is known, and If the standard deviation is unknown. But in the first ...
0
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1answer
121 views

Upper bound for the error on the Fourier series for $e^{x}$

I have been given the following problem: Find the Fourier series for $e^{x}$ over the interval $-\pi \le t \le \pi$. Hence find the upper bound of its error. To spare me typing a huge expression and ...
1
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1answer
60 views

An asymptotic estimate for density of eigenvalues

this is the screenshot of the useful part of the cited book Let $\{\lambda_n\}$ be constants such that ${\lambda_n}=n^2\pi^2+\int_{0}^{1} q(t)\,dt +c_n \qquad \text{for} \quad n\rightarrow \infty$ ...
1
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1answer
32 views

problem using MLE on gamma distributed variable

I am making some kind of systematic error(s), while working with maximum likelihood estimations. Could someone please point these out to me? In my last assignment, I tried to find the MLE of $\beta$ ...
-1
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1answer
32 views

Finding number pattern

I have a program which I originally thought was linear, meaning that if I had twice the data, the program would take twice as long to process. I was wrong, and it seems to be somewhat exponential. ...
0
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1answer
16 views

on a step of a proof of the Levinson density theorem

Let $n(r): \mathbb{R}\rightarrow\mathbb{N}$ be a monotone (increasing) function such that $\int_{1}^{r} \frac{n(u)}{u}du \leq \frac{1}{2}\log (r)+ A$ where $A$ is a certain constant. I should deduce ...
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0answers
18 views

Finding the closest vector to an observation

I have a collection of vectors (a codebook in hand) which are presented within a matrix $A$ $$ A = [ a (\theta_1), \, a (\theta_2), \, a (\theta_3), \, a (\theta_4), \,\cdots]$$ We have obtained ...
1
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1answer
42 views

Estimate of a (integral) function

I should show that function $H(w)=\int_{-\pi}^{\pi}f(x) e^{iwx}dx$, where $f(x)\in L^2(-\pi,\pi)$, is such that $H(re^{i\theta})=O(e^{\pi r |sin(\theta)|})$. Any suggestion?
1
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1answer
18 views

Required number of simulation runs

I have the following problem: One wants to estimate the expectation of a random variable X. A set of 16 data values (i.e. simulation outputs) is given, and one should determine roughly how many ...
1
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2answers
21 views

Significant figures addition/subtraction rounding?

I thought you round to the same place as the number with the addend with the least precision. For example, if you had $25.63+ 42.3$ the answer would be rounded to the tens place ($67.9$). However, my ...
1
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1answer
61 views

Estimate the Green function for the Laplace equation in 2D

The Green function for the Dirichlet problem for the Laplace equation in the unit disk in $\mathbb R^2$ has the following form: $$ G(x,y) = \frac{1}{2\pi}\ln \frac{|x-y|}{|x| \bigl|y - \frac{x}{|x|^...
3
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2answers
41 views

Is it possible to determine which yields a better approximation for $\pi$?

If I use the trapezoidal rule, using two equal partitions, to estimate $$\int_{0.5}^{1} \sqrt{1-x^2}dx$$ I can obtain an approximation of $\pi$, as the integral's exact value can be found without much ...
0
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0answers
20 views

Calculating a 95% confidence interval for µ

A test was conducted to determine the length of time required for joggers to run a particular path. The joggers were instructed to run the course at the maximum speed at which they could without ...
2
votes
1answer
88 views

Expectation of largest and smallest order statistic from uniform distribution

Given is a random sample of size n from a uniform distribution with parameters $-\theta$ and $\theta$, $\theta>0$. I'm asked to find a constant $c$ such that $c(X_{n:n}-X_{1:n})$ is an unbiased ...
0
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1answer
32 views

Deriving a bound on an integral

I'm trying to follow a result in a paper and I can't get it to work out. The result requires deriving the bound \begin{equation} \int_{\mathbb{R}^N\times\mathbb{R}_+} \left|\frac{\mathrm{d}\psi}{\...
2
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3answers
112 views

An interview question: $2.1^{3.1}$ vs $3.1^{2.1}$,$ 2.1^{4.1}$ vs $4.1^{2.1}$, which is larger?

While Mathematica told me that $2.1^{3.1} - 3.1^{2.1} = -0.786932$ and $2.1^{4.1} - 4.1^{2.1} = 1.58855$, I wonder how to compare them quickly, by hand. I see $2^3 < 3^2$, so perhaps we have $2.1^{...
0
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0answers
21 views

When regularity conditions for the CRLB to hold are not met, can a lower bound be computed numerically with the MLE?

I have a probability density $p(\hat{d}|d)$ such that $\int_d^\infty p(\hat{d}|d) = 1$ where $d$ is the parameter to be estimated based on observations $\hat{d}$ so the regularity conditions for a ...
3
votes
2answers
202 views

Finding an error estimation for the De Moivre–Laplace theorem

Context for my question: For one part of my thesis I try to find an upper bound for the error in the normal approximation of the binomial distribution following the standard proof of the De Moivre–...
1
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1answer
38 views

$\int^{\eta}_{\epsilon} \frac{cos(\lambda x)-cos(\mu x)}{x}dx$

Let $\lambda,\mu$ be positive constants. Let $I_{\epsilon,\eta}=\int^{\eta}_{\epsilon} \frac{cos(\lambda x)-cos(\mu x)}{x}dx\:\:\epsilon,\eta >0$. Show that, by integration by part, $|I_{\epsilon,\...
1
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0answers
60 views

Estimate on the exponential integral of a complex argument (a reference needed)

Consider the exponential integral of the complex argument defined by $$ Ei( z ) = \gamma + \ln(-z) +\sum\limits_{ n = 1 }^{ \infty } \frac{ z^n }{n n!}, $$ where $ z \in \mathbb{C} \backslash ( \...
2
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1answer
66 views

Estimating a sum related to a short Euler product

The Question Is $$\sum_{\substack{n>y\\ p\mid n\Rightarrow p\leq y}}\frac{\Lambda(n)}{n^s\log n}=O(1/\log T)$$ where $y=(\log T)^{100}$ and $T$ is large? Background Assume that $$\log\zeta(\...
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0answers
16 views

Least Square solution for approximating a sequence

Suppose I have a sequence of length N $a_1,...,a_N$ I want to approximate this sequence by $k^1,...,k^N$ where $k$ is my variable. What is the least square solution of this? is there a closed ...
1
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1answer
35 views

Improving estimatied parameters of a known distribution

Assume there is a Set of data which follows a known distribution (e.g. normal distribution). $$S = \left\{ a_0,a_1 ... a_n \right\}$$ When taking a subset from S $$S_k = \left\{ a_0,a_1 ... a_k \...
0
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0answers
21 views

Histogram estimator, why dividing by smoothing parameter?

In histogram estimator: $$\widehat{f}(x)=\frac{1}{nh} \sum_i^n I\left(-\frac{1}{2} \le \frac{x_i-x}{h} \le \frac{1}{2}\right)$$ why do we have to divide by smoothing parameter $h$ in first term $\...
0
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0answers
22 views

Is this estimation of an double integral right?

I'm in the setting, where $\Omega$ is a bounded domain, $t_0 > 0$ is fix and I have $g, \ h \in L^\infty((0,t_0);L^1(\mathbb{R}^n \setminus \Omega))$. So this should say, that $g$ and $h$ are $L^\...
0
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0answers
33 views

Big Oh notation problem

Retaining the three terms in the series, estimate the remaining series using "Big Oh" notation with the best integer value possible, as $x\to 0$. The series is $$\ln (\tan (x)) =\ln(x)+ \frac{x^2}{3}...
0
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0answers
25 views

Variable value estimation for given product/fracture values

I have a data set (time series) with given values for certain fractions xy = x/y (where x,y are not constant over time) Thus, there are following fractions: AB = A/B CB = C/B AD = A/D CD = C/D AE = ...
2
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0answers
40 views

Method of moment estimator. Deconvolution

The distributions $Y, X, Z$ and $W$ are related as follows: $$Y_1 = X + Z$$ $$Y_2 = X + W,$$ that is $X$ (random variable) is a common factor to the random variables $Y_1$ and $Y_2$, which ...
0
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0answers
11 views

How to interpret the Quantified properties of estimator?

https://en.wikipedia.org/wiki/Estimator The link provides a very good explanation of the estimator. I am beginner to statistics and inference , so i have some confusion about the quantified properties ...
1
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1answer
30 views

Sufficient statistics problem

$X_1, X_2, \ldots, X_n$ are iid $N(0,\theta), 0 < \theta < \infty$ Show $$\sum_{i=1}^{n} X_i^2$$ is a sufficient statistic for $\theta$. My attempt at this is $S = (X_1^2 + X_2^2+\cdots+X_n^...
4
votes
1answer
47 views

Cramer-Rao lower bound for any unbiased estimator

The first part of a question I am trying to solve asked to find the maximum likelihood estimator for $\theta$ for a pdf $f_X(x)=\frac{2x}{\theta^2}$, $0 < x \le \theta$ , $0$ otherwise. ($X_1, X_2,...
0
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0answers
50 views

Minimum mean square estimator for closing value of the NYSE

I am attempting to develop an estimator for the closing value of the NYSE $x(n)$ based on previous $N$ closing values, $x(n-1), x(n-2), ... x(n-N)$. I want to find the Minimum Mean-Square estimator ...
1
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1answer
153 views

Cost Function of Neural Network (Forward Propagation)

This question is related to Andrew Ng's machine learning course on Coursera. Basically, when I calculate the cost function of a neural network, I use the following formula that was described by Ng: $$ ...
1
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1answer
129 views

Determining the MVUE of $ f(x;\theta) = \theta^x (1-\theta)$.

The Statement of the Problem: Let $X_1, X_2, ... , X_n$ be a random sample from $$ f(x;\theta) = \theta^x (1-\theta) \quad x = 0,1,2,... $$ (a) Find the ML estimator of $\theta$. (b) Show that $T =...
1
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1answer
51 views

How many bathtubs in a min?

During the summer about $750,000$ gallons of water fall over the edge of Niagara Falls every second. If an Olympic sized swimming pool holds about $660,000$ gallons of water, how many Olympic sized ...
2
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0answers
35 views

Estimating the Average and Standard Deviation of a Population based on a Sample with Missing Data with Known Ranks

I need a way to shows me how the parameters of PDF, log-normal in this case, can be estimated based on a set with missing data points at the tail end of a sample. For example, Consider we had 20 ...
1
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2answers
28 views

How to estimate magnitude of expontent?

When given an exponent, such as 6^12, is there a simple way to approximate how large(magnitude) the result is, without performing the calculation? Is this method accurate for large exponents?
0
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0answers
218 views

Confidence Interval for Incidence Rates

I have a huge sample of patients followed up for a certain Event. I would like to calculate the following crude incident rates: #{Events}/(1000*PersonYear). My sample is big enough to assume that this ...
2
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2answers
98 views

Is $\lim S_{n,m}=\sum_{k=1}^n({-1})^k{n\choose k}k^{-m}<\infty $ for $ n \to \infty$ and $m$ large?

Let $m$ be a fixed positive integer ($m>1)$ and let $$S_{n,m}=\sum_{k=1}^n({-1})^k{n\choose k}k^{-m}$$ be a partial sum of real series. My question here is : Is $\lim S_{n,m} <\infty $ as $ n \...
0
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1answer
150 views

How to calculate entropy from a set of samples?

entropy (information content) is defined as: $$ H(X) = \sum_{i} {\mathrm{P}(x_i)\,\mathrm{I}(x_i)} = -\sum_{i} {\mathrm{P}(x_i) \log_b \mathrm{P}(x_i)} $$ This allows to calculate the entropy of a ...
0
votes
1answer
130 views

In terms of $a, b,$ and $\theta$, what is the biased $b(\hat \theta)$?

The Statement of the Problem: Let $\{P_{\theta}: \theta \in \Theta \}$ be a statistical model. Suppose that $\hat \theta$ is an estimator for a parameter $\theta$ and $E_{\theta}(\hat \theta) = a\...
1
vote
2answers
42 views

Expected value and the standard simple regression model

Given the standard simple regression model: $y_i = β_0 + β_1 x_i + u_i$ What is the expected value of the estimator $\hat\beta_1$in terms of $x_i, \beta_0$ and $\beta_1$ when $\hat\beta_1=\sum x_i ...
1
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0answers
26 views

Proof of differentialbility in mean square calculus?

let $x_t$ be a mean squared Riemann integrable over $[a, t]$ for every $t\in[a,b]$. Then $y_t=\int\limits_a^t x_\tau d\tau\ $ is mean squared continuous on $[a, b]$. Furthermore, if $x_t $ is mean ...
3
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0answers
45 views

Estimating the sum of a series within arbitrary certainty.

Find the sum of the series $\displaystyle \sum_{n=1}^{\infty} \frac{1}{n^5} = a_n$ within three decimal places. The sum is estimated by $\displaystyle a_n \approx \sum_{k=1}^{n}\frac{1}{k^5}+R(n)$ ...
6
votes
1answer
94 views

Finding a better upper bound for an integral of a product of $n$ terms

So I'm trying to find and upper bound for the integral $$ \int\limits_{a}^b \! (x-x_1)^2 \cdots (x-x_n)^2\, \mathrm{d}x, $$ where $x_i \in [a,b], \enspace \forall i=1,\dots ,n.$ I've tried ...
0
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0answers
24 views

Establishing consistency

I need to establish the (weak) consistency of an estimator of the mean, $T=a+b\bar{X}$. I tried to apply Chebyshev's inequality, but I couldn't do much because the parameter that subtract in the ...
3
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2answers
68 views

Confidence interval for sample

I have a sample of size $n=19593$ of count data ...
1
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1answer
34 views

construct confidence interval from proportions

Suppose you have a population of count data, i.e., $1,2,3, \dots, k$, you have a sample of the population of size $n$, and you have a confidence interval for the proportion of $1$'s , $2$'s,\dots$n$'s ...