The area of statistics that focuses on taking information from samples of a population, in order to derive information on the entire population.

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F-Testing ; constant returns to scale

$lnQ=1.37+0.632lnK+0.452lnL$ (0.257). (0.219) $cov(bk,bl)=0.055, R^2=0.98$ $H_0: bk+bl=1$ How can I proceed f-test even though I can't find df and RSS?
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1answer
23 views

Least Squares estimate [on hold]

at the elements of statistical learning page 47: $$ \hat{\sigma}^2=\frac{1}{N - p - 1}\sum_{j=1}^N(y_i-\hat{y}_i)^2 $$ there are sum of N variables. how can I inference this formula $$ (N - p - ...
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1answer
21 views

Two types of errors, type-$1$ error and type-$2$ error, can not be minimized simultaneously when the sample size $n$ is already fixed. How?

I read in some of the books that the two types of errors, type-$1$ error and type-$2$ error, can not be minimized simultaneously in Neyman Pearson Theory of testing of hypothesis when the sample size ...
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1answer
23 views

How to find which treatment is most effective in gene data given one standard method and 3 variations

Sorry I am a biologist and it appears am not quite confident enough for statistical analysis. I have datasets that represent different treatments on a biological system. It records how many genes have ...
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1answer
23 views

Almost sure convergence of $\hat{\sigma^2}$

Let $Y \sim N(X\beta,\sigma^2I)$ where $Rank(X_{n\times p})=p \leq n$. The least square estimate of $\sigma^2$ is $\hat{\sigma^2}=\frac{Y'(I-P)Y}{n-p}$ where $P=X(X'X)^{-1}X'$ is the projection matrix ...
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1answer
27 views

Expected value of a sample

I am unsure of how to solve this question. I know from examples questions that expected value of a sample is usually very close to the population mean. However, it says to compute the expected value ...
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23 views

Recursive Variance

What will be the distribution or features about the following $x$? $x=\mu+\epsilon$ where $\epsilon\sim N(0,x^{-1})$. It seems interesting in econometrics if we allow $x$ being a time series and ...
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18 views

How to measure the stability of datas

The background: I have a server handling $n$ kinds of requests, denoted by $k_1, ..., k_n$, at a certain time, many requests has been processed, the average time it takes to process $k_i$ is $t_i$, ...
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14 views

Approximation of Mahalanobis distance

If $A$ is a symmetric positive definite $n\times n$ matrix then the square Mahalanobis norm of a vector $v\in \mathbb{R}^n$ is given by $$\lVert v \rVert_A^2=v^t A^{-1} v.$$ Now I have a situation ...
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22 views

variance of a sum of independent random variables

I don't get why here https://en.wikipedia.org/wiki/Standard_error, T/n = 1/n²*(n*sig²) Is there a side knowledge to have here ?
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3answers
49 views

Confidence interval for sample

I have a sample of size $n=19593$ of count data ...
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35 views

Same Expected Value but different variances. Is $E[U(X)] \ge E[U(Y)]$?

Let $U: \mathbb R -> \mathbb R$ be a concave function, and let $X$ be a random variable with a normal distribution, expected value $\mu$, and standard deviation $\sigma$. Let $\lambda \gt 1$, and ...
2
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1answer
42 views

Prove that $E[U(X)] \ge E[U(Z)]$

Let U: $\mathbb R$ -> $\mathbb R$ be a concave function, let X be a random variable with a finite expected value, and let Y be a random variable that is independent of X and has an expected value 0. ...
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1answer
23 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 ...
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2answers
42 views

Confidence interval for Poisson distribution coefficient

This is an exam question, testing if water is bad - that is if a sample has more than 2000 E.coli in 100ml. We have taken $n$ samples denoted $X_i$, and model the samples as a Poisson distribution ...
2
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1answer
29 views

The asymptotic equivalence of LR, Wald and score tests

Suppose that $Y_1, \ldots, Y_{n}$ are iid from a Bernoulli distribution with parameter $p$ and consider $H_0 : p = p_0\,.$ The test statistics are $$ T_W = \frac{n ({\widehat p} - p_0)^2}{{\widehat ...
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0answers
11 views

What is the asymptotic value of the smoothed probability in a HMM model?

If I have a HMM model with a hidden markov chain $(S_t)_t$ with 3 states and if I assume that the distribution of the observation knowing in which state it is, is a normal. Do I know what is the value ...
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0answers
38 views

Sample median of Cauchy distribution is consistent. How?

When we use chebyshev's inequality to show whether an estimator is consistent or not, we require the mean square error of the estimator and I do not know sample median's probability distribution. So ...
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2answers
25 views

Estimating grader bias/variance and MLE test scores given multiple graders assigned to grade each test

Suppose we have $m$ graders and $n$ students, and we want to grade a test so that $k$ graders are assigned to grade to each test, and all graders grade the same number of tests. (I realize $m,n,k$ ...
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1answer
21 views

Sum of Squares From Regression Formula in Matrix Form

I am trying to show that the regression sum of squares, $$SS_{reg}=\sum(\hat{Yi} - \bar{Y})^2 = Y'(H - \frac 1 nJ)Y$$ where $H$ is the hat matrix and $J$ is a matrix of ones. I can do this using the ...
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0answers
13 views

How does one estimate the order of a Markov chain empirically (given the data)?

I have a string of symbols $x_1, x_2, ...., x_n$, ($n$ very large), belonging to a finite alphabet. I know that they are a result of a Markov process, but I want to find out the order of the process. ...
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1answer
18 views

How do I show that the sum of residuals of OLS are always zero using matrices

I am trying to show that $$\sum_{i=1}^ne_i = 0$$ using matrices (or vectors). I have two hints, so to speak: $$ HX = X$$ where $H$ is the hat matrix, and that $$\sum_{i=1}^ne_i = e'1$$ My previous ...
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2answers
27 views

A problem in method of moment - in my Quiz

Let $X_1,...,X_n$ be an i.i.d. sample from the uniform distribution on ($-\theta$, $\theta$). (a) Find a method of moments estimator of $\theta$. By integration of second moment, ...
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15 views

Sufficient Statistics Multi Conditional values

I am trying to find $\mathbb{E}\{X_1| X_1+X_2, X_1+X_3\}$ where all are non negative independent r.v.'s (e.g. Poisson). I am not clear about the concept of sufficient statistics, is't it enough in ...
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1answer
18 views

In OLS is the vector of residuals always 0? [duplicate]

I am trying to show that $$\sum_{i=1}^ne_i = 0$$ I have two hints, so to speak: $$ HX = X$$ where $H$ is the hat matrix, and that $$\sum_{i=1}^ne_i = e'1$$ My solution is as follows: $$e'1 = ...
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4answers
41 views

Line Of Regression given x? [closed]

You have found the regression line for a set of data points to be: ŷ = 30.23x + 173.52. Use the line to predict the value of y when x = 48.
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13 views

Estimate Sample [duplicate]

You wish to estimate, with 99% confidence, the proportion of drivers who want the speed limit raised to 130 kph. Your estimate must be accurate to within 5%. How many drivers must you survey, if your ...
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1answer
14 views

Point Estimates Using C.I.

0.680 < p < 0.800 What is the point estimate for p, and the margin of error from which the C.I. was formed? I am confused as to what "p̂" and "E" are equal to. Normally, I would use the ...
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23 views

$p$-value in hypothesis testing

Find $p$-value, make appropriate conclusion about $H_0$. Left tailed test ($H_a$ is $<$), $z= -1.28$, $\alpha= 0.05$ Two-tailed test ($H_a$ is $\neq$), $z= 1.28$, $\alpha=.01$ ...
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1answer
20 views

Statistical Estimation

There is an example problem in my book that doesn't explain how they got to this answer: sample: $217$ sample mean: $132.5$ standard deviation: $10$ "The $95$ part of the $68-95-99.7$ rule for ...
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1answer
39 views

Calculating MSE of the estimate $T=\max\{X_1,X_2,\ldots,X_n\}$ of $\theta$.

The variables $X_1,X_2,\ldots X_n$ are i.i.d uniform distributed on $[0,\theta]$. $$T=\max\{X_1,\ldots,X_n\}$$ is the estimate of $\theta$. I need to calculate MSE. I know that ...
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21 views

How does a median have a value that is a decimal which isn't exactly half of an integer if the data should consist of only integer values?

I real an article which said the average man accumulated 6.1 sexual partners while the average woman accumulates 3.6. If the statistic talked about the average, surely the numbers would be equal-so it ...
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1answer
9 views

Prediction intervals with OLS and indicator variables

Suppose I have a model like so, call it the first model: $$E[y] = \beta_0+\beta_1x+\beta_2x_m+\beta_3(x\cdot x_m) $$ where $x_m$ is an indicator variable. I fit it using ordinary least squares. ...
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26 views

Estimating Prevalence of Disease, Defects, or Spam With Screening Tests

Background. A screening test is a relatively quick and easy or inexpensive preliminary test that gives preliminary warning of undesirable condition $D$, such as disease in a patient, defect in a ...
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2answers
66 views

Reasoning for confidence interval

Suppose $$X_1,\dots,X_{20} \sim f_X(x;\beta)$$ where $$f_X(x;\beta) = \frac{1}{\beta} e^{-\frac{x}{\beta}},\quad x>0;\beta>0$$ It can shown that ("details omitted") $$P(0.52 \bar{X} \leq \beta ...
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82 views

Homogeneous polynomials on sphere - need an example that is used in machine learning.

My question is about an example of use of homogeneous polynomials on sphere as a hypothesis space in learning problem. In order to ask a question I need to make a quick introduction: I'm reading an ...
3
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0answers
27 views

Pareto distribution confidence interval

$X$ is distributed by Pareto with $$f_X (x) = \frac{\alpha k^{\alpha}}{x^ {\alpha +1}},\alpha,k>0,x>k.$$ Derive a 95% confidence interval for $k $. My friend said I gotta do this $$Pr ...
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2answers
36 views

General method to find exact confidence interval?

Say we have a beta random variable with pdf, $f_X (x) = e^{-x/\theta}/\theta$ for positive $x $. Find the exact confidence interval of $\theta$ with 95% confidence. A solution to this says ...
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55 views

How to use Laplace method to get the asymptotic expansion of multiple integral

I meet difficulty when I try to get the asymptotic behaviour of multiple integral as x tends to plus infinity. And $-1<$p$<1$ $$\int_x^{+\infty}\int_x^{+\infty}e^{-{\frac{1}{2\sigma^2(1-p^2)}\ \ ...
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1answer
28 views

Confidence interval and symmetric distribution

Let $X_1,...,X_n$ random sample of $X$~$U[\theta-\frac{1}{2};\theta+\frac{1}{2}]$.Consider $[X_{(1)};X_{(n)}]$ a confidence interval for $\theta$. Find their confidence level and show that ...
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65 views

Probability about confidence interval

Let $X_1,...X_n$ be iid $N(\theta,1)$. A 95% confidence interval for $\theta$ is $\overline{X}\pm\frac{1.96}{\sqrt{n}}$.Let p denote the probability that an additional independent ...
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7 views

Pitman Asymptotic Relative Efficiency

Let $T_1$ and $T_2$ are two test for testing the same $H_0$ versus $H_1$. Let $t_1$ and $t_2$ are test statistic correspond to $T_1$ and $T_2$, respectively. If $t_1$ and $t_2$ have noncentral chi ...
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2answers
39 views

Reference request, statistical inference

Good morning, I'm looking for a good reference for study on statistical inference, the main topics that will study are Tests of Hypotheses Interval estimation I recommend taking a look at Mood ...
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19 views

Method of moments estimation for not-idd rv:s

I'm trying to solve a problem in which I have a sample from to random variables X and Y with $X\sim Exp(a)$ and $Y \sim Exp(\frac a2)$ (or actually it says $X \in Exp(a)$ and $Y \in exp(\frac a2)$ but ...
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57 views

How to write R program to solve the confidence interval?

The problem: let $X_1,\ldots,X_n$ be random variable from $\mathrm{Poisson}(\theta)$. Under $H_0: \theta=\theta_0$, we want to find the $(1-\alpha)100\%$ confidence interval for $\theta$ by using the ...
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14 views

Log-likelihood approaches +Inf for a Gaussian process

I am trying to do a standard likelihood maximization for the hyperparameters of a Gaussian Process (details in Chapter 5, Rasmussen & Williams: ...
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1answer
39 views

Sampling distribution of sample trimmed (truncated) mean

It is elementary probability theory that the sample mean of an i.i.d. sample follows normal distribution, if the background distribution is normal. But what about the trimmed mean? Is there any result ...
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12 views

testing significant difference on species richness

I measured the species richness (number species) in three different sites. Now, I want to test whether there is significant difference between each site in terms of species richness. The species ...
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21 views

Relation between the Coefficient of Multiple Correlation and Coefficient of Simple Correlation

Consider the regression model $Y=\beta_1 X_1+\beta_2 X_2+\epsilon$, with a sample of size $n$, $Y_i=\beta_1 X_{i1}+\beta_2 X_{i2}+\epsilon_i$, $\epsilon_i \in N(0,\sigma^2)$. Suppossing ...
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1answer
10 views

Proof that the least square estimators are normally distributed

In my book I have the following proof showing that one of the least square estimators is normally distributed: $\hat\beta_i$ = $\frac {S_{xy}}{S_{xx}}$ = $\frac {1}{S_{xx}}\sum_1^n({x_i}- ...