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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Difference among the same distribution , identical distribution and similar distribution.

$X\sim N(\mu_1,\sigma)$ and $Y\sim N(\mu_2,\sigma)$ are similar but not identical. $X\sim N(\mu,\sigma)$ and $Y\sim N(\mu,\sigma)$ are identical. But what is same distribution? Do same and ...
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11 views

Improvement of Minimum description length (MDL) estimate.

I earnestly request apology if this question is inappropriate for the forum. The question has two parts one technical and the other is not technical. I would appreciate any response. Let me consider ...
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0answers
7 views

Sampling via SRSWOR and biasedness of estimates

In a survey to estimate the proportion "p" of votes that a party will poll in an election, the voter list is divided into male and female lists. A sample of 100 from each list by simple random ...
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1answer
20 views

Suppose $40\%$ of the population possess a given characteristic … What is the probability $44\%$ or fewer possess the characteristic?

I have the following question: Suppose $40\%$ of the population possess a given characteristic. If a random sample of size $300$ is drawn from the population, then the probability that $44\%$ or ...
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1answer
13 views

Correlation Query

I can find $\mu_1, \mu_2, \operatorname{Var}(Y_1) \ \text{and } \operatorname{Var}(Y_2)$ but I am not sure how to get the co-variance of $Y_1$ and $Y_2$ in order to find the correlation of $Y_1$ ...
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1answer
15 views

Moment generating function of a uniform random variable.

My first attempt to this question was to find the first few moments about the mean and try to rearranging the those moments to obtain the general function as desired. However, when I tried to ...
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15 views

Finding variance with method of moments

In part (a), I know Var(Ni)=p(p-1) but how do I find the variance of the estimator with this result? And for part (b) of this question, I have to clue on how to tackle this question. I don't get ...
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25 views

$E(g(X)), E(g'(X)) <\infty $ implies $\lim_{x\rightarrow \infty} f(x)g(x)= 0$ ($f$ is the density of $X$)?

I am trying to figure out the Stein's identity which asserts that for r.v $X$ having pdf $$p_\theta(x)=\exp\{ \theta T(x)-A(\theta)\}h(x)$$ where $ T$ is differentiable and $g>0$ is ...
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1answer
19 views

Finding method of moments

Question: In order to solve find the method of moments estimator, I know I need to first find the expected value of Y. But after finding $E(Y)= \frac{\theta}{2}$, what should I do next?
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27 views

Statistical research [closed]

I'm preparing for an exam and I came across this question in a book. Below, I understand that we have to assumed a split plot in time analysis with 3 way factorial with 3 drug $\times$ 2 sex $\times$ ...
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2answers
26 views

How to show U1 and U2 are independent (Stats)

Prompt: Assume that $Y_1, Y_2, Y_3$ and $Y_4$ are independently and identically distributed $N(\mu,\sigma^2)$ random variables. Show that $Y_1 + Y_2 – Y_3 – Y_4$ and $Y_1 – Y_2 + Y_3 – Y_4$ are ...
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28 views

Prove that risk function is analytic?

I'm considering the statistical minimax estimation problem of the bounded normal mean: Specifically, the problem is to find the minimax estimator of $X \sim N(\theta,1)$ where $\theta \in ...
2
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1answer
15 views

How to estimate the variance of several populations when every population mean and variance is different?

I'm currently using the Pooled Variance method, but in my case I cannot assume that every population variance is the same. Is there a method for these cases ?
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0answers
24 views

$100(1-\alpha)$% confidence interval

$100(1-\alpha)$% confidence interval for population mean $\bar Y$ $$\bar y\pm Z_{\frac{\alpha}{2}}\sqrt{\mathbb v(\bar y)}$$ Why is this $Z$ value depending on $\alpha$ important for constructing ...
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1answer
22 views

proof that a density proportional to Gaussian is Gaussian

I try to develop bayesian estimation for one dimensional Gaussian with unknown $\mu$ and known $\sigma$. I got $$p(x\mid D) = \int p(x\mid\mu)p(\mu\mid D) \, d\mu =\int \frac{1}{\sigma ...
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0answers
7 views

Calculate probability distribution $p\left(\left.X_{1:T}\right|Z_{1:T},y_{1:T}\right)$ in linear- non-Gaussian state space model.

I have a linear, non-Gaussian state space model. Observation equation: $y_{t}=a+bX_{t}+cZ_{t}+\epsilon_{t}$ $\,\,\,\,$ $\epsilon_{t}\sim\mathcal{N}\left(0,\omega^{2}\right)$ Transition equations: ...
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1answer
30 views

Find probabilities given moment generating function.

The moment generating function of $X$ is $$m(t)= \exp( -6t+32t^2)$$ Find A. $\;\;P(-4<X<16)$ B. $\;\;P(-10<X<0)$ I have known that the first derivative for moment generating function ...
1
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1answer
28 views

Bound for Outlyingness

Given a sample of $n$ data, $x_1, \dots, x_n$. Define the sample mean $$\bar x := \frac{1}{n}(x_1+\cdots+x_n),$$ and sample variance $$s^2 := \frac{1}{n-1} \sum_{i=1}^n (x_i-\bar x)^2.$$ To measure ...
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1answer
25 views

Proof of Transformed Binomial Converging to $N(0,1)$

I'm trying to (efficiently) prove that a transformation of the sum of a sequence of $n$ Bernoulli$(p)$ trials ($X_1,\dots,X_n$) converges in distribution to $N(0,1)$. Specifically, if we denote $B = ...
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1answer
21 views

Maximum likelihood estimator for general multinomial

Let $(X_1,\ldots,X_r)\sim\text{multinomial}(n,(p_1,\ldots,p_r))$, where $p_r=1-p_1-\cdots-p_{r-1}$. The random likelihood is $Ap_1^{X_1}\ldots p_r^{X_r}$, for some non-zero $A$. The random ...
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0answers
9 views

direct connection between gradient descent and follow the (perturbed) leader algorithm or weighted majority? [migrated]

Is there a direct conversion between gradient descent ([1], Alg 1 ) and any of the following algorithms? 1) Weighted Majority: http://onlineprediction.net/?n=Main.WeightedMajorityAlgorithm 2) ...
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1answer
47 views

Assistance with Bayesian Random Effects and Mixed Effects Models

I am looking to build either a random effects or mixed effects model for a project I am working on, but have had trouble finding good sources and understanding the general structure of the model. ...
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1answer
27 views

Consistency of two measurements including means and standard deviations

This is a simplified version of a real life experiment where we have done two experiments attempt to measure the same quantity and we obtained the results $0.8 \pm 0.1$ and $1.2 \pm 0.2.$ (That's all ...
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1answer
20 views

Evenly selected order statistic?

Recently, I'm studying Nonparametric methods, especially Order Statistics. And I saw a sentence which says that Order statistic has to be evenly selected. And now I'm wondering what 'evenly ...
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0answers
18 views

Relative Error of Normal Approximation

I have this math statistics assignment that I worked out and came out with a couple of graphs. How do I interpret the graphs? Task: If you consider the average of 20 iid rvs, normal distribution is ...
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0answers
7 views

How to prove that noncentral chi-square distribution is stachastically increasing?

How to prove that noncentral chi-square distribution is stachastically increasing in n (degree of freedom) and sigma (noncentrality parameter)
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8 views

How to interpret Standard Residual Deviation?

have fitted regression lines for 200 data sets where each data set contains 40 data points.As a result of that now I have 200 Standard Residual Deviation values and need to interpret that results. How ...
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1answer
12 views

Computing likelihood for data corrupted by zero mean noise

The following statement is from a text on Statistical Estimation. I am trying to figure out how the likelihood function was arrived at. By definition of likelihood, ...
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0answers
39 views

Inference point estimation method of moments [closed]

Consider the probability density function for the uniform distribution on the interval $(a, b), b \gt a$: $$ f_X(x) = (b − a)^{−1} , a \lt x \lt b $$ (a) Sketch the probability density function ...
2
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1answer
21 views

Bayesian versus Classical (frequentist) Statistics

Very often in text-books the comparison of Bayesian vs. Classical Statistics are presented upfront in a very abstract way. For example, in the current book I'm studying there's the following ...
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10 views

Correct choice of analytical statistic method for time series problem

I want to make a statement about corruption (y) influenced by the ratification of the UN-convention-Against-Corruption "contract" (x). Luckily, most of them signed it within 3 years. So, I have ...
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1answer
21 views

What does likelihood density f(y|theta)=5 imply?

I just dont understand what dose constant likelihood density imply, e.g. f(y|theta)=5? In addition, when I use this likelihood density to derive posterior function, it cancels out so my posterior ...
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1answer
37 views

This textbook on time series analysis says something wacky

This is from a discussion of analyzing a time series with a sinusoid + noise model. The troublesome statement is: ...data values near the beginning and end of a record are most important for ...
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1answer
28 views

Solve Integral $ \int_{-\infty}^{+\infty} \log[F(y_k)]*F(y_k)^{\frac{1-\theta}{\theta}}*f(y_k) $ for MVUE proof

Let $ \theta > 0 $ a parameter, $ Y_1 ... Y_n $ is a set of iid observations with marginal distribution function $F_\theta(y) = [F(y)]^{1/\theta} ,-\infty < y < +\infty $ Show that $ t = ...
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0answers
14 views

Test whether the proportion of days which do not have “Good” air quality has changed.

I have the following data of AQ-value $$36,52,54,60,60,31,22,13,67,59,52,54,30,12,33,39,50,51,34,56,62,51,33,39,23,12,35,28,41,54,44,66,42,33,28,22,36,57,46,39etc.$$ I have total $364$ such values in ...
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Statistical sample with age ranges. How to extrapolate it using the real age distribution over population.

I have a data set consisting in the classification of the numbers of suicides by age range. I want to figure out if there is or not association between the number of suicides and the age range. But, ...
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0answers
29 views

Why does $\int f(x)(y-r(x))\;dP(y,x) = 0$?

My question is, why does: $$\int f(x)(y-r(x))\;dP(y,x) = 0,$$ where $r(x) = \int y \;dP(y|x)$ and $P$ is a probability distribution function. It was also given (in my book) that: $$\int ...
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0answers
68 views

$\mathsf kth$ moment of the standard deviation about the origin from a $\mathsf N(\mu,\sigma^2)$ population

Let T be the standard deviation of a random sample of size n from a $\mathsf N(\mu,\sigma^2)$ normal population. Find the $\mathsf kth$ moment of T about the origin, and state the condition for the ...
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1answer
15 views

One-to-one correspondence between mean value and parameters

I am currently taking a course in statistics, and in this course we are considering linear models $\mu = X\beta$ where $\mu \in L$ and $L = col(X)$ is a linear subspace of $\mathbb{R}^n$, $X$ is the ...
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0answers
17 views

Simulate from a distribution using Metropolis-Hastings and Rejection Sampling?

We have covered the basics behind rejection sampling as well as Metropolis-Hastings from class, but I am not sure how to use the two in conjunction to solve the following problem: Given $\pi(x) = ...
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0answers
12 views

Find the UMP test for Hypothesis testing

Let $X_1,...,X_n$ be a random sample from the $N(μ,σ^2)$. Assume $μ=0$ (a) Find the UMP test for $H_0 : σ^2=σ_0^2$ versus $H1 : σ^2 < σ_0^2$ at signicant level $\alpha$. (b) For the UMP test in ...
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0answers
5 views

Asymptotic confidence interval

Let x1, x2, ..., xn be a random sample with a density function given by $ f(x) = \frac{3}{\theta^3} x^2 I_{(0,\theta]}(x)$ where $I_{(0,\theta)}(x)$ is the indicator function and $\theta > 0$ ...
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How to arrive at this conditional expectation $E[Y|A]= E[Y] + \frac{Cov(A,Y)}{Var(A)}(A-E[A])$

My professor arrived at the formula (1) but I cannot figure out how he got there. Is (1) from an identity after some manipulation? The setup is: X,Y are correlated, jointly gaussian random ...
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15 views

Boxplot comparisons

How do I compare the following boxplots? Comparing locality; I would say that, the median of the second boxplot (to the right) is higher than the median of the first boxplot (to the left). ...
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15 views

optimising likelihood function in R

I want to try and optimise the function below in R to find best estimates of $\tau^2$ and $\mu$. $L(\mu, \tau^2) = -\frac{1}{2}\sum\limits_{i=1}^k \mbox{log}(2\pi(\sigma_i^2 + \tau^2)) - ...
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1answer
22 views

How can I uniformly draw points from an ellipsoid?

Specifically, given a positive definite matrix $A \in \mathbb{R}^{n \times n}$, how can I efficiently generate points $x \in \mathbb{R}^n$ that satisfy $x^TAx \leq 1$? I know how to do this when the ...
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22 views

Find a complete sufficient statistic

Here is my problem : Suppose theta is a nonrandom parameter satisfying theta > 1. Suppose further that, given theta, Y1 , Y2, ... , Yn are i.i.d. observations with each density f_\theta(y) = (\theta - ...
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39 views

Indepenent variables and functions

Random variables $x_1, x_2,...,x_n$ are independent. Then, how to prove whether these functions $$y_1=f_1(x) \\ y_2=f_2(x) \\ ... \\ y_n=f_n(x)$$ are independent or not . where, $x=(x_1,...x_n)$ ...
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6 views

Effect of coding technique used for qualitative variables, on the variance

In regression modeling, can using a certain coding technique for qualitative variables, be better than others. As in, say can using effect coding instead of dummy coding it reduce the total Error sum ...
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0answers
11 views

Sufficient statistics for $\mu_1 -\mu_2$

If $ X_1, ..., X_n$ is a random sample from $ X \sim N(\mu_1, \sigma^2)$ and $Y_1,..., Y_n$ is a random sample from $Y \sim N(\mu_2, \sigma^2),$ if the samples are independent and $ \sigma^2$ is ...