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Questions tagged [mathematical-statistics]

Mathematical theory of statistics, concerned with formal definitions and general results.

8
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2answers
141 views

Conditional expectation of uniform random variable given order statistics

Assume X = $(X_1, ..., X_n)$ ~ $U(\theta, 2\theta)$, where $\theta \in \Bbb{R}^+$. How does one calculate the conditional expectation of $E[X_1|X_{(1)},X_{(n)}]$, where $X_{(1)}$ and $X_{(n)}$ are ...
0
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0answers
8 views

'Shrink' Step occuring often in Nelder Mead optimization?

I have an implementation of Nelder Mead, which is giving me good results. I had a bug, though, and while I managed to fix that bug, I noticed during my debugging that the 'shrink' step is occurring ...
0
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0answers
11 views

Rescaling model coefficients

I have the following model $y \sim b_0 + Z \theta_0 + X \beta + \sum((X_j \circ Z) \theta_j)$ Where... $Z$ is (n, k) $X$ is (n, p) $\beta$ is (p,) $\theta$ is (p, k) $\theta_0$ is (k,) $\circ$ is ...
0
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0answers
17 views

Show that the intersection of two sets involving symmetric PMF is empty

Consider the stepwise cumulative distribution function $$ \Delta(x; \lambda, \mu)=\sum_{j=1}^J \lambda_j 1\{x\geq \mu_j\} \hspace{1cm} \forall x \in \mathbb{R} $$ where $J<\infty$ $\lambda\equiv (\...
0
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0answers
34 views

Asymptotic distribution using the Delta method

Let $X_1, \ldots, X_n$ be i.i.d. normal random variables, where $X_i \sim N(\theta, \theta^2)$ with an unknown $\theta > 0$. We could for example estimate $\theta$ using the average $$\hat\theta_n ...
0
votes
0answers
7 views

Likelihood deviation on diseased test

I feel confused with the following problem: the population $n=237$ diseased people are required to perform a 6-successive-day diagnostic test on cancer. And the random variables are given as follows: ...
0
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0answers
10 views

Derive taylor series expansion of df

I was trying to understand ito's lemma. When I came across the taylor series expansion of df(x). df(x) = f'(x) dx + (1/2!) f''(x) (dx)^2 + ... I searched everywhere for the derivation of this but ...
1
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1answer
20 views

Generate random sample of X1 and X2

If X2 is dependent on X1, how to generate the random sample of (X1,X2)? One scenario is that we know the prior distribution of X1 and functional relationship between X1 and X2, how to generate the ...
0
votes
1answer
42 views

How to find confidence interval for variance?

Well, I'm taking a statistics class online, and the problem goes like this: Drug concentrations​ (measured as a​ percentage) for 25 randomly selected tablets are shown in the accompanying table. For ...
2
votes
0answers
35 views

Sum of Random Variables — An Example with R [closed]

So I was trying to simulate these formulas from my lecture with R. $$E[\bar{X_n}] = \mu_x\text{, where $X_i \sim i.i.d$}\\Var(\bar{X_n}) = \frac{1}{n}*\sigma^2_x$$ My code in R: ...
0
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0answers
14 views

Variance of a Sum of Random Variable [duplicate]

Unsure about my teachers solution as she was not very convincing when she presented the solution and was quite confused herself. $$Y = a+bX+cZ \\ X \sim \mathcal{N}(4,\,\sigma^{2}_x)\\ Z \sim \...
1
vote
2answers
55 views

Estimator for $\frac{1}{\lambda}$ using $\min_i X_i$ when $X_i$ are i.i.d $\mathsf{Exp}(\lambda)$

Let $X_1,\ldots,X_n$ be i.i.d. $\mathsf{Exp}(\lambda)$ random variables, where $\lambda$ is unknown. Consider $f_{\min}(x) = \min_{i}(X_i)=$ $ n \lambda $ Exp$(n\lambda x)$. I am told that $\hat \...
1
vote
0answers
37 views

UMP test for $H_0:p=0.5$ vs $H_1:p\neq0.5$?

Let $X_1,\dots, X_n$ Bernoulli trials. I know that the UMP tests for $$H_0:p=0.5 \quad\text{vs}\quad H_1:p>0.5$$ and $$H_0:p=0.5 \quad\text{vs}\quad H_1:p<0.5$$ can be obtained with the Neyman ...
0
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0answers
16 views

How to calculate median and mode from continuous data? [duplicate]

If data is continuous (not discrete) how to calculate median and mode?
4
votes
3answers
208 views

What does conditioning on a random variable mean?

What does onditioning on a random variable mean? For example: in p(X|Y), X and Y are the random variables, so does the conditioning on Y mean Y is fixed (or non-random)?
0
votes
1answer
29 views

Conditional expectation of two independent RV

The expectation of the product of two independent random variables $X$ and $Y$ is the product of the expectations: \begin{align} E(XY) = E(X)E(Y) \end{align} Let's add another random variable $Z$ in ...
1
vote
2answers
31 views

Is this formula for the Law of Iterated Expectations correct?

I saw two versions of the law of iterated expectations, this one: \begin{align} E(E(Y\vert X)) = E(Y) \end{align} and this one: \begin{align} E(E(Y\vert X_1, X_2)\vert X_1) = E(Y \vert X_1) \end{align}...
2
votes
0answers
46 views

How to calculate $\int_{S}^{T} \Phi \Big(\frac{l_0-f(t)}{\sigma}\Big)g(t)dt$? [migrated]

How can I calculate the integral- $\int_{S}^{T} \Phi \Big(\frac{l_0-f(t)}{\sigma}\Big)g(t)dt$ where $f(t)= \mu_0+\mu_1 e^{-\gamma (7+logt)^\delta}$ and $g(t)= \frac{1}{(t+h)^k}$ . Here $\Phi \Big(\...
0
votes
1answer
36 views

How to decide which Probability distribution to use on a specific problem?

I need some guidance. Which probability distribution I can use to understand the number of orders one person can handle in a company? It's like each person has orders which takes minimum of 50 days to ...
1
vote
0answers
35 views

Could you derive the mean and variance in this theorem? [duplicate]

How to derive the mean and variance by using the given conditional pdf? please help me ~
13
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5answers
2k views

Why Normality assumption in linear regression

My question is very simple: why we choose normal as the distribution that error term follows in the assumption of linear regression? Why we don't choose others like uniform, t or whatever?
0
votes
2answers
32 views

The difference between our sample estimate and the true population value is known as: [closed]

The difference between our sample estimate and the true population value is known as: A. Mean Value B. Standard Deviation C. Standard Error D. Variance With explanation please
0
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0answers
19 views

How are R2 and adjusted R2 mathematically related to the idea of explained variance?

I am trying to understand in what sense, $R^2$ and $R_{adj}^2$ represent the "explained variance." I can't find any similar question that explores the connection in mathematical detail. My current ...
0
votes
0answers
17 views

How to statistically split a set of numbers into two groups for Kaplan-Meier plot?

I want to perform a survival analysis for the cohort of breast cancer patients. For each patient, I know whether he was right-censored or not and what was his survival time (or the end-of-study time ...
1
vote
0answers
23 views

Using R to maximize a two parameter Weibull model via multivariate extension of Newton-Raphson method

I am just getting back into using R for the first time in a while, and wrote some code to perform the aforementioned task in the title. I was wondering if anyone could take a look at it and see if ...
4
votes
1answer
77 views

How is it possible for both the likelihood and log-likelihood to be asymptotically normal?

I was trying to understand asymptotic normality of the posterior better, and came across a confusing point. So let's say we have a likelihood, $L(\theta | X) = \Pi_{i=1}^n p(X_i | \theta)$, so the log-...
0
votes
0answers
11 views

Expected number of visits to the state in a Markov chain

We know that in a Markov chain the taboo probability, $P^n_{i,i+1}(H)=Pr\{ X_n=i+1,X_k$ doesn't belong to H for $0 < k < n| X_0=i \}$, where H is an arbitrary set of states in a Markov chain $...
0
votes
0answers
43 views

Sufficient Statistic and MLE

Suppose $X_1, \dots, X_n \sim B(1,p)$. Show that a sufficient statistic for $\theta = (1-p)^2$ is $T(x) = \sum X_i$ and that the MLE for $\theta$ is $(1-\frac{1}{n}T)^2$. I am having a lot of ...
4
votes
1answer
57 views

distribution for scaled Maximum of n independent Weibulls for $n \to \infty$

Assume that $X_1, X_2,...\sim Weibull(\lambda, k) \quad iid.$, i.e. $F(X_1\leq x) = 1-e^{-(\lambda x)^k}$ define $M_n:= \max\{X_1, ..., X_n\}$ and $\tilde{M}_n:=\frac{M_n-b_n}{a_n}$ according to ...
0
votes
0answers
6 views

Finding Fisher information [duplicate]

Let $X$ distribution belongs for the family $\mathcal{P}\{P_{\theta}, \theta \in \Theta \}$. We need to find Fisher information $I(\theta)$ according $n$ simple sample, when $P_{\theta}$ is $N(\mu,\...
0
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0answers
5 views

Find the same components between two distributions

GMM assumes that a distribution consists of multiple Gaussian Distributions. Is it possible to find the same components between the two distribution? For example, there are two distribution D1 and D2. ...
0
votes
0answers
13 views

Monte Carlo simulation from autocorrelated empirical distribution

So I'm trying to perform a Monte Carlo simulation an empirical distribution of log-return time series data. However, Ljung-Box test indicates that there is significant autocorrelation at multiple lags....
1
vote
1answer
22 views

Why does the cumulative distribution function for discrete random variable right continuous?

If the random variable is discrete, then the cumulative value should also be discrete because the variable can only take on discrete values, right?
0
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0answers
26 views

Introductory books on bayesian statistics with focus on normal distribution

I am searching for introductory books on bayesian statistics. Which Focus on normal distribution (Most of the books I came across through this answer focus on binomial distribution) Practical ...
1
vote
1answer
38 views

Distribution of a one realization of a stochastic process [closed]

Suppose $X$ is a stochastic process such that $X(t) \sim N\left(\mu(t), \sigma^2(t)\right)$ for all $t$ and $\mu$ and $\sigma$ are some smooth functions and we are given one realization of this ...
1
vote
1answer
33 views

Question about prediction: if a variable can predict another variable [closed]

I have a question about the prediction. This question stopped me for a whole afternoon, still do not have an idea on how to solve it. Would you please give me a clue. The question is: ...
1
vote
1answer
17 views

How to show that the error variance of the best linear predictor is inferior to the proportional predictor?

Let's consider the 1D case. How do we prove that the error variance of the Best Linear Predictor (BLP) is inferior than the Proportional Predictor (i.e. the Linear Predictor without the intercept)? ...
0
votes
1answer
28 views

Calculating variance of process with time-varying variance

This is a question stemming off a previous post I had regarding calculating portfolio volatility. For a portfolio consisting of multiple assets, I understand that there are multiple ways to calculate ...
2
votes
1answer
34 views

Calculating portfolio volatility from portfolio returns vs. from covariance matrix

I'm having trouble understanding the difference in calculating portfolio volatility via the portfolio returns vs. via the covariance matrix. To be more specific: I understand that on the individual ...
-3
votes
1answer
34 views

Statistics and Probability. Please show solution [closed]

Miss Romero noted that the mean scores of a random sample of 15 grade 8 students who had taken a special test were 80.5. If the standard deviation of the scores was 3.1 and the sample came from an ...
0
votes
0answers
24 views

Log Likelihood and parameter space

I'm taking a 3rd year statistical theory class in which we're introducing the likelihood function with a bit more mathematical rigor. Under the discussion for maximizing the likelihood function $\...
0
votes
0answers
7 views

Specifying the bandwidth parameter for Local Whittle estimation

In every Local Whittle function I have seen I have to set bandwith parameter usually denoted m such that m = floor(1+T^delta). I am interested in the delta, how do I choose what value to put in the ...
0
votes
0answers
40 views

How to compare two dispersion measures in a specific probability density function?

I am very interested in the properties of a specific probability density function (pdf) proposed in this article. It seems to me that this pdf is a very general one to capture the properties of wide ...
0
votes
2answers
107 views

How to modify the mean and variance/dispersion of a given distribution

I am trying to find a parametric adjustment that allows modifying the mean and variance/dispersion of a given distribution. Ideally, this adjustment would be implemented through a parametric function ...
0
votes
0answers
10 views

Dependency of variable P on multiple factors, and factors interactions

In my data, I have a variable/vector P which is dependent on variables X, Y, Z, K and gender. The factors X, Y, Z are interacting. What kind of statistical tool/method I can use to explore dependency ...
1
vote
1answer
55 views

Algorithm for modified knapsack problem

We have n-players in a game. We have a population of players we can choose from. Each player score is a normally distributed random variable and each player has a cost to add to the team. We are ...
3
votes
1answer
55 views

Are the law of iterated expectation and the law of total expectations the same?

On the Wikipedia page of the Law of total expectations it is said that The proposition in probability theory known as the law of total expectation, the law of iterated expectations, the tower rule, ...
2
votes
1answer
41 views

Accounting Student taking Master in Stat, need your help regarding taking subjects

I had a Bachelor in Accounting and now I'm doing a Masters in Statistics. This is my first semester and I have to Choose at least 3 out of 6 offered subjects for this semester. These subjects Are: ...
2
votes
1answer
20 views

AR(1) Finding $\gamma_l$

I have $\gamma_l = Cov(r_t, r_{t-l})$ as a definition in my notes and now I need to find $gamma_l$ for a series $r_t- m = p(r_{t-1} - m) + a_t$ where $r_t$ is a linear time series with expected value $...