All Questions
Tagged with likelihood-ratio asymptotics
22
questions
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Local Linearity vs Regularity Conditions for the asymptotic distribution of the Likelihood Ratio
In his book 'Asymptotic Statistics,' Aad van der Vaart when discussing the asymptotic distribution of the log-likelihood-ratio says:
"The most important conclusion of this chapter is that, under ...
- 11
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0
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61
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Exact Likelihood ratio statistic for discrete distribution
Suppose that the random variables in a sample $Y_1, Y_2, \ldots, Y_n$ are iid with values in $[0,1]$, and that an investigator knows that the underlying probability density $f_Y(y)$ has the form
$f_Y(...
- 133
2
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1
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66
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Likelihood ratio test for model specification with boundary Null
I am interested in understanding the asymptotic distribution of Likelihood ratio (LR) test statistic for model specification. I am focusing on the case in which the null hypothesis is of the form (i.e....
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Asymptotical convergence of the Likelihood ratio test in general hypotheses testing
I'm aware that the 2-loglikelihood ratio is asymptotically distributed as a Chi-squared distribution under the Null hypotheses for nested hypotheses. My question is, there is any generalized formula ...
- 101
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118
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Does there always exist for n small, a non-chi-squared test-statistic for the likelihood-ratio (neyman-pearson, karlin-rubin), score, and wald-tests?
An additional reason that the chi-squared distribution is widely used is that it turns up as the large sample distribution of generalized likelihood ratio tests (LRT).[6] LRTs have several desirable ...
user318514
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Asymptotic chi-squared distribution of likelihood ratio statistic in regression problem
There is a famous result, going back to Wilks (1938) "The large-sample distribution of the likelihood ratio for testing composite hypotheses" (Ann. Math. Stat., 9, 60-62) that states that ...
- 26.1k
3
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1
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Failing to obtain $\chi^2(1)$ asymptotic distribution under $H_0$ in a likelihood ratio test: example 2
I have a large sample (a vector) $\mathbf{x}$ from a random variable $X\sim N(\mu,\sigma^2)$. The variance $\sigma^2$ is known, but the expectation $\mu$ is unknown. I would like to test the null ...
- 68.6k
4
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1
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261
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Failing to obtain $\chi^2(1)$ asymptotic distribution under $H_0$ in a likelihood ratio test: example 1
I have a large sample (a vector) $\mathbf{x}$ from a random variable $X\sim N(\mu,\sigma^2)$. The variance $\sigma^2$ is known, but the expectation $\mu$ is unknown. I would like to test the null ...
- 68.6k
5
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2
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394
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Asymptotic null distribution of the LR statistic with point null and point alternative
I have a large sample (a vector) $\mathbf{x}$ from a random variable $X\sim N(\mu,\sigma^2)$. The variance $\sigma^2$ is known, but the expectation $\mu$ is unknown. I would like to test the null ...
- 68.6k
6
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1
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688
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Asymptotic equivalence of Likelihood Ratio Statistic and Wald Statistic?
When we say the likelihood ratio statistic and the Wald statistic of a set of binomial distributions are asymptotically equivalent, do we mean that the sampling distributions of the two statistics ...
- 249
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Can Wilk's $-2\log(\Lambda)\sim \chi^2_d$ rule be used with a sample size $n=2$?
Suppose I have $X \sim \text{Poisson}(\lambda_x)$ and $Y \sim \text{Poisson}(\lambda_y)$ and they are independent. Suppose $H_0: \lambda_x =\lambda_y$ and $H_A: \lambda_x\ne\lambda_y$.
My likelihood ...
- 4,423
1
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242
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Asymptotic Distribution of Wald test
If $X_1 , X_2 , ... X_n$ are iid and they have the same pdf $f_θ(x)$ .
Consider testing $H_0 : θ= θ_0$ vs $H_n : θ = θ_0 +Δ * n^{-1/2}$.
where , $Δ = (Δ_1 , Δ_2 , .. Δ_k)^T$
We want to find the ...
- 213
4
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1
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Understanding simple LRT test asymptotic using Taylor expansion?
I am trying to understand the proof that the LRT test for
$$H_0: \theta = \theta_0 \quad vs \quad H_1: \theta \neq \theta_0$$
is asymptotically $\chi_1^2$. I am reading the proof presented in Casella ...
- 2,564
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Consistency and Asymptotic Normality for MLE of Independent NON-identically distributed normals
I have the following setting:
$$ x_k \sim N(\mu,\sigma^2 + \hat{\delta}^2_k),k=1,\dots,K, $$
where $\{x_k,k=1,\dots,K\}$ - observed data, $\{\hat{\delta}^2_k\,k=1,\dots,K \}$ are known parameters (...
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2
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What happens to the likelihood ratio as more and more data is gathered?
Let $f$, $g$ and $h$ be densities and suppose you have $x_i \sim h$, $i \in \mathbb{N}$. What happens to the likelihood ratio
$$
\prod_{i=1}^n \frac{f(x_i)}{g(x_i)}
$$
as $n \rightarrow \infty$ ? (...
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