Let $\hat { \theta } _ { 1 } , \ldots \ldots , \hat { \theta } _ { m }$ be the maximum likelihood estimators of the unknown parameters $\theta _ { 1 } , \ldots \ldots , \theta _ { m }$ , and let $\chi ^ { 2 }$ denote the test statistic value based on these estimators. If the data are classified into k categories, then the critical value $c _ { \alpha }$ that specifies a level $\alpha$ upper-tailed test satisfies
A) $\hat { \chi } _ { α - 1 } ^ { 2 } \leq c _ α \leq \hat { \chi } _ { α , k - 1 - m } ^ { 2 }$
B) $\chi _ { α , k - 1 - m } ^ { 2 } \leq c _ { α } \leq \hat { \lambda } _ { α , k - 1 } ^ { 2 }$
C) $c _ { α } \geq { X } _ {α , k - 1 } ^ { 2 }$
D) $c _ { α } \leq \hat { \lambda } _ { α , k - 1 - m } ^ { 2 }$
E) $\chi _ {α , m - 1 } ^ { 2 } \leq c _ { α} \leq \chi_ {α , k - 1 } ^ { 2 }$

Question

Let $\hat { \theta } _ { 1 } , \ldots \ldots , \hat { \theta } _ { m }$ be the maximum likelihood estimators of the unknown parameters $\theta _ { 1 } , \ldots \ldots , \theta _ { m }$ , and let $\chi ^ { 2 }$ denote the test statistic value based on these estimators. If the data are classified into k categories, then the critical value $c _ { \alpha }$ that specifies a level $\alpha$ upper-tailed test satisfies
A) $\hat { \chi } _ { α - 1 } ^ { 2 } \leq c _ α \leq \hat { \chi } _ { α , k - 1 - m } ^ { 2 }$
B) $\chi _ { α , k - 1 - m } ^ { 2 } \leq c _ { α } \leq \hat { \lambda } _ { α , k - 1 } ^ { 2 }$
C) $c _ { α } \geq  { X } _ {α , k - 1 } ^ { 2 }$
D) $c _ { α } \leq \hat { \lambda } _ { α , k - 1 - m } ^ { 2 }$
E) $\chi _ {α , m - 1 } ^ { 2 } \leq c _ { α} \leq \chi_ {α , k - 1 } ^ { 2 }$

Quizplus · Accepted Answer

The correct critical value $c_{\alpha}$ for a chi-square test with $k$ categories and $m$ estimated parameters is determined by the degrees of freedom, which is $k - 1 - m$. This accounts for the number of categories minus one (for the total constraint) minus the number of estimated parameters. The notation $\chi_{\alpha, k-1-m}^{2}$ correctly represents the chi-square distribution critical value at significance level $\alpha$ with $k-1-m$ degrees of freedom.

Let θ^1,……,θ^m\hat { \theta } _ { 1 } , \ldots \ldots , \hat { \theta } _ { m }θ^1​,……,θ^m​

Let $\hat { \theta } _ { 1 } , \ldots \ldots , \hat { \theta } _ { m }$