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This page is basically complete except that the PDF activies only have the non-interactive versions for now.
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( Non-interactive Version )
As we mentioned at the end of the Introduction to Unit 4B , we will focus only on two-sided tests for the remainder of this course. One-sided tests are often possible but rarely used in clinical research.
CO-4: Distinguish among different measurement scales, choose the appropriate descriptive and inferential statistical methods based on these distinctions, and interpret the results.
LO 4.35: For a data analysis situation involving two variables, choose the appropriate inferential method for examining the relationship between the variables and justify the choice.
LO 4.36: For a data analysis situation involving two variables, carry out the appropriate inferential method for examining relationships between the variables and draw the correct conclusions in context.
CO-5: Determine preferred methodological alternatives to commonly used statistical methods when assumptions are not met.
REVIEW: Unit 1 Brooks Heritage Mens Fusion Light Grey/Dark Grey/Blue/Teal ejdcMk40
Video: k > 2 Independent Samples (21:15)

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Introduction

In this part, we continue to handle situations involving one categorical explanatory variable and one quantitative response variable, which is case C→Q.

Here is a summary of the tests we have covered for the case where k = 2. Methods in BOLD are our main focus in this unit.

So far we have discussed the two samples and matched pairs designs, in which the categorical explanatory variable is two-valued. As we saw, in these cases, examining the relationship between the explanatory and the response variables amounts to comparing the mean of the response variable (Y) in two populations, which are defined by the two values of the explanatory variable (X). The difference between the two samples and matched pairs designs is that in the former, the two samples are independent, and in the latter, the samples are dependent.

We now move on to the case where k > 2 when we have independent samples. Here is a summary of the tests we will learn for the case where k > 2. Notice we will not cover the dependent samples case in this course.

Notice we will not cover the dependent samples case in this course.

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Random weighting, proposed by Zheng (1987) [ Caterpillar Mens Brode SteelToe Work Shoe Espresso iNYM9Q
], is an emerging computational method in statistics, and has been used to solve different problems [ 11 ]. The random weighting method has following advantages: (1) it is sample in computation; (2) it does not require the previous knowledge on the distribution, and the estimation results are unbiased; (3) the estimation error of the random weighting method is smaller than that of Bootstrap in the case of small samples; (4) it is independent and identically distributed, and robust; (5) statistic determined by the random weighting method has the density function, so it is particularly suitable for the problem described in the density function. This paper is to develop a robust estimation method of combining analogy with random weighting method for the {\mathcal{G}}_I^0 model, which has the good properties of unbiased, the small mean-squared error and its ability to resist contamination. Even in small samples and low computational cost, its performance is still very robust.

The paper is organized as follows: Section II presents the random weighting estimation for parameters of {\mathcal{G}}_I^0 distribution. In Section III, we will present and discuss the main numerical results. Finally, Conclusions and future work are presented in Section IV.

Random weighting estimation for {\mathcal{G}}_I^0 distribution

{f}_X(x)=\frac{L^L\varGamma \left( L-\alpha \right)}{\gamma^{\alpha}\varGamma (L)\varGamma \left(-\alpha \right)}\frac{x^{L-1}}{{\left(\gamma + Lx\right)}^{L-\alpha}},\kern0.5em x>0
(1)
\gamma >0
(2)
{f}_X(x)=\frac{L^L\varGamma \left( L-\alpha \right)}{\gamma^{\alpha}\varGamma (L)\varGamma \left(-\alpha \right)}\frac{x^{L-1}}{{\left(\gamma + Lx\right)}^{L-\alpha}},\kern0.5em x>0
(3)

where −  α  > 0 is the roughness parameter, γ  > 0 is the scale parameter and L  ≥ 1 is the number of looks [ 1 ].

E\left({X}^r\right)={\left(\frac{\gamma}{L}\right)}^r\frac{\varGamma \left(-\alpha - r\right)\varGamma \left( L+ r\right)}{\varGamma \left(-\alpha \right)\varGamma (L)}
(4)

The {\mathcal{G}}_I^0 distribution is very attractive for modeling data with speckle noise, due to its mathematical tractability and ability to describe information from most types of areas, for given α  < − 1 and L . These densities are presented in semi-logarithmic scale, showing that they have heavy (linear) tails with respect to the Gaussian distribution which displays quadratic behavior. It is noticeable that the larger values of α , the larger the variances have; in fact, the variance is not finite when α  ≥ − 1.

\begin{array}{@{}[email protected]{}} J(W^{(t+1)}) \leq J(W^{(t)}), \end{array}
(32)
\begin{array}{@{}[email protected]{}} J(W) = tr\left(A-XWH^{T}Y^{T}\right)^{T}D\left(A-XWH^{T}Y^{T}\right)\\ +\lambda_{1} tr\left(W^{T}PW\right) +\lambda_{2} tr\left(H^{T}QH\right) \end{array}
(33)

And, according to the statement of Lemma 6, under the W update rule in Algorithm 3, J ( W ) monotonically decreases. In order to prove the statement, we follow the approaches utilizing auxiliary functions [ 13 , TDA Womens Single Strap Bows High Heel Salsa Tango Ballroom Latin Modern Dance Wedding Shoes Red pi3TVKTk
]. □

Definition 2

(, ) is an auxiliary function for the function () if (, )≥() for all and (,)=().

\begin{array}{@{}[email protected]{}} W^{(t+1)} = \underset{W}{\text{argmin}} \quad G\left(W,W^{(t)}\right) \end{array}
\begin{array}{@{}[email protected]{}} J\left(W^{(t+1)}\right) = G\left(W^{(t+1)},W^{(t+1)}\right) \leq G\left(W^{(t+1)},W^{(t)}\right)\\ \leq G\left(W^{(t)},W^{(t)}\right) = J\left(W^{(t)}\right) \end{array}

This proves that J ( W ( t ) ) is monotonically decreasing.

Lemma 9
\begin{array}{*{20}l} G(W,W^{\prime}) = tr\left(A^{T}DA\right) - 2tr\left(YHW^{T}X^{T}DA\right)\\ \quad +\lambda_{1} tr\left(W^{T}PW\right)+ \lambda_{2} tr\left(H^{T}QH\right) \\ \quad + \sum_{i=1}^{m}\sum_{j=1}^{r} \frac{\left(X^{T}DXW^{\prime}H^{T}Y^{T}YH\right)_{{ij}}W^{2}_{{ij}}}{{W^{\prime}}_{{ij}}} \end{array}
(34)
\begin{array}{*{20}l} J(W) = tr\left(A^{T}DA\right) - 2tr\left(YHW^{T}X^{T}DA\right)\\ \quad + \lambda_{1} tr\left(W^{T}PW\right)+ \lambda_{2} tr\left(H^{T}QH\right)\\ \quad + tr\left(W^{T}X^{T}DXWH^{T}Y^{T}YH\right) \end{array}
(35)
\begin{array}{@{}[email protected]{}} tr\left(W^{T}\Lambda WB\right) \leq \sum_{i}\sum_{j}\left(\Lambda W^{\prime} B\right)_{{ji}}\frac{W^{2}_{{ij}}}{W^{\prime}_{{ij}}}, \end{array}
(36)

where, Λ , B , W are non-negative matrices, and Λ , B are symmetric matrices. And obviously the equality holds in Eq. AllhqFashion Womens Zipper Round Closed Toe HighHeels Pu Solid Boots Claret 38 DljIi
when W = W .

In Eq. PUMA Mens Tazon 6 Mesh Dotd CrossTrainer Shoe Asphalt/Puma Silver akf9YHo0J
, if we do the substitutions: Λ = X T D X , B = H T Y T Y H , W = W , W = W , we see that the fifth term of Eq. Caterpillar Colorado RD Walala Womens Leather Boots Black 9Q87pC
is smaller than the fifth term of Eq. Stacy Adams Mens Ruther SlipOn Loafer Sandstone Suede pEUABCI8g
. However, the equality holds when W = W . Thus G ( W , W ) in Eq. fereshte Womens Rivet Canvas Lace Up High Heel Fashion Sneakers Ankle Boots White GK5oJX3dRF
is an auxiliary function of J ( W ). □

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