Publications by Al-Ahmadgaid Asaad
Parametric Inference: Likelihood Ratio Test Problem 2
More on Likelihood Ratio Test, the following problem is originally from Casella and Berger (2001), exercise 8.12.ProblemFor samples of size $n=1,4,16,64,100$ from a normal population with mean $mu$ and known variance $sigma^2$, plot the power function of the following LRTs (Likelihood Ratio Tests). Take $alpha = .05$. $H_0:muleq 0$ versus $H_1:mu...
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Parametric Inference: Karlin-Rubin Theorem
A family of pdfs or pmfs ${g(t|theta):thetainTheta}$ for a univariate random variable $T$ with real-valued parameter $theta$ has a monotone likelihood ratio (MLR) if, for every $theta_2>theta_1$, $g(t|theta_2)/g(t|theta_1)$ is a monotone (nonincreasing or nondecreasing) function of $t$ on ${t:g(t|theta_1)>0;text{or};g(t|theta_2)>0}$. Note that $c...
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R, Python, and SAS: Getting Started with Linear Regression
Consider the linear regression model, $$ y_i=f_i(boldsymbol{x}|boldsymbol{beta})+varepsilon_i, $$ where $y_i$ is the response or the dependent variable at the $i$th case, $i=1,cdots, N$ and the predictor or the independent variable is the $boldsymbol{x}$ term defined in the mean function $f_i(boldsymbol{x}|boldsymbol{beta})$. For simplicity, cons...
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R and Python: Theory of Linear Least Squares
In my previous article, we talked about implementations of linear regression models in R, Python and SAS. On the theoretical sides, however, I briefly mentioned the estimation procedure for the parameter $\boldsymbol{beta}$. So to help us understand how software does the estimation procedure, we’ll look at the mathematics behind it....
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R and Python: Gradient Descent
One of the problems often dealt in Statistics is minimization of the objective function. And contrary to the linear models, there is no analytical solution for models that are nonlinear on the parameters such as logistic regression, neural networks, and nonlinear regression models (like Michaelis-Menten model). In this situation, we h...
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