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1. Consider a GLM (generalised linear model) for a Poisson random sample Y1,. .. , Y, with \Vi each Yi having a pdf or...
2. Consider a simple linear regression model for a response variable Yi, a single predictor variable...
2. Consider a simple linear regression model for a response variable Yi, a single predictor variable ri, i-1,... , n, and having Gaussian (i.e. normally distributed) errors Ý,-BzitEj, Ejį.i.d. N(0, σ2) This model is often called "regression through the origin" since E(Yi) 0 if xi 0 (a) Write down the likelihood function for the parameters β and σ2 (b) Find the MLEs for β and σ2, explicitly showing that they are unique maximizers of the likelihood function. (Hint: The function...
2. Suppose Y ~ Exp(a), which has pdf f(y)-1 exp(-y/a). (a) Use the following R code to generate data from the model Yi...
2. Suppose Y ~ Exp(a), which has pdf f(y)-1 exp(-y/a). (a) Use the following R code to generate data from the model Yi ~ Exp(0.05/Xi), and provide the scatterplot of Y against X set.seed(123) n <- 500 <-rnorm (n, x 3, 1) Y <- rexp(n, X) (b) Fit the model Yi-Ao + Ax, + ε¡ using the lm function in R and provide a plot of the best fit line on the scatterplot of Y vs X, and the residual...
4. Let Yi, . .. ,y, denote a random sample from the pdf 0-1 0Ky1, elsewhere. y"(1- y)0-1 0, (a) Find the method of moments estimator of θ. (b) Find a sufficient statistics for θ 4. Let Yi, ....
4. Let Yi, . .. ,y, denote a random sample from the pdf 0-1 0Ky1, elsewhere. y"(1- y)0-1 0, (a) Find the method of moments estimator of θ. (b) Find a sufficient statistics for θ
4. Let Yi, . .. ,y, denote a random sample from the pdf 0-1 0Ky1, elsewhere. y"(1- y)0-1 0, (a) Find the method of moments estimator of θ. (b) Find a sufficient statistics for θ
4. Let Yi, ½, . . . , Yn be a random sample from some pdf/pmf...
4. Let Yi, ½, . . . , Yn be a random sample from some pdf/pmf f(y; θ)·Let W be a point estimator h(y, Y2, . . . , Yn) for θ. The bias of W as a point estimator for θ is defined as W Blase(W) = E(W)- The mean square error of W is defined as MSEe(W) = E(W-0)2 (a) Using properties of expected values, and the definition of variance from PSTAT 120A/B, show that MSEe(W) = Vare(W)...
1. Let Y1, . . . ,Y,, be a random sample from a population with density...
1. Let Y1, . . . ,Y,, be a random sample from a population with density function 0, otherwise (a) Find the method of moments estimator of θ (b) Show that Yan.-max(Yi, . . . ,%) is sufficient for 02] (Hint: Recall the indicator function given by I(A)1 if A is true and 0 otherwise.) (c) Determine the density function of Yn) and hence find a function of Ym) that is an unbiased estimator of θ (d) Find c so...
7. Suppose the data consist of repeated observations (y;it, X), t = 1, -.. ,T, for each in- dividual i = 1,... ,n. Here...
7. Suppose the data consist of repeated observations (y;it, X), t = 1, -.. ,T, for each in- dividual i = 1,... ,n. Here yit is the response and xt is a covariate vector. A linear mixed-effects model for analysing the population-averaged and subject-specific effects of Xit is of the following form Z;B + W;b; + €;, yi = where y (yi1;* . ,ViT)T; Z; is a T x p design matrix built from {xji} for the fixed effects B;...
linear stat modeling & regression 1) Consider n data points with 3 covariates and observations {xn, ^i2, xi3,yid; i,,n, and you fit the following model, y Bi+Br2+Br+e that is yi A) +Ari,1 +Ari...
linear stat modeling & regression
1) Consider n data points with 3 covariates and observations {xn, ^i2, xi3,yid; i,,n, and you fit the following model, y Bi+Br2+Br+e that is yi A) +Ari,1 +Ari,2 +Buri,3 + єї where є,'s are independent normal distribution with mean zero and variance ơ2 . H the vectors of (Y1, . . . ,Yn). Assume the covariates are centered: Σίχί,,-0, k = 1,2,3. ere, n = 50, Let L are Assume, X'X is a diagonal matrix...
5. Consider a random sample Y1, . . . , Yn from a distribution with pdf f(y|θ) = 1 θ 2 xe−x/θ , 0 < x < ∞. Calcula...
5. Consider a random sample Y1, . . . , Yn from a distribution with pdf f(y|θ) = 1 θ 2 xe−x/θ , 0 < x < ∞. Calculate the ML estimator of θ. 6. Consider the pdf g(y|α) = c(1 + αy2 ), −1 < y < 1. (a) Show that g(y|α) is a pdf when c = 3 6 + 2α . (b) Calculate E(Y ) and E(Y 2 ). Referencing your calculations, explain why M1 can’t be...
2. Consider a simple linear regression i ion model for a response variable Y, a single...
2. Consider a simple linear regression i ion model for a response variable Y, a single predictor variable ,i1.., n, and having Gaussian (i.e. normally distributed) errors: This model is often called "regression through the origin" since E(X) = 0 if xi = 0 (a) Write down the likelihood function for the parameters β and σ2 (b) Find the MLEs for β and σ2, explicitly showing that they are unique maximizers of the likelihood function Hint: The function g(x)log(x) +1-x...
FR2 (4+4+4 12 points) (a) Let XI, X2, X10 be a randoin sample from N(μι,σ?) and Yi, Y2, 10 , Y 15 be a random sample from N (μ2, σ2), where all parameters are unknown. Sup- pose Σ 1 (Xi X 2 0 321 (Y-...
FR2 (4+4+4 12 points) (a) Let XI, X2, X10 be a randoin sample from N(μι,σ?) and Yi, Y2, 10 , Y 15 be a random sample from N (μ2, σ2), where all parameters are unknown. Sup- pose Σ 1 (Xi X 2 0 321 (Y-Y )2-100. obtain a 99% confidence interval for σ of having the form b, 0o) for some number b (No derivation needed). (b) 60 random points are selected from the unit interval (r:0 . We want...
2. Consider a simple linear regression model for a response variable Yi, a single predictor variable ri, i-1,... , n, and having Gaussian (i.e. normally distributed) errors Ý,-BzitEj, Ejį.i.d. N(0, σ2) This model is often called "regression through the origin" since E(Yi) 0 if xi 0 (a) Write down the likelihood function for the parameters β and σ2 (b) Find the MLEs for β and σ2, explicitly showing that they are unique maximizers of the likelihood function. (Hint: The function...
2. Suppose Y ~ Exp(a), which has pdf f(y)-1 exp(-y/a). (a) Use the following R code to generate data from the model Yi ~ Exp(0.05/Xi), and provide the scatterplot of Y against X set.seed(123) n <- 500 <-rnorm (n, x 3, 1) Y <- rexp(n, X) (b) Fit the model Yi-Ao + Ax, + ε¡ using the lm function in R and provide a plot of the best fit line on the scatterplot of Y vs X, and the residual...
4. Let Yi, . .. ,y, denote a random sample from the pdf 0-1 0Ky1, elsewhere. y"(1- y)0-1 0, (a) Find the method of moments estimator of θ. (b) Find a sufficient statistics for θ
4. Let Yi, . .. ,y, denote a random sample from the pdf 0-1 0Ky1, elsewhere. y"(1- y)0-1 0, (a) Find the method of moments estimator of θ. (b) Find a sufficient statistics for θ
4. Let Yi, ½, . . . , Yn be a random sample from some pdf/pmf f(y; θ)·Let W be a point estimator h(y, Y2, . . . , Yn) for θ. The bias of W as a point estimator for θ is defined as W Blase(W) = E(W)- The mean square error of W is defined as MSEe(W) = E(W-0)2 (a) Using properties of expected values, and the definition of variance from PSTAT 120A/B, show that MSEe(W) = Vare(W)...
1. Let Y1, . . . ,Y,, be a random sample from a population with density function 0, otherwise (a) Find the method of moments estimator of θ (b) Show that Yan.-max(Yi, . . . ,%) is sufficient for 02] (Hint: Recall the indicator function given by I(A)1 if A is true and 0 otherwise.) (c) Determine the density function of Yn) and hence find a function of Ym) that is an unbiased estimator of θ (d) Find c so...
7. Suppose the data consist of repeated observations (y;it, X), t = 1, -.. ,T, for each in- dividual i = 1,... ,n. Here yit is the response and xt is a covariate vector. A linear mixed-effects model for analysing the population-averaged and subject-specific effects of Xit is of the following form Z;B + W;b; + €;, yi = where y (yi1;* . ,ViT)T; Z; is a T x p design matrix built from {xji} for the fixed effects B;...
linear stat modeling & regression
1) Consider n data points with 3 covariates and observations {xn, ^i2, xi3,yid; i,,n, and you fit the following model, y Bi+Br2+Br+e that is yi A) +Ari,1 +Ari,2 +Buri,3 + єї where є,'s are independent normal distribution with mean zero and variance ơ2 . H the vectors of (Y1, . . . ,Yn). Assume the covariates are centered: Σίχί,,-0, k = 1,2,3. ere, n = 50, Let L are Assume, X'X is a diagonal matrix...
2. Consider a simple linear regression i ion model for a response variable Y, a single predictor variable ,i1.., n, and having Gaussian (i.e. normally distributed) errors: This model is often called "regression through the origin" since E(X) = 0 if xi = 0 (a) Write down the likelihood function for the parameters β and σ2 (b) Find the MLEs for β and σ2, explicitly showing that they are unique maximizers of the likelihood function Hint: The function g(x)log(x) +1-x...
FR2 (4+4+4 12 points) (a) Let XI, X2, X10 be a randoin sample from N(μι,σ?) and Yi, Y2, 10 , Y 15 be a random sample from N (μ2, σ2), where all parameters are unknown. Sup- pose Σ 1 (Xi X 2 0 321 (Y-Y )2-100. obtain a 99% confidence interval for σ of having the form b, 0o) for some number b (No derivation needed). (b) 60 random points are selected from the unit interval (r:0 . We want...
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