AIS data

Description

The set of AIS data involves recorded body factors of 202 athletes including 100 women 102 men, see Cook (2009). Among factors, two variables body mass index (BMI) and body fat percentage (Bfat) are chosen for cluster analysis.

Usage

data(AIS)

Format

A text file with 3 columns.

References

R. D. Cook and S. Weisberg, (2009). An Introduction to Regression Graphics, John Wiley & Sons, New York.

Examples

data(AIS)

Extracting diagonal and upper off-diagonal elements of a squre matrix.

Description

Suppose x is a d \times d square matrix. This function extracts the diagonal and upper off-diagonal elements of a given square matrix.

Usage

arrange_lambda(x)

Arguments

Value

a list of addresses for elements of the given matrix x as {x}_{11},{x}_{12},\cdots,{x}_{1n},{x}_{21},{x}_{22},\cdots,{x}_{2n},\cdots,{x}_{mn} .

Author(s)

Mahdi Teimouri

Examples

x <- matrix( c( 0.4, -0.20, -0.20, 0.5 ), nrow = 2, ncol = 2 )
arrange_lambda(x)

Extracting diagonal and upper off-diagonal elements of a squre matrix.

Description

Suppose x is a d \times d square matrix. This function extracts the diagonal and upper off-diagonal elements of a given square matrix.

Usage

arrange_sigma(x)

Arguments

Value

diagonal and upper off-diagonal elements of {x} that consist of {x}_{11},{x}_{12},\cdots,{x}_{1d},{x}_{21},{x}_{22},\cdots,{x}_{2d},\cdots,{x}_{dd} .

Author(s)

Mahdi Teimouri

Examples

x <- matrix( c( 0.4, -0.20, -0.20, 0.5 ), nrow = 2, ncol = 2 )
arrange_sigma(x)

bankruptcy data

Description

The bankruptcy dataset involves ratio of the retained earnings (RE) to the total assets, and the ratio of earnings before interests and the taxes (EBIT) to the total assets of 66 American firms, see Altman (1969).

Usage

data(bankruptcy)

Format

A text file with 3 columns.

References

E. I. Altman, 1969. Financial ratios, discriminant analysis and the prediction of corporate bankruptcy, The Journal of Finance, 23(4), 589-609.

Examples

data(bankruptcy)

Output configuration.

Description

Output design for standard error of the standard error for parameters of the finite mixture models.

Usage

configuration1(Y, G, weight, mu, sigma, lambda, family, skewness, param, theta,
 ofim1_solve, sigma_arrange1, level)

Arguments

Value

designated form for output of parameters and their standard errors.

Author(s)

Mahdi Teimouri

Examples

      n <- 200
      G <- 2
 weight <- rep( 0.5, 2 )
    mu1 <- rep(-5  , 2 )
    mu2 <- rep( 5  , 2 )
 sigma1 <- matrix( c( 0.4, -0.20, -0.20, 0.5 ), nrow = 2, ncol = 2 )
 sigma2 <- matrix( c( 0.5,  0.20,  0.20, 0.4 ), nrow = 2, ncol = 2 )
lambda1 <- c(-5, -5 )
lambda2 <- c( 5,  5 )
 theta1 <- c( 10, 12 )
 theta2 <- c( 10, 20 )
     mu <- list(mu1, mu2)
  sigma <- list( sigma1 , sigma2 )
 lambda <- list( lambda1, lambda2)
  theta <- list( theta1 , theta2 )
  param <- c("a","b")
    PDF <- quote( (b/2)^(a/2)*x^(-a/2 - 1)/gamma(a/2)*exp( -b/(x*2) ) )
  tick  <- c(1, 1)
      Y <- rmix(n, G, weight, model = "restricted", mu, sigma, lambda, family = "igamma", theta)
  ofim  <- ofim1(Y[, 1:2], G, weight, mu, sigma, lambda, family = "igamma",
  skewness = "TRUE", param, theta, tick, h = 0.001, N = 3000, level = 0.05, PDF)
configuration1(Y[, 1:2], G, weight = weight, mu, sigma, lambda, family = "igamma",
skewness = "TRUE", param, theta, ofim1_solve = ofim$Fisher,
sigma_arrange1 = ofim$index_sigma, level = 0.05)

Output configuration.

Description

Output design for standard error of the standard error for parameters of the finite mixture models.

Usage

configuration2(Y, G, weight, model, mu, sigma, lambda, family, skewness, param,
theta, ofim2_solve, sigma_arrange2, level)

Arguments

Value

designated form for output of parameters and their standard errors.

Author(s)

Mahdi Teimouri

Examples

      n <- 100
      G <- 2
 weight <- rep( 0.5, 2 )
    mu1 <- rep(  -5, 2 )
    mu2 <- rep(   5, 2 )
 sigma1 <- matrix( c( 0.4, -0.20, -0.20, 0.5 ), nrow = 2, ncol = 2 )
 sigma2 <- matrix( c( 0.5,  0.20,  0.20, 0.4 ), nrow = 2, ncol = 2 )
lambda1 <- diag( c(-5, -5) )
lambda2 <- diag( c( 5,  5) )
 theta1 <- c( 10, 12 )
 theta2 <- c( 5, 20 )
     mu <- list( mu1, mu2 )
  sigma <- list( sigma1 , sigma2 )
 lambda <- list( lambda1, lambda2)
  theta <- list( theta1 , theta2 )
  param <- c("a","b")
    PDF <- quote( (b/2)^(a/2)*x^(-a/2 - 1)/gamma(a/2)*exp( -b/(x*2) ) )
  tick  <- c(1, 1)
      Y <- rmix(n, G, weight, model = "unrestricted", mu, sigma, lambda, family = "igamma", theta)
  ofim  <- ofim2(Y[, 1:2], G, weight, model = "unrestricted", mu, sigma, lambda,
  family = "igamma", skewness = "TRUE", param, theta, tick, h = 0.01, N = 3000, level = 0.05, PDF)
configuration2(Y[, 1:2], G, weight = weight, model = "unrestricted", mu, sigma, lambda,
family = "igamma", skewness = "TRUE", param, theta, ofim2_solve = ofim$Fisher,
sigma_arrange2 = ofim$index_sigma, level = 0.05)

Approximating the density function of the finite mixture models applied for model-based clustering.

Description

The density function of a G-component finite mixture model can be represented as

g({\bold{y}}|\Psi)=\sum_{g=1}^{G} \omega_{g} f_{\bold{Y}}({\bold{y}}, \Theta_g),

where \bold{\Psi} = \bigl(\bold{\Theta}_{1},\cdots, \bold{\Theta}_{G}\bigr)^{\top} with \bold{\Theta}_g=\bigl({\bold{\omega}}_g, {\bold{\mu}}_g, {{\Sigma}}_g, {\bold{\lambda}}_g\bigr)^{\top}. Herein, f_{\bold{Y}}(\bold{y}, \bold{\Theta}_g) accounts for the density function of random vector \bold{Y} within each component. In the restricted case, f_{\bold{Y}}(\bold{y}, \bold{\Theta}_g) admits the representation given by

{\bold{Y}} \mathop=\limits^d {\bold{\mu}}_{g}+\sqrt{W}{\bold{\lambda}}_{g}\vert{Z}_0\vert + \sqrt{W}{\Sigma}_{g}^{\frac{1}{2}} {\bold{Z}}_1,

where {\bold{\mu}}_{g} \in {R}^{d} is location vector, {\bold{\lambda}}_{g} \in {R}^{d} is skewness vector, \Sigma_{g} is a positive definite symmetric dispersion matrix for g=1,\cdots,G. Further, W is a positive random variable with mixing density function f_W(w| \bold{\theta}_{g}), {Z}_0\sim N(0, 1) , and {\bold{Z}}_1\sim N_{d}\bigl( {\bold{0}}, \Sigma_{g}\bigr) . We note that W, Z_0, and {\bold{Z}}_1 are mutually independent. In the canonical or unrestricted case, f_{\bold{Y}}(\bold{y}, \bold{\Theta}_g) admits the representation as

{\bold{Y}} \mathop=\limits^d {\bold{\mu}}_{g}+\sqrt{W}{\bold{\Lambda}}_{g} \vert\bold{Z}_0\vert + \sqrt{W}{\Sigma}_{g}^{\frac{1}{2}} {\bold{Z}}_1,

where \bold{\Lambda}_{g} is the skewness matrix and random vector \bold{Z}_0 follows a zero-mean normal random vector truncated to the positive hyperplane R^{d} whose independent marginals have variance unity. We note that in the unrestricted case \bold{\Lambda}_{g} is a d \times d diagonal matrix whereas in the canonical case, it is a d\times q matrix and so, random vector \bold{Z}_0 follows a zero-mean normal random vector truncated to the positive hyperplane R^{q}.

Usage

dmix(Y, G, weight, model = "restricted", mu, sigma, lambda, family = "constant",
skewness = "FALSE", param = NULL, theta = NULL, tick = NULL, N = 3000, log = "FALSE")
 

Arguments

Value

Monte Carlo approximated values of mixture model density function.

Author(s)

Mahdi Teimouri

Examples

      Y <- c(1, 2)
      G <- 2
 weight <- rep( 0.5, 2 )
    mu1 <- rep(  -5, 2 )
    mu2 <- rep(   5, 2 )
 sigma1 <- matrix( c( 0.4, -0.20, -0.20, 0.5 ), nrow = 2, ncol = 2 )
 sigma2 <- matrix( c( 0.5,  0.20,  0.20, 0.4 ), nrow = 2, ncol = 2 )
lambda1 <- c( 5, -5 )
lambda2 <- c(-5,  5 )
     mu <- list( mu1, mu2 )
  sigma <- list( sigma1 , sigma2 )
 lambda <- list( lambda1, lambda2)
    out <- dmix(Y, G, weight, model = "restricted", mu, sigma, lambda, family =
           "constant", skewness = "TRUE", param = NULL, theta = NULL, tick =
           NULL, N = 3000)

Monte Carlo approximation for density function of positive alpha-stable distribution.

Description

The density function f_{P}(p|\alpha), of positive \alpha-stable distribution is given by (Kanter, 1975):

f_{P}(p|\alpha)=\frac{1}{\pi}\int_{0}^{\pi}{\frac{\alpha}{2-\alpha}}a(\theta) p^{-\frac{\alpha}{2-\alpha}-1}a(\theta) \exp\Bigl\{-p^{-\frac{\alpha}{2-\alpha}}a(\theta)\Bigr\}d\theta,

where 0<\alpha \leq 2 is tail thickness parameter or index of stability and

a(\theta)=\frac{\sin\Bigl(\bigl(1-\frac{\alpha}{2}\bigr)\theta\Bigr)\Bigl[\sin \bigl(\frac{\alpha \theta}{2}\bigr)\Bigr]^{\frac{\alpha}{2-\alpha}}}{[\sin(\theta)]^{\frac{2}{2-\alpha}},}

for 0<\theta < \pi. We use the Monte Carlo method for approximating f_{P}(p|\alpha).

Usage

dpstable(x, param)

Arguments

Value

The density function of positive \alpha-stable distribution at point x.

Author(s)

Mahdi Teimouri

References

M. Kanter, (1975). Stable densities under change of scale and total variation inequalities, Annals of Probability, 3(4), 697-707.

Examples

x <- 2
param <- 1.5
dpstable(x, param)

Monte Carlo approximation for density function of polynomially tilted alpha-stable distribution.

Description

The density function f_{T}(t|\alpha, \beta), of polynomially tilted \alpha-stable distribution is given by (Devroye, 2009):

f_{T}(t | \alpha, \beta)=\frac{\Gamma(1+\beta)}{\Gamma\Bigl(1+\frac{\beta}{\alpha}\Bigr)}t^{-\beta}f_{P}(t|\alpha),

where 0<\alpha \leq 2 is tail thickness parameter or index of stability and \beta> 0 is tilting parameter. We note that f_{P}(t|\alpha) is the density function of a positive \alpha-stable distribution that has an integral representation (Kanter, 1975):

f_{P}(t|\alpha)=\frac{1}{\pi}\int_{0}^{\pi}{\frac{\alpha}{2-\alpha}}a(\theta) t^{-\frac{\alpha}{2-\alpha}-1}a(\theta) \exp\Bigl\{-t^{-\frac{\alpha}{2-\alpha}}a(\theta)\Bigr\}d\theta,

where

a(\theta)=\frac{\sin\Bigl(\bigl(1-\frac{\alpha}{2}\bigr)\theta\Bigr)\Bigl[\sin \bigl(\frac{\alpha \theta}{2}\bigr)\Bigr]^{\frac{\alpha}{2-\alpha}}}{[\sin(\theta)]^{\frac{2}{2-\alpha}}},

for 0 < \theta < \pi.

Usage

dptstable(x, param, Dim)

Arguments

Value

The density function of polynomially tilted \alpha-stable distribution at point x.

Author(s)

Mahdi Teimouri

References

M. Kanter, (1975). Stable densities under change of scale and total variation inequalities, Annals of Probability, 3(4), 697-707.

L. Devroye, (2009). Random variate generation for exponentially and polynomially tilted stable distributions, ACM Transactions on Modeling and Computer Simulation, 19(4), doi: 10.1145/1596519.1596523.

Examples

    x <- 2
param <- 1.5
  Dim <- 2
dptstable(x, param, Dim)

Computing posteriors expected value.

Description

For a finite mixture model with density function

{\cal{M}}(\bold{y}|\bold{\Psi})=\sum_{g=1}^{G} \omega_{g} f_{\bold{Y}}(\bold{y}, \bold{\Theta}_g),

we use method of Basford et al. (1997) for Computing observed Fisher information matrix. Based on above representation for density function of a finite mixture model, we can write

\bold{Y}| {T}, W \sim N_{d}\bigl(\bold{\mu}+\bold{\lambda}{T}, {W} \Sigma\bigr), {T}| W \sim HN({0}, {W}), W \sim f_W(w| \bold{\theta}).

The required posteriors expectations are E\bigl(W^{-1}|\bold{y},\bold{\Psi}\bigr), E\bigl(W^{-1}T|\bold{y},\bold{\Psi}\bigr), and E\bigl(W^{-1}T^2|\bold{y},\bold{\Psi}\bigr).

Usage

estep1(Y, G, weight, mu, sigma, lambda, family, skewness, param, theta, tick, h, N, PDF)

Arguments

Value

The required posteriors expectations for approximating the observed Fisher information matrix for restricted finite mixture model. These include \tau_{ig}=E\bigl( Z_{ig}=1 \vert {\bold{y}}_{i}, \hat{\bold{\Psi}} \bigr) = {\omega}_g f_{\bold{Y}}\bigl({\bold{y}}_{i}\big|{\bold{\Theta}_g}\bigr)/\bigl[ \sum_{g=1}^{G} {{\omega}}_g f_{\bold{Y}}\bigl({\bold{y}}_{i}\big|{\bold{\Theta}_g}\bigr) \bigr], E\bigl( W^{-1}\big \vert {\bold{y}}_{i}, \hat{\bold{\Psi}} \bigr), E\bigl( {{U}}W^{-1}\big \vert {\bold{y}}_{i}, \hat{\bold{\Psi}} \bigr), E\bigl( U^2 W^{-1}\big \vert {\bold{y}}_{i}, \hat{\bold{\Psi}} \bigr), and E\bigl( \partial f_W(w| {\bold{\theta)}})/\partial {\bold{\theta)}} \big \vert {\bold{y}}_{i}, \hat{{\bold{\Psi}}} \bigr).

Author(s)

Mahdi Teimouri

References

K. E. Basford, D. R. Greenway, G. J. McLachlan, and D. Peel, (1997). Standard errors of fitted means under normal mixture, Computational Statistics, 12, 1-17.

Examples

      n <- 100
      G <- 2
 weight <- rep( 0.5, 2 )
    mu1 <- rep(-5  , 2 )
    mu2 <- rep( 5  , 2 )
 sigma1 <- matrix( c( 0.4, -0.20, -0.20, 0.5 ), nrow = 2, ncol = 2 )
 sigma2 <- matrix( c( 0.5,  0.20,  0.20, 0.4 ), nrow = 2, ncol = 2 )
lambda1 <- c( 5, -5 )
lambda2 <- c(-5,  5 )
 theta1 <- c( 10 )
 theta2 <- c( 20 )
     mu <- list( mu1, mu2 )
  sigma <- list( sigma1 , sigma2 )
 lambda <- list( lambda1, lambda2)
  theta <- list( theta1 , theta2 )
  param <- c( "a" )
    PDF <- quote( (a/2)^(a/2)*x^(-a/2 - 1)/gamma(a/2)*exp( -a/(2*x) ) )
  tick  <- rep( 1, 2 )
theta10 <- c( 10, 10 )
theta20 <- c( 20, 20 )
 theta0 <- list( theta10 , theta20 )
      Y <- rmix( n, G, weight, model = "restricted", mu, sigma, lambda, family = "igamma",
      theta0)
    estep <- estep1( Y[, 1:2], G, weight, mu, sigma, lambda, family = "igamma",
    skewness = "TRUE", param, theta, tick, h = 0.001, N = 3000, PDF)

Computing posteriors expected value.

Description

For a finite mixture model with density function

{\cal{M}}(\bold{y}|\bold{\Psi})=\sum_{g=1}^{G} \omega_{g} f_{\bold{Y}}(\bold{y}, \bold{\Theta}_g),

we use method of Basford et al. (1997) for Computing observed Fisher information matrix. Based on above representation for density function of a finite mixture model, we can write

\bold{Y}| {T}, W \sim N_{d}\bigl(\bold{\mu}+\bold{\lambda}{T}, {W} \Sigma\bigr), {T}| W \sim HN({0}, {W}), W \sim f_W(w| \bold{\theta}).

The required posteriors expectations are E\bigl(W^{-1}|\bold{y},\bold{\Psi}\bigr), E\bigl(W^{-1}T|\bold{y},\bold{\Psi}\bigr), and E\bigl(W^{-1}T^2|\bold{y},\bold{\Psi}\bigr).

Usage

estep2(Y, G, weight, mu, sigma, lambda, family, skewness, param, theta, tick, h, N, PDF)

Arguments

Value

The required posteriors expectations for approximating the observed Fisher information matrix for canonical or unrestricted finite mixture model. These include \tau_{ig}=E\bigl( Z_{ig}=1 \vert {\bold{y}}_{i}, \hat{\bold{\Psi}} \bigr) = {\omega}_g f_{\bold{Y}}\bigl({\bold{y}}_{i}\big|{\bold{\Theta}_g}\bigr)/\bigl[ \sum_{g=1}^{G} {{\omega}}_g f_{\bold{Y}}\bigl({\bold{y}}_{i}\big|{\bold{\Theta}_g}\bigr) \bigr], E\bigl( W^{-1}\big \vert {\bold{y}}_{i}, \hat{\bold{\Psi}} \bigr), E\bigl( {\bold{U}}W^{-1}\big \vert {\bold{y}}_{i}, \hat{\bold{\Psi}} \bigr), E\bigl( {\bold{U}}{\bold{U}}^{\top}W^{-1}\big \vert {\bold{y}}_{i}, \hat{{\bold{\Psi}}} \bigr), and E\bigl( \partial f_W(w| {\bold{\theta)}})/\partial {\bold{\theta)}} \big \vert {\bold{y}}_{i}, \hat{{\bold{\Psi}}} \bigr)

Author(s)

Mahdi Teimouri

References

K. E. Basford, D. R. Greenway, G. J. McLachlan, and D. Peel, (1997). Standard errors of fitted means under normal mixture, Computational Statistics, 12, 1-17.

Examples

      n <- 100
      G <- 2
 weight <- rep( 0.5, 2 )
    mu1 <- rep(-5  , 2 )
    mu2 <- rep( 5  , 2 )
 sigma1 <- matrix( c( 0.4, -0.20, -0.20, 0.5 ), nrow = 2, ncol = 2 )
 sigma2 <- matrix( c( 0.5,  0.20,  0.20, 0.4 ), nrow = 2, ncol = 2 )
lambda1 <- diag( c(-5, -5) )
lambda2 <- diag( c( 5,  5) )
 theta1 <- c( 10 )
 theta2 <- c( 20 )
     mu <- list( mu1, mu2 )
  sigma <- list( sigma1 , sigma2 )
 lambda <- list( lambda1, lambda2)
  theta <- list( theta1 , theta2 )
  param <- c( "a" )
    PDF <- quote( (a/2)^(a/2)*x^(-a/2 - 1)/gamma(a/2)*exp( -a/(2*x) ) )
  tick  <- rep( 1, 2 )
theta10 <- c( 10, 10 )
theta20 <- c( 20, 20 )
 theta0 <- list( theta10 , theta20 )
      Y <- rmix( n, G, weight, model = "unrestricted", mu, sigma, lambda,
      family = "igamma", theta0)
    out <- estep2( Y[, 1:2], G, weight, mu, sigma, lambda, family = "igamma",
    skewness = "TRUE", param, theta, tick, h = 0.001, N = 3000, PDF)

Computing observed Fisher information matrix for restricted finite mixture model.

Description

This function computes the observed Fisher information matrix for a given restricted finite mixture model. For this, we use the method of Basford et al. (1997). The density function of each G-component finite mixture model is given by

g({\bold{y}}|\Psi)=\sum_{g=1}^{G} \omega_{g} f_{\bold{Y}}({\bold{y}}, \Theta_g),

where \bold{\Psi} = \bigl(\bold{\Theta}_{1},\cdots, \bold{\Theta}_{G}\bigr)^{\top} with \bold{\Theta}_g=\bigl({\bold{\omega}}_g, {\bold{\mu}}_g, {{\Sigma}}_g, {\bold{\lambda}}_g\bigr)^{\top}. Herein, f_{\bold{Y}}(\bold{y}, \bold{\Theta}_g) accounts for the density function of random vector \bold{Y} within g-th component that admits the representation given by

{\bold{Y}} \mathop=\limits^d {\bold{\mu}_g}+\sqrt{W}{\bold{\lambda}_g}\vert{Z}_0\vert + \sqrt{W}{\Sigma}_{g}^{\frac{1}{2}} {\bold{Z}}_1,

where {\bold{\mu}_g} \in {R}^{d} is location vector, {\bold{\lambda}_g} \in {R}^{d} is skewness vector, \Sigma_g is a positive definite symmetric dispersion matrix for g =1,\cdots,G. Further, W is a positive random variable with mixing density function f_W(w| \bold{\theta}_g), {Z}_0\sim N(0, 1) , and {\bold{Z}}_1\sim N_{d}\bigl( {\bold{0}}, \Sigma_g \bigr) . We note that W, Z_0, and {\bold{Z}}_1 are mutually independent. For approximating the observed Fisher information matrix of the finite mixture models, we use the method of Basford et al. (1997). Based on this method, using observations {\bold{y}}=({{\bold{y}}_{1},{\bold{y}}_{2},\cdots,{\bold{y}}_{n}})^{\top}, an approximation of the expected information

-E\Bigl\{ \frac{\partial^2 \log L({\bold{\Psi}}) }{\partial \bold{\Psi} \partial \bold{\Psi}^{\top} } \Bigr\},

is give by the observed information as

\sum_{i =1}^{n} \hat{{\bold{h}}}_{i} \hat{{\bold{h}}}^{\top}_{i},

where

\hat{\bold{h}}_{i} = \frac{\partial}{\partial \bold{\Psi} } \log L_{i}(\hat{\bold{\Psi} })

and \log L(\hat{\bold{\Psi} })= \sum_{i =1}^{n} \log L_{i}(\hat{\bold{\Psi} })= \sum_{i =1}^{n} \log \Bigl\{ \sum_{g=1}^{G} \widehat{{\omega}}_g f_{\bold{Y}}\bigl({\bold{y}}_{i}|\widehat{\bold{\Theta}_g}\bigr)\Bigr\}. Herein \widehat{\omega}_g and \widehat{\bold{\Theta}_g} denote the maximum likelihood estimator of \omega_g and \bold{\Theta}_g, for g=1,\cdots,G, respectively.

Usage

 ofim1(Y, G, weight, mu, sigma, lambda, family = "constant", skewness = "FALSE",
      param = NULL, theta = NULL, tick = NULL, h = 0.001, N = 3000, level = 0.05,
    PDF = NULL )

Arguments

Value

A two-part list whose first part is the observed Fisher information matrix for finite mixture model.

Author(s)

Mahdi Teimouri

References

K. E. Basford, D. R. Greenway, G. J. McLachlan, and D. Peel, (1997). Standard errors of fitted means under normal mixture, Computational Statistics, 12, 1-17.

Examples

      n <- 100
      G <- 2
 weight <- rep( 0.5, 2 )
    mu1 <- rep(-5  , 2 )
    mu2 <- rep( 5  , 2 )
 sigma1 <- matrix( c(0.4, -0.20, -0.20, 0.5 ), nrow = 2, ncol = 2 )
 sigma2 <- matrix( c(0.5,  0.20,  0.20, 0.4 ), nrow = 2, ncol = 2 )
lambda1 <- c( 5, -5 )
lambda2 <- c(-5,  5 )
     mu <- list( mu1, mu2 )
 lambda <- list( lambda1, lambda2 )
  sigma <- list( sigma1 , sigma2  )
    PDF <- quote( (b/2)^(a/2)*x^(-a/2 - 1)/gamma(a/2)*exp( -b/(x*2) ) )
  param <- c( "a","b")
 theta1 <- c( 10, 12 )
 theta2 <- c( 10, 20 )
  theta <- list( theta1, theta2 )
  tick  <- c( 1, 1 )
         Y <- rmix(n, G, weight, model = "restricted", mu, sigma, lambda, family = "igamma", theta)
      out <- ofim1(Y[, 1:2], G, weight, mu, sigma, lambda, family = "igamma", skewness = "TRUE",
            param, theta, tick, h = 0.001, N = 3000, level = 0.05, PDF)

Computing observed Fisher information matrix for unrestricted or canonical finite mixture model.

Description

This function computes the observed Fisher information matrix for a given unrestricted or canonical finite mixture model. For this, we use the method of Basford et al. (1997). The density function of each G-component finite mixture model is given by

g({\bold{y}}|\Psi)=\sum_{g=1}^{G} \omega_{g} f_{\bold{Y}}({\bold{y}}, \Theta_g),

where \bold{\Psi} = \bigl(\bold{\Theta}_{1},\cdots, \bold{\Theta}_{G}\bigr)^{\top} with \bold{\Theta}_g=\bigl({\bold{\omega}}_g, {\bold{\mu}}_g, {{\Sigma}}_g, {\bold{\lambda}}_g\bigr)^{\top}. Herein, f_{\bold{Y}}(\bold{y}, \bold{\Theta}_g) accounts for the density function of random vector \bold{Y} within g-th component that admits the representation given by

{\bold{Y}} \mathop=\limits^d {\bold{\mu}}_g+\sqrt{W}{\bold{\lambda}}_g\vert{Z}_0\vert + \sqrt{W}{\Sigma}_g^{\frac{1}{2}} {\bold{Z}}_1,

where {\bold{\mu}}_g \in {R}^{d} is location vector, {\bold{\lambda}}_g \in {R}^{d} is skewness vector, \Sigma_g is a positive definite symmetric dispersion matrix for g =1,\cdots,G. Further, W is a positive random variable with mixing density function f_W(w| \bold{\theta}_g), {Z}_0\sim N(0, 1) , and {\bold{Z}}_1\sim N_{d}\bigl( {\bold{0}}, \Sigma\bigr) . We note that W, Z_0, and {\bold{Z}}_1 are mutually independent. For approximating the observed Fisher information matrix of the finite mixture models, we use the method of Basford et al. (1997). Based on this method, using observations {\bold{y}}=({{\bold{y}}_{1},{\bold{y}}_{2},\cdots,{\bold{y}}_{n}})^{\top}, an approximation of the expected information

-E\Bigl\{ \frac{\partial^2 \log L({\bold{\Psi}}) }{\partial \bold{\Psi} \partial \Psi^{\top} } \Bigr\},

is give by the observed information as

\sum_{i =1}^{n} \hat{{\bold{h}}}_{i} \hat{{\bold{h}}}^{\top}_{i},

where

\hat{\bold{h}}_{i} = \frac{\partial}{\partial \bold{\Psi} } \log L_{i}(\hat{\bold{\Psi} })

and \log L(\hat{\bold{\Psi} })= \sum_{i =1}^{n} \log L_{i}(\hat{\bold{\Psi} })= \sum_{i =1}^{n} \log \Bigl\{ \sum_{g=1}^{G} \widehat{{\omega}}_g f_{\bold{Y}}\bigl({\bold{y}}_{i}|\widehat{\bold{\Theta}_g}\bigr)\Bigr\}. Herein \widehat{\omega}_g and \widehat{\bold{\Theta}_g} denote the maximum likelihood estimator of \omega_g and \bold{\Theta}_g, for g=1,\cdots,G, respectively.

Usage

 ofim2(Y, G, weight, model, mu, sigma, lambda, family = "constant", skewness = "FALSE",
      param = NULL, theta = NULL, tick = NULL, h = 0.001, N = 3000, level = 0.05,
    PDF = NULL )

Arguments

Value

A two-part list whose first part is the observed Fisher information matrix for finite mixture model.

Author(s)

Mahdi Teimouri

References

K. E. Basford, D. R. Greenway, G. J. McLachlan, and D. Peel, (1997). Standard errors of fitted means under normal mixture, Computational Statistics, 12, 1-17.

Examples

      n <- 100
      G <- 2
 weight <- rep( 0.5, 2 )
    mu1 <- rep(-5  , 2 )
    mu2 <- rep( 5  , 2 )
 sigma1 <- matrix( c(0.4, -0.20, -0.20, 0.5 ), nrow = 2, ncol = 2 )
 sigma2 <- matrix( c(0.5,  0.20,  0.20, 0.4 ), nrow = 2, ncol = 2 )
lambda1 <- diag( c( 5, -5 ) )
lambda2 <- diag( c(-5,  5 ) )
     mu <- list( mu1, mu2 )
 lambda <- list( lambda1, lambda2 )
  sigma <- list( sigma1 , sigma2  )
    PDF <- quote( (b/2)^(a/2)*x^(-a/2 - 1)/gamma(a/2)*exp( -b/(x*2) ) )
  param <- c( "a","b")
 theta1 <- c( 10, 12 )
 theta2 <- c( 10, 20 )
  theta <- list( theta1, theta2 )
  tick  <- c( 1, 1 )
         Y <- rmix(n, G, weight, model = "unrestricted", mu, sigma, lambda, family = "igamma",
            theta)
    out <- ofim2(Y[, 1:2], G, weight, model = "unrestricted", mu, sigma, lambda,
            family = "igamma", skewness = "TRUE", param, theta, tick, h = 0.001, N = 3000,
           level = 0.05, PDF)

Creating name for columns and rows of OFI matrix.

Description

This function creates name for columns and rows of OFI matrix.

Usage

ofim_name(G, weight, Dim, lambda, model = "restricted", family = "constant",
skewness = "FALSE", param )

Arguments

Value

Vector of names for columns and rows of OFI matrix.

Author(s)

Mahdi Teimouri

Simulating from inversse Gaussian random variable.

Description

Using method of Michael and Schucany (1976), we can generate from inversse Gaussian random variable. The density function of an inversse Gaussian distribution is given by

f_W(w\vert{\bold{\theta}}) =\sqrt{\frac{\beta}{2 \pi w^3}}\exp\biggl\{-\frac{\beta(w - \alpha)^2}{2\alpha^2 w}\biggr\},

where w>0 and {\bold{\theta}}=(\alpha, \beta)^{\top}. Herein \alpha>0 is the mean and \beta> 0 are the first (mean) and second (shape) parameter of this family, respectively.

Usage

rigaussian(n, alpha, beta)

Arguments

Value

simulated realizations of size n from inversse Gaussian random variable.

Author(s)

Mahdi Teimouri

References

J. R. Michael and Schucany, (1976). Generating Random Variates Using Transformations with Multiple Roots, The American Statistician, 30(2), 88-90, doi: 10.1080/00031305.1976.10479147.

Examples

    n <- 100
alpha <- 4
 beta <- 2
rigaussian(n, alpha, beta)

Generating realization from finite mixture models.

Description

The density function of a restricted G-component finite mixture model can be represented as

{\cal{M}}(\bold{y}|\bold{\Psi})=\sum_{g=1}^{G} \omega_{g} f_{\bold{Y}}(\bold{y}, \bold{\Theta}_g),

where positive constants \omega_{1}, \omega_{2},\cdots,\omega_{G} are called weight (or mixing proportions) parameters with this properties that \sum_{g=1}^{G}\omega_{g}=1 and \bold{\Psi} = \bigl(\bold{\Theta}_{1},\cdots, \bold{\Theta}_{G}\bigr)^{\top} with \bold{\Theta}_g=\bigl({\bold{\omega}}_g, {\bold{\mu}}_g, {{\Sigma}}_g, {\bold{\lambda}}_g\bigr)^{\top}. Herein, f_{\bold{Y}}(\bold{y}, \bold{\Theta}_g) accounts for the density function of random vector \bold{Y} within g-th component that admits the representation given by

{\bf{Y}} \mathop=\limits^d {\bold{\mu}}_{g}+\sqrt{W}{\bold{\lambda}}_{g}\vert{Z}_0\vert + \sqrt{W}{\Sigma}_{g}^{\frac{1}{2}} {\bf{Z}}_1,

where {\bold{\mu}}_{g} \in {R}^{d} is location vector, {\bold{\lambda}}_{g} \in {R}^{d} is skewness vector, and \Sigma_{g} is a positive definite symmetric dispersion matrix for g=1,\cdots,G. Further, W is a positive random variable with mixing density function f_W(w| \bold{\theta}_{g}), {Z}_0\sim N(0, 1) , and {\bold{Z}}_1\sim N_{d}\bigl( {\bold{0}}, \Sigma_{g}\bigr) . We note that W, Z_0, and {\bf{Z}}_1 are mutually independent.

Usage

rmix(n, G, weight, model = "restricted", mu, sigma, lambda, family = "constant",
        theta = NULL) 

Arguments

Value

a matrix with n rows and d + 1 columns. The first d columns constitute n realizations from random vector \bold{Y}=(Y_1,\cdots,Y_d)^{\top} and the last column is the label of realization \bold{Y}_i ( for i = 1, \cdots n ) indicating the component that \bold{Y}_i is coming from.

Author(s)

Mahdi Teimouri

Examples

 weight <- rep( 0.5, 2 )
    mu1 <- rep(-5  , 2 )
    mu2 <- rep( 5  , 2 )
 sigma1 <- matrix( c( 0.4, -0.20, -0.20, 0.4 ), nrow = 2, ncol = 2 )
 sigma2 <- matrix( c( 0.4,  0.10,  0.10, 0.4 ), nrow = 2, ncol = 2 )
lambda1 <- matrix( c( -4, -2,  2,  5 ), nrow = 2, ncol = 2 )
lambda2 <- matrix( c(  4,  2, -2, -5 ), nrow = 2, ncol = 2 )
 theta1 <- c( 10, 10 )
 theta2 <- c( 20, 20 )
     mu <- list( mu1, mu2 )
  sigma <- list( sigma1 , sigma2 )
 lambda <- list( lambda1, lambda2)
  theta <- list( theta1 , theta2 )
      Y <- rmix( n = 100, G = 2, weight, model = "canonical", mu, sigma, lambda,
           family = "igamma", theta )

Generating from multivariate normal distribution with location vector \bold{\mu} and covariance matrix \Sigma_{d \times d}.

Description

Using the well-recognized Cholesky decomposition, this function simulates from the density function of a d-dimensional random vector \bold{Y}=(Y_1,\cdots,Y_d)^{T} following a normal distribution with mean vector \bold{\mu} and covariance matrix \Sigma_{d \times d} is

f_{\bold{Y}}(\bold{y})=\frac{1}{(2 \pi)^{\frac{d}{2}}\vert \Sigma\vert ^{-\frac{1}{2} } } \exp\biggl\{-\frac{(\bold{y}-\bold{\mu})^t\Sigma^{-1}(\bold{y}-\bold{\mu})}{2}\bigg\},

Usage

rmvnorm(n, Mu, Sigma) 

Arguments

Value

an n \times d matrix of realizations from multivariate normal distribution with mean vector \bold{\mu} and covariance matrix \Sigma_{d \times d}.

Author(s)

Mahdi Teimouri

Examples

      n <- 100
    Mu  <- rep(0, 2)
 Sigma  <- matrix( c( 2, 0.50, 0.50, 2 ), nrow = 2, ncol = 2 )
 rmvnorm(n, Mu, Sigma)

Simulating from polynomially tilted \alpha-stable (ptstable) random variable.

Description

Using method of Devroye (2009), we can generate from ptstable random variable. The density function of a ptstable distribution is given by

f_{T}(t|{\bold{\theta}})=\frac{\Gamma(1+\beta)}{\Gamma\Bigl(1+\frac{\beta}{\alpha}\Bigr)}t^{-\beta}f_{P}(t|\alpha),

where {\bold{\theta}}=(\alpha,\beta)^{\top} in which 0<\alpha \leq 2 is tail thickness parameter or index of stability and \beta> 0 is tilting parameter. We note that f_{P}(t|\alpha) is the density function of a positive \alpha-stable distribution that has an integral representation (Kanter, 1975):

f_{P}(t|\alpha)=\frac{1}{\pi}\int_{0}^{\pi}{\frac{\alpha}{2-\alpha}}a(\theta) t^{-\frac{\alpha}{2-\alpha}-1}a(\theta) \exp\Bigl\{-t^{-\frac{\alpha}{2-\alpha}}a(\theta)\Bigr\}d\theta,

for

a(\theta)=\frac{\sin\Bigl(\bigl(1-\frac{\alpha}{2}\bigr)\theta\Bigr)\Bigl[\sin \bigl(\frac{\alpha \theta}{2}\bigr)\Bigr]^{\frac{\alpha}{2-\alpha}}}{[\sin(\theta)]^{\frac{2}{2-\alpha}}}.

Usage

rptstable(n, alpha, beta)

Arguments

Value

simulated realizations of size n from ptstable distribution.

Author(s)

Mahdi Teimouri

References

M. Kanter, (1975). Stable densities under change of scale and total variation inequalities, Annals of Probability, 3(4), 697-707.

L. Devroye, (2009). Random variate generation for exponentially and polynomially tilted stable distributions, ACM Transactions on Modeling and Computer Simulation, 19(4), doi: 10.1145/1596519.1596523.

Examples

    n <- 100
alpha <- 1.4
beta  <- 0.5
rptstable(n, alpha, beta)

Approximating the asymptotic standard error for parameters of the finite mixture models based on the observed Fisher information matrix.

Description

The density function of each finite mixture model can be represented as

{\cal{M}}(\bold{y}|\bold{\Psi})=\sum_{g=1}^{G} \omega_{g} f_{\bold{Y}}(\bold{y}, \bold{\Theta}_g),

where positive constants \omega_{1}, \omega_{2},\cdots,\omega_{G} are called weight (or mixing proportions) parameters with this properties that \sum_{g=1}^{G}\omega_{g}=1 and \bold{\Psi} = \bigl(\bold{\Theta}_{1},\cdots, \bold{\Theta}_{G}\bigr)^{\top} with \bold{\Theta}_g=\bigl({\bold{\omega}}_g, {\bold{\mu}}_g, {{\Sigma}}_g, {\bold{\lambda}}_g\bigr)^{\top}. Herein, f_{\bold{Y}}(\bold{y}, \bold{\Theta}_g) accounts for the density function of random vector \bold{Y} within g-th component that admits the representation given by

{\bold{Y}} \mathop=\limits^d {\boldsymbol{\mu}}_g+\sqrt{W}{\boldsymbol{\lambda}}_g\vert{Z}_0\vert + \sqrt{W}{\Sigma}_g^{\frac{1}{2}}{\bold{Z}}_1,

where {\bold{\mu}_g} \in {R}^{d} is location vector, {\bold{\lambda}_g} \in {R}^{d} is skewness vector, \Sigma_g is a positive definite symmetric dispersion matrix for g=1,\cdots,G. Further, W is a positive random variable with mixing density function f_W(w| \bold{\theta}_g), {Z}_{0}\sim N({0},1), and {\bold{Z}}_{1}\sim N_{d}\bigl({\bold{0}}, \Sigma_g\bigr). We note that W, Z_0, and {\bold{Z}}_{1} are mutually independent. For approximating the asymptotic standard error for parameters of the finite mixture model based on observed Fisher information matrix, we use the method of Basford et al. (1997). In fact, the covariance matrix of maximum likelihood (ML) estimator \hat{\bold{\Psi}}, can be approximated by the inverse of the observed information matrix as

\sum_{i = 1}^{n} \hat{{\bold{h}}}_{i} \hat{{\bold{h}}}^{\top}_{i},

where

\hat{\bold{h}}_{i} = \frac{\partial}{\partial \bold{\Psi} } \log L_{i}(\hat{\bold{\Psi}}),

and \log L(\hat{\bold{\Psi}}) = \sum_{i =1}^{n} \log L_{i}(\hat{\bold{\Psi}}) = \sum_{i =1}^{n} \log \Bigl\{ \sum_{g=1}^{G} \widehat{\omega_{g}} f_{\bold{Y}}\bigl({\bold{y}}_{i}|\widehat{\bold{\Theta}_{g}}\bigr)\Bigr\}. Herein \widehat{\omega}_g and \widehat{\bold{\Theta}_g}, for g=1,\cdots,G, denote the ML estimator of \omega_g and \bold{\Theta}_g, respectively.

Usage

sefm(Y, G, weight, model = "restricted", mu, sigma, lambda, family = "constant",
    skewness = "FALSE", param = NULL, theta = NULL, tick = NULL, h = 0.001, N = 3000,
  level = 0.05, PDF = NULL)

Arguments

Details

Mathematical expressions for density function of mixing distributions f_W(w\vert{\bold{\theta}}), are "bs" (for Birnbaum-Saunders), "burriii" (for Burr type iii), "chisq" (for chi-square), "exp" (for exponential), "f" (for Fisher), "gamma" (for gamma), "gig" (for generalized inverse-Gaussian), "igamma" (for inverse-gamma), "igaussian" (for inverse-Gaussian), "lindley" (for Lindley), "loglog" (for log-logistic), "lognorm" (for log-normal), "lomax" (for Lomax), "pstable" (for positive \alpha-stable), "ptstable" (for polynomially tilted \alpha-stable), "rayleigh" (for Rayleigh), and "weibull" (for Weibull). We note that the density functions of "pstable" and "ptstable" families have no closed form and so are not represented here. The pertinent and given by the following, respectively.

f_{bs}(w\vert{\bold{\theta}}) = \frac {\sqrt{\frac{w}{\beta}}+\sqrt{\frac {\beta}{w}}}{2\sqrt{2\pi}\alpha w}\exp\Biggl\{-\frac {1}{2\alpha^2}\Bigl[\frac{w}{\beta}+\frac{\beta}{w}-2\Bigr]\Biggr\},

where {\bold{\theta}}=(\alpha,\beta)^{\top}. Herein \alpha> 0 and \beta> 0 are the first and second parameters of this family, respectively.

f_{burrii}(w\vert {\bold{\theta}}) = \alpha \beta w^{-\beta-1} \bigl( 1+w^{-\beta} \bigr) ^{-\alpha-1},

where w>0 and {\bold{\theta}}=(\alpha, \beta)^{\top}. Herein \alpha> 0 and \beta> 0 are the first and second parameters of this family, respectively.

f_{chisq}(w\vert{{\theta}}) = \frac{2^{-\frac {\alpha}{2}}}{\Gamma\bigl(\frac{\alpha}{2}\bigr)} w^{\frac{\alpha}{2}-1}\exp\Bigl\{-\frac {w}{2} \Bigr\},

where w>0 and {{\theta}}=\alpha. Herein \alpha> 0 is the degrees of freedom parameter of this family.

f_{exp}(w\vert{{\theta}}) =\alpha \exp \bigl\{-\alpha w\bigr\},

where w>0 and {{\theta}}=\alpha where \alpha> 0 is the rate parameter of this family.

f_{f}(w\vert{\bold{\theta}}) = B^{-1}\Bigl(\frac {\alpha}{2}, \frac {\beta}{2}\Bigr)\Bigl( \frac {\alpha}{\beta} \Bigr)^{\frac {\alpha}{2}} w^{\frac {\alpha}{2}-1}\Bigl(1 + \alpha\frac {w}{\beta} \Bigr)^{-\left(\frac {\alpha+\beta}{2} \right)},

where w>0 and B(.,.) denotes the ordinary beta function. Herein {\bold{\theta}}=(\alpha, \beta)^{\top} where \alpha> 0 and \beta> 0 are the first and second degrees of freedom parameters of this family, respectively.

f_{gamma}(w\vert{\bold{\theta}}) = \frac {\beta^{\alpha}}{\Gamma(\alpha)} \Bigl( \frac{w}{\beta}\Bigr)^{\alpha-1}\exp\bigl\{ - \beta w \bigr\},

where w>0 and {\bold{\theta}}=(\alpha, \beta)^{\top}. Herein \alpha> 0 and \beta> 0 are the shape and rate parameters of this family, respectively.

f_{gig}(w\vert{\bold{\theta}}) =\frac{1}{2{\cal{K}}_{\alpha}( \sqrt{\beta \delta})}\Bigl(\frac{\beta}{\delta}\Bigr)^{\alpha/2}w^{\alpha-1} \exp\biggl\{-\frac{\delta}{2w}-\frac{\beta w}{2}\biggr\},

where {\cal{K}}_{\alpha}(.) denotes the modified Bessel function of the third kind with order index \alpha and {\bold{\theta}}=(\alpha, \delta, \beta)^{\top}. Herein -\infty <\alpha <+\infty, \delta> 0, and \beta> 0 are the first, second, and third parameters of this family, respectively.

f_{igamma}(w\vert{\bold{\theta}}) = \frac{1}{\Gamma(\alpha)} \Bigl( \frac{w}{\beta}\Bigr)^{-\alpha-1}\exp\Bigl\{ - \frac{\beta}{w} \Bigr\},

where w>0 and {\bold{\theta}}=(\alpha, \beta)^{\top}. Herein \alpha> 0 and \beta> 0 are the shape and scale parameters of this family, respectively.

f_{igaussian}(w\vert{\bold{\theta}}) =\sqrt{\frac{\beta}{2 \pi w^3}} \exp\biggl\{-\frac{\beta(w - \alpha)^2}{2\alpha^2 w}\biggr\},

where w>0 and {\bold{\theta}}=(\alpha, \beta)^{\top}. Herein \alpha>0 and \beta> 0 are the first (mean) and second (shape) parameters of this family, respectively.

f_{lidley}(w\vert{{\theta}}) =\frac{\alpha^2}{\alpha+1} (1+w)\exp \bigl\{-\alpha w\bigr\},

where w>0 and {{\theta}}=\alpha where \alpha> 0 is the only parameter of this family.

f_{loglog}(w\vert{\bold{\theta}}) =\frac{\alpha}{ \beta^{\alpha}} w^{\alpha-1}\left[ \Bigl( \frac {w}{\beta}\Bigr)^\alpha +1\right] ^{-2},

where w>0 and {\bold{\theta}}=(\alpha, \beta)^{\top}. Herein \alpha> 0 and \beta> 0 are the shape and scale (median) parameters of this family, respectively.

f_{lognorm}(w\vert{\bold{\theta}}) = \bigl(\sqrt{2\pi} \sigma w \bigr)^{-1} \exp\biggl\{ -\frac{1}{2}\left( \frac {\log w - \mu}{\sigma}\right) ^2\biggr\},

where w>0 and {\bold{\theta}}=(\mu, \sigma)^{\top}. Herein -\infty<\mu<+\infty and \sigma> 0 are the first and second parameters of this family, respectively.

f_{lomax}(w\vert{\bold{\theta}}) = \alpha \beta \bigl( 1+\beta w\bigr)^{-(\alpha+1)},

where w>0 and {\bold{\theta}}=(\alpha, \beta)^{\top}. Herein \alpha>0 and \beta> 0 are the shape and rate parameters of this family, respectively.

f_{rayleigh}(w\vert{{\theta}}) = 2\frac {w}{\beta^2}\exp\biggl\{ -\Bigl( \frac {w}{\beta}\Bigr)^2 \biggr\},

where w>0 and {{\theta}}=\beta. Herein \beta>0 is the scale parameter of this family.

f_{weibull}(w\vert{\bold{\theta}}) = \frac {\alpha}{\beta}\Bigl( \frac {w}{\beta} \Bigr)^{\alpha - 1}\exp\biggl\{ -\Bigl( \frac{w}{\beta}\Bigr)^\alpha \biggr\},

where w>0 and {\bold{\theta}}=(\alpha, \beta)^{\top}. Herein \alpha>0 and \beta> 0 are the shape and scale parameters of this family, respectively.

In what follows, we give four examples. In the first, second, and third examples, we consider three mixture models including: two-component normal, two-component restricted skew t, and two-component restricted skew sub-Gaussian \alpha-stable (SSG) mixture models are fitted to iris, AIS, and bankruptcy data, respectively. In order to approximate the asymptotic standard error of the model parameters, the ML estimators for parameters of skew t and SSG mixture models have been computed through the R packages EMMIXcskew (developed by Lee and McLachlan (2018) for skew t) and mixSSG (developed by Teimouri (2022) for skew sub-Gaussian \alpha-stable). To avoid running package mixSSG, we use the ML estimators correspond to bankruptcy data provided by Teimouri (2022). The package mixSSG is available at https://CRAN.R-project.org/package=mixSSG. In the fourth example, we apply a three-component generalized hyperbolic mixture model to Wheat data. The ML estimators of this mixture model have been obtained using the R package MixGHD available at https://cran.r-project.org/package=MixGHD. Finally, we note that if parameter h is very small (less than 0.001, say), then the approximated observed Fisher information matrix may not be invertible.

Value

A list consists of the maximum likelihood estimator, approximated asymptotic standard error, upper, and lower bounds of 100(1-\alpha)\% asymptotic confidence interval for parameters of the finite mixture model.

Author(s)

Mahdi Teimouri

References

K. E. Basford, D. R. Greenway, G. J. McLachlan, and D. Peel, (1997). Standard errors of fitted means under normal mixture, Computational Statistics, 12, 1-17.

S. X. Lee and G. J. McLachlan, (2018). EMMIXcskew: An R package for the fitting of a mixture of canonical fundamental skew t-distributions, Journal of Statistical Software, 83(3), 1-32, doi: 10.18637/jss.v083.i03.

M. Teimouri, (2022). Finite mixture of skewed sub-Gaussian stable distributions, https://arxiv.org/abs/2205.14067.

C. Tortora, R. P. Browne, A. ElSherbiny, B. C. Franczak, and P. D. McNicholas, (2021). Model-based clustering, classification, and discriminant analysis using the generalized hyperbolic distribution: MixGHD R package. Journal of Statistical Software, 98(3), 1-24, doi: 10.18637/jss.v098.i03.

Examples

# Example 1: Approximating the asymptotic standard error and 95 percent confidence interval
#            for the parameters of fitted three-component normal mixture model to iris data.
      Y <- as.matrix( iris[, 1:4] )
colnames(Y) <- NULL
rownames(Y) <- NULL
      G <- 3
 weight <- c( 0.334, 0.300, 0.366         )
    mu1 <- c( 5.0060, 3.428, 1.462, 0.246 )
    mu2 <- c( 5.9150, 2.777, 4.204, 1.298 )
    mu3 <- c( 6.5468, 2.949, 5.482, 1.985 )
 sigma1 <- matrix( c( 0.133, 0.109, 0.019, 0.011, 0.109, 0.154, 0.012, 0.010,
                      0.019, 0.012, 0.028, 0.005, 0.011, 0.010, 0.005, 0.010 ), nrow = 4 , ncol = 4)
 sigma2 <- matrix( c( 0.225, 0.076, 0.146, 0.043, 0.076, 0.080, 0.073, 0.034,
                      0.146, 0.073, 0.166, 0.049, 0.043, 0.034, 0.049, 0.033 ), nrow = 4 , ncol = 4)
 sigma3 <- matrix( c( 0.429, 0.107, 0.334, 0.065, 0.107, 0.115, 0.089, 0.061,
                      0.334, 0.089, 0.364, 0.087, 0.065, 0.061, 0.087, 0.086 ), nrow = 4 , ncol = 4)
     mu <- list(    mu1,    mu2,    mu3 )
  sigma <- list( sigma1, sigma2, sigma3 )
  sigma <- list( sigma1, sigma2, sigma3 )
 lambda <- list( rep(0, 4), rep(0, 4), rep(0, 4) )
   out1 <- sefm( Y, G, weight, model = "restricted", mu, sigma, lambda, family = "constant",
    skewness = "FALSE")
# Example 2: Approximating the asymptotic standard error and 95 percent confidence interval
#            for the parameters of fitted two-component restricted skew t mixture model to
#            AIS data.
      data( AIS )
      Y <- as.matrix( AIS[, 2:3] )
      G <- 2
 weight <- c(  0.5075,  0.4925 )
    mu1 <- c( 19.9827, 17.8882 )
    mu2 <- c( 21.7268,  5.7518 )
 sigma1 <- matrix( c(3.4915, 8.3941, 8.3941, 28.8113 ), nrow = 2, ncol = 2 )
 sigma2 <- matrix( c(2.2979, 0.0622, 0.0622,  0.0120 ), nrow = 2, ncol = 2 )
lambda1 <- ( c( 2.5186, -0.2898 ) )
lambda2 <- ( c( 2.1681,  3.5518 ) )
 theta1 <- c( 68.3088 )
 theta2 <- c(  3.8159 )
     mu <- list(     mu1,     mu2 )
  sigma <- list(  sigma1,  sigma2 )
 lambda <- list( lambda1, lambda2 )
  theta <- list(  theta1,  theta2 )
  param <- c( "nu" )
    PDF <- quote( (nu/2)^(nu/2)*w^(-nu/2 - 1)/gamma(nu/2)*exp( -nu/(w*2) ) )
  tick  <- c( 1, 1 )
   out2 <- sefm( Y, G, weight, model = "restricted", mu, sigma, lambda, family = "igamma",
            skewness = "TRUE", param, theta, tick, h = 0.001, N = 3000, level = 0.05, PDF )
# Example 3: Approximating the asymptotic standard error and 95 percent confidence interval
#            for the parameters of fitted two-component restricted skew sub-Gaussian
#            alpha-stable mixture model to bankruptcy data.
      data( bankruptcy )
      Y <- as.matrix( bankruptcy[, 2:3] );  colnames(Y) <- NULL; rownames(Y) <- NULL
      G <- 2
 weight <- c(  0.553,  0.447 )
    mu1 <- c( -3.649, -0.085 )
    mu2 <- c( 40.635, 19.042 )
 sigma1 <- matrix( c(1427.071, -155.356, -155.356, 180.991 ), nrow = 2, ncol = 2 )
 sigma2 <- matrix( c( 213.938,    9.256,    9.256,  74.639 ), nrow = 2, ncol = 2 )
lambda1 <- c( -41.437, -21.750 )
lambda2 <- c(  -3.666,  -1.964 )
 theta1 <- c( 1.506 )
 theta2 <- c( 1.879 )
     mu <- list(     mu1,     mu2 )
  sigma <- list(  sigma1,  sigma2 )
 lambda <- list( lambda1, lambda2 )
  theta <- list(  theta1,  theta2 )
  param <- c( "alpha" )
  tick  <- c( 1 )
   out3 <- sefm( Y, G, weight, model = "restricted", mu, sigma, lambda, family = "pstable",
            skewness = "TRUE", param, theta, tick, h = 0.01, N = 3000, level = 0.05 )
# Example 4: Approximating the asymptotic standard error and 95 percent confidence interval
#            for the parameters of fitted two-component restricted generalized inverse-Gaussian
#            mixture model to AIS data.
      data( wheat )
      Y <- as.matrix( wheat[, 1:7] ); colnames(Y) <- NULL; rownames(Y) <- NULL
      G <- 3
 weight <- c( 0.325, 0.341, 0.334 )
    mu1 <- c( 18.8329, 16.2235, 0.9001, 6.0826, 3.8170, 1.6604, 6.0260 )
    mu2 <- c( 11.5607, 13.1160, 0.8446, 5.1873, 2.7685, 4.9884, 5.2203 )
    mu3 <- c( 13.8071, 14.0720, 0.8782, 5.5016, 3.1513, 0.6575, 4.9111 )
lambda1 <- diag( c( 0.1308, 0.2566,-0.0243, 0.2625,-0.1259, 3.3111, 0.1057) )
lambda2 <- diag( c( 0.7745, 0.3084, 0.0142, 0.0774, 0.1989,-1.0591,-0.2792) )
lambda3 <- diag( c( 2.0956, 0.9718, 0.0042, 0.2137, 0.2957, 3.9484, 0.6209) )
 theta1 <- c( -3.3387, 4.2822 )
 theta2 <- c( -3.6299, 4.5249 )
 theta3 <- c( -3.9131, 5.8562 )
 sigma1 <- matrix( c(
 1.2936219, 0.5841467,-0.0027135, 0.2395983, 0.1271193, 0.2263583, 0.2105204,
 0.5841467, 0.2952009,-0.0045937, 0.1345133, 0.0392849, 0.0486487, 0.1222547,
-0.0027135,-0.0045937, 0.0003672,-0.0033093, 0.0016788, 0.0056345,-0.0033742,
 0.2395983, 0.1345133,-0.0033093, 0.0781141, 0.0069283,-0.0500718, 0.0747912,
 0.1271193, 0.0392849, 0.0016788, 0.0069283, 0.0266365, 0.0955757, 0.0002497,
 0.2263583, 0.0486487, 0.0056345,-0.0500718, 0.0955757, 1.9202036,-0.0455763,
 0.2105204, 0.1222547,-0.0033742, 0.0747912, 0.0002497,-0.0455763, 0.0893237 ), nrow = 7, ncol = 7 )
 sigma2 <- matrix( c(
 0.9969975, 0.4403820, 0.0144607, 0.1139573, 0.1639597,-0.2216050, 0.0499885,
 0.4403820, 0.2360065, 0.0010769, 0.0817149, 0.0525057,-0.0320012, 0.0606147,
 0.0144607, 0.0010769, 0.0008914,-0.0023864, 0.0049263,-0.0122188,-0.0042375,
 0.1139573, 0.0817149,-0.0023864, 0.0416206, 0.0030268, 0.0490919, 0.0407972,
 0.1639597, 0.0525057, 0.0049263, 0.0030268, 0.0379771,-0.0384626,-0.0095661,
-0.2216050,-0.0320012,-0.0122188, 0.0490919,-0.0384626, 4.0868766, 0.1459766,
 0.0499885, 0.0606147,-0.0042375, 0.0407972,-0.0095661, 0.1459766, 0.0661900 ), nrow = 7, ncol = 7 )
 sigma3 <- matrix( c(
 1.1245716, 0.5527725,-0.0005064, 0.2083688, 0.1190222,-0.4491047, 0.2494994,
 0.5527725, 0.3001219,-0.0036794, 0.1295874, 0.0419470,-0.1926131, 0.1586538,
-0.0005064,-0.0036794, 0.0004159,-0.0034247, 0.0019652,-0.0026687,-0.0044963,
 0.2083688, 0.1295874,-0.0034247, 0.0715283, 0.0055925,-0.0238820, 0.0867129,
 0.1190222, 0.0419470, 0.0019652, 0.0055925, 0.0243991,-0.0715797, 0.0026836,
-0.4491047,-0.1926131,-0.0026687,-0.0238820,-0.0715797, 1.5501246,-0.0048728,
 0.2494994, 0.1586538,-0.0044963, 0.0867129, 0.0026836,-0.0048728, 0.1509183 ), nrow = 7, ncol = 7 )
    mu <- list( mu1, mu2, mu3 )
 sigma <- list( sigma1 , sigma2, sigma3 )
lambda <- list( lambda1, lambda2, lambda3 )
 theta <- list( theta1 , theta2, theta3 )
  tick <- c( 1, 1, 0 )
 param <- c( "a", "b" )
   PDF <- quote( 1/( 2*besselK( b, a ) )*w^(a - 1)*exp( -b/2*(1/w + w) ) )
  out4 <- sefm( Y, G, weight, model = "unrestricted", mu, sigma, lambda, family = "gigaussian",
            skewness = "TRUE", param, theta, tick, h = 0.001, N = 3000, level = 0.05, PDF )

wheat data

Description

These data are about 210 wheat grains belonging to three different varieties (including: Kama, Rosa, and Canadian) on which 7 quantitative variables related to these kernel structures detected by using a soft X-ray visualization technique have been measured. These variables are: area, perimeter, compactness, length of kernel, width of kernel, asymmetry coefficient, length of kernel groove, and class label variable variety.

Usage

data(wheat)

Format

A text file with 8 columns.

References

P. Giordani, M. B. Ferraro and F. Martella, (2020). An Introduction to Clustering with R, Springer, Singapore.

Examples

data(wheat)