Package pearson7. June 22, 2016
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1 Version Date Package pearson7 June 22, 2016 Title Maximum Likelihood Inference for the Pearson VII Distribution with Shape Parameter 3/2 Author John Hughes Maintainer John Hughes Supports maximum likelihood inference for the Pearson VII distribution with shape parameter 3/2 and free location and scale parameters. This distribution is relevant when estimating the velocity of processive motor proteins with random detachment. License GPL (>= 2) Collate 'pearson7.r' 'zzz.r' RoxygenNote NeedsCompilation no Repository CRAN Date/Publication :42:12 R topics documented: dpearson pearson7.fit pearson7.objective ppearson qpearson rpearson summary.pearson Index 9 1
2 2 dpearson7 dpearson7 Evaluate the density for the Pearson VII distribution with shape parameter 3/2. Evaluate the density for the Pearson VII distribution with shape parameter 3/2. dpearson7(x, mu = 0, sigma = 1, log = FALSE) x mu sigma log vector of quantiles. vector of means. vector of scales. logical; if TRUE, probabilities p are given as log(p). If mu is not specified, it assumes the default value of 0. If sigma is not specified, it assumes the default value of 1. The Pearson VII distribution with location µ, scale σ, and shape 3/2 has density f(x) = 1/(2σ)[1 + {(x µ)/σ} 2 ] 3/2. the density. Pearson, K. (1916) Mathematical contributions to the theory of evolution. xix. second supplement to a memoir on skew variation. Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character, 216, ppearson7, qpearson7, rpearson7
3 pearson7.fit 3 curve(dpearson7(x), -5, 5, lwd = 2, n = 500, ylab = "f(x)") curve(dnorm(x), lwd = 2, lty = 2, n = 500, add = TRUE) pearson7.fit Find the MLE for a sample from the Pearson VII distribution with shape parameter 3/2. Find the MLE for a sample from the Pearson VII distribution with shape parameter 3/2. pearson7.fit(y, mu0 = median(y), sigma0 = sqrt(3) * median(abs(y - median(y))), tol = 1e-08) y a vector of observations. mu0 an initial value for µ. sigma0 an initial value for σ. tol the convergence tolerance. This function uses a Newton-Raphson algorithm to find the MLE. The starting values for µ and σ are the sample median and 3 times the sample MAD, respectively. See the reference for details. pearson7.fit returns an object of class pearson7, which is a list containing the following components. theta.hat the estimates of µ and σ. hessian iterations value the Hessian matrix evaluated at theta.hat. the number of iterations required to attain convergence. the value of the log likelihood at theta.hat.
4 4 pearson7.objective pearson7.objective y = rpearson7(100, 100, 10) fit = pearson7.fit(y) fit summary(fit) pearson7.objective Compute the negative log likelihood for a sample. Compute the negative log likelihood for a sample. pearson7.objective(params, y) params y a vector of parameter values. a vector of observations. This function computes the negative log likelihood for (µ, σ) given a sample. This function can be optimized using optim, but it is better to use pearson7.fit. the negative log likelihood. dpearson7, pearson7.fit
5 ppearson7 5 ppearson7 Evaluate the distribution function for the Pearson VII distribution with shape parameter 3/2. Evaluate the distribution function for the Pearson VII distribution with shape parameter 3/2. ppearson7(q, mu = 0, sigma = 1, lower.tail = TRUE, log.p = FALSE) q mu sigma lower.tail log.p vector of quantiles. vector of means. vector of scales. logical; if TRUE (default), probabilities are P (X x), otherwise P (X > x). logical; if TRUE, probabilities p are given as log(p). If mu is not specified, it assumes the default value of 0. If sigma is not specified, it assumes the default value of 1. The Pearson VII distribution with location µ, scale σ, and shape 3/2 has cdf F (x) = {1 + (x µ)/ σ 2 + (x µ) 2 }/2. the probability. dpearson7, qpearson7, rpearson7 curve(ppearson7(x), 0, 5, lwd = 2, ylim = c(0.8, 1), ylab = "F(x)") curve(pnorm(x), lwd = 2, lty = 2, add = TRUE)
6 6 qpearson7 qpearson7 Evaluate the quantile function for the Pearson VII distribution with shape parameter 3/2. Evaluate the quantile function for the Pearson VII distribution with shape parameter 3/2. qpearson7(p, mu = 0, sigma = 1, lower.tail = TRUE, log.p = FALSE) p mu sigma lower.tail log.p vector of probabilities. vector of means. vector of scales. logical; if TRUE (default), probabilities are P (X x), otherwise P (X > x). logical; if TRUE, probabilities p are given as log(p). If mu is not specified, it assumes the default value of 0. If sigma is not specified, it assumes the default value of 1. The Pearson VII distribution with location µ, scale σ, and shape 3/2 has quantile function F 1 (x) = µ + (σ/2)(2x 1)/ x(1 x). the quantile. dpearson7, ppearson7, rpearson7 curve(qpearson7(x), 0, 1, lwd = 2, ylab = expression(f^{-1}*(x))) curve(qnorm(x), lwd = 2, lty = 2, n = 500, add = TRUE)
7 rpearson7 7 rpearson7 Generate random deviates from a Pearson VII distribution with shape parameter 3/2. Generate random deviates from a Pearson VII distribution with shape parameter 3/2. rpearson7(n, mu = 0, sigma = 1) n mu sigma number of observations. vector of means. vector of scales. If mu is not specified, it assumes the default value of 0. If sigma is not specified, it assumes the default value of 1. random deviates. Devroye, L. (1986) Non-Uniform Random Variate Generation. New York: Springer-Verlag. dpearson7, ppearson7, qpearson7 y = rpearson7(1000) hist(y, prob = TRUE, breaks = 100, col = "gray") curve(dpearson7(x), lwd = 2, col = "blue", add = TRUE)
8 8 summary.pearson7 summary.pearson7 Print a summary of a Pearson VII fit. Print a summary of a Pearson VII fit. ## S3 method for class 'pearson7' summary(object, alpha = 0.05, digits = 4,...) object alpha an object of class pearson7, the result of a call to pearson7.fit. the significance level used to compute the confidence intervals. The default is digits the number of significant digits to display. The default is additional arguments. This function displays (1) a table of estimates, (2) the value of the log likelihood, and (3) the number of Newton-Raphson iterations. Each row of the table of estimates shows the parameter estimate and the approximate (1 α)100% confidence interval for the parameter. pearson7.fit
9 Index dpearson7, 2, 4 7 optim, 4 pearson7.fit, 3, 4, 8 pearson7.objective, 4, 4 ppearson7, 2, 5, 6, 7 qpearson7, 2, 5, 6, 7 rpearson7, 2, 5, 6, 7 summary.pearson7, 8 9
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