# whuber

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bio website quantdec.com location Northeastern US age 13 member for 3 years, 4 months seen 2 mins ago profile views 15,203

Consultant (environmental and spatial stats a specialty), expert witness, and teacher. I can be reached through (outdated but still valid) links posted on my web site.

 Dec4 comment Correlation between two Decks of cards? To help us better understand what you want, perhaps you could be a little more precise about what you mean by "the order of cards is correlated." Dec4 comment Joint pdf of functions of order statistics The restrictions are immediate from the definitions of order statistics and the $v_i$: we know that two of the bins between $x_2$ and $x_{10}$ have widths of $v_1$ and $v_2$ (while the rest can be arbitrarily small), so $x_2+v_1+v_2$ cannot exceed $x_{10}$. Dec4 comment T-test with sample standard deviation of zero. Possible? @Elvis perhaps a tiny bit: see stats.stackexchange.com/a/35056/919 for instance. Dec4 comment Comparison of two error distributions to determine “goodness of fit” A natural and powerful way to compare two models is to compute the (log) likelihood of the data with respect to each model: the model with the greatest likelihood wins. Your likelihood calculation will be easiest in a QM setting, where the models directly provide transition probabilities. For more information research literature about "likelihood ratios" and maximum likelihood estimation. Dec4 comment T-test with sample standard deviation of zero. Possible? There's more to it than that, which is why I left my remark as a comment: the interpretation of $\bar{x}$ is a little delicate when data have been rounded (or otherwise binned) to the point of making $s=0.$ In addition to using proxies for $s$ based on measurement error or bin width, one can also apply methods of interval arithmetic to this problem. Dec4 comment T-test with sample standard deviation of zero. Possible? I agree with @Nick but would not be quite as pessimistic: often when $s=0$ it is because measurement error was greater than natural variation (and the data set is small); but if by using the SD of measurement error in the formula in place of $s$ you obtain a significant result, then a fortiori you would have an even more significant result without the measurement error. Dec4 comment Compute the inverse of function This question appears to be off-topic because it is only about coding. Dec4 comment The rule of the 3 sigmas -— how many times multiply the sigma to get 85%? Your values are obtained from the cumulative standard Normal distribution. For instance, in R you can use pnorm by typing (say) pnorm(1.2). The answer you will get is 88.5%. The function qnorm is the inverse; e.g., qnorm(0.85) yields the answer 1.036: that's the multiplier to use for 85% coverage. Dec4 comment Joint pdf of functions of order statistics And that's all the information you need: $F$ is the probability integral transform so indeed $F(Y_i)$ is distributed as the $i^\text{th}$ order statistic from a uniform distribution. For a non-rigorous (but nevertheless accurate) heuristic, think of this order statistic as dividing the interval $[0,1]$ into three bins of length $F(Y_i)$, $dY_i$ (an infinitesimal), and $1-F(Y_i)$, into which precisely $i-1$, $1$, and $n-i$ of the values fall (that's where the multinomial coefficients come from). Dec4 comment How to test whether the variance of two distributions is different if the distributions are not normal It would help immensely to have a model, or at least some suggestive theory, that attempts to explain why some peaks would be narrower and others wider. Because you are interested in the widths of these peaks, you must have at least a conceptual model, if not a quantitative one. What mechanisms do you suppose produce such peaks and govern their widths? Do you have independent information that suggests when the peaks ought to occur? (This reduces uncertainty in peak identification.) Do peaks occur contemporaneously or at different times? Dec4 comment Nonlinear regression: best transformation when getting very different parameter estimates I have a couple of practical suggestions then for achieving stable and reliable solutions. (1) Use $\exp(a)$ in model 1 and $a$ in model 2 to obtain comparable results without NaN problems. (2) Run model 2 first and feed its estimates as starting values to model 1. (3) Break model 2 into two parts: only $x_0$ is nonlinear; $a$ and $b$ are efficiently determined by least squares (e.g., use lm). You can then use a one variable application of nls to estimate $x_0$. If you want the full output of nlsLM, feed it those estimates as starting values. Dec4 comment Intuition or explanation of “rule of thumb” for loss of degrees of freedom in chi-square for model selection/knot placement? It might be related to my answer at stats.stackexchange.com/a/17148, but I am at a loss to come up with a quantitative rule of thumb. Dec3 comment Doubt in derivative of logarithm The edit appears to confuse "$x$" with the parameter "$d$". As such it is nonsensical. Take a look at related questions on our site, such as stats.stackexchange.com/questions/32103, which illustrate the procedure generally, or stats.stackexchange.com/questions/4052, which examines a specific probability model. Dec3 comment Does more variables mean tighter confidence intervals? This is a wonderful answer, full of useful information--but it's wrong, for several reasons: (1) you overlook the effect of the reduced degrees of freedom in $t_{n-3}.$ This is profound when $n$ is small (which is exactly when $t$ distributions should be used instead of Normal distributions). (2) You overlook the effect of $\tilde{x}^T(X^TX)^{-1}\tilde{x}$: prediction bands can actually widen with more variables, such as when the new ones have little additional predictive value. Dec3 comment (non) linear regression on graphs with multiple y per x values An example in which a scientist's claims were wholly discredited due to the use of average values in a regression is documented in this 2007 EFSA Review. For recent (amazing) developments see retractionwatch.com/2013/11/28/…: the same scientist misused statistical analyses so badly that six French scientific academies joined to denounce his work. Dec3 comment Doubt in derivative of logarithm The derivative is zero, because the right hand side does not include any $x$, which is the variable with respect to which you are differentiating. If the $x_i$ are assumed to be functions of $x$, then this derivative is incorrect because it does not account for the $d(x_i)/dx.$ Dec3 comment Utility of Probability Generating Function This is not the place to post long polemics or arguments: we expect clear questions related to actual problems you face. Please consult our help center for more guidance about what is on topic here. Dec3 comment Utility of Probability Generating Function In my role as a site moderator, I am merely inviting you to edit your question to make it understandable. Otherwise, it will get no answers or if it does the answers might not appear relevant to you. Dec3 comment Nonlinear regression: best transformation when getting very different parameter estimates The differences in $b$ and $x_0$ do not appear large: check their standard errors. The difference in $a$ is due to a basic mistake with logs: $\log(y) = \log(a) + b\log(x-x_0)$ (which I guess is compensated for in the last line), whence the "a" in the log model should be close to the log of the "a" in the original model. When this change is made (by replacing "a" in the first model with "exp(a)") the changes in all three coefficients are within the ranges indicated by their standard errors. In short, all appears fine: is there really a problem to be resolved here? Dec3 comment Probability and Sampling distribution Have you first searched our site for answers to this question? I see some threads that look useful.