Re: Central Limit Theorem Proof using MGF
In article 9c0djs$9vs$[EMAIL PROTECTED], Glen Barnett [EMAIL PROTECTED] wrote: Herman Rubin [EMAIL PROTECTED] wrote in message 9bv951$[EMAIL PROTECTED]">news:9bv951$[EMAIL PROTECTED]... Any good book on probability using measure theory is likely to have the necessary proofs. I get the impression he has seen the proof in the case of the MGF, and is after a book that gives more explanation of the steps in the argument than is usually found. [He may well not have any measure theory. He may not have any complex variables. That's probably why he's working with MGFs] I know lots of proofs of the Central Limit Theorem. But I do not know any using the MGF directly. The easiest one follows Lindeberg, and shows that there is a bound on the difference between the cdf and the normal cdf by a substitution process. It can be given in an undergraduate course at the advanced calculus level. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399 [EMAIL PROTECTED] Phone: (765)494-6054 FAX: (765)494-0558 = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
Re: Simple ? on standardized regression coeff.
I now think that the betas would have to be within [-1,+1]. Suppose you do a standarized regression with response Y, and have p variables (X matrix) already in. Call the estimates for the current X matrix is beta_hat. Then by induction thru the standard two-step least-squares betas should be [-1,+1]: In the case p=1, beta_hat is [-1,+1]. If it's true for a certain p, then if you add in another variable Z, beta_hat for this new varible (gamma_hat)=inv(Z'RZ)Z'RY. This must also be [-1,+1]. When Z is added in, beta_hat for X becomes inv(X'X)X'(Y-Z*gamma_hat), which are again [-1,+1]. Then it's clear that beta-hats will be [-1,+1]. Hi, thanks for the reply. But is beta really just b/SD_b? In the standardized case, the X and Y variables are centered and scaled. If Rxx is the corr matrix [ ... ] = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
Re: ANCOVA vs. sequential regression
On Mon, 23 Apr 2001, jim clark wrote: On 22 Apr 2001, Donald Burrill wrote: If I were doing it, I'd begin with a full model (or augmented model, in Judd McClelland's terms) containing three predictors: y = b0 + b1*X + b2*A + b3*(AX) + error where A had been recoded to (0,1) and (AX) = A*X.[1] A number of sources (e.g., Aiken West's Multiple regression: testing and interpreting interactions) would recommend centering X first (i.e., subtracting out its mean to produce deviation scores). Yes, this is always an option. Usually recommended to avoid certain computational problems that may arise if the distribution of X has a particularly low coefficient of variation, for example, and if the model contains many variables (and in particular interactions among them). Such problems are unlikely to arise in so simple a model as [1], and are more effectively dealt with when they do arise by deliberately orthogonalizing the predictors. I've never quite understood why deviations from a sample mean, which is after all a random function of the particular sample one has, should be preferred either to the original values of X (unless there ARE distributional problems) or to deviations from some value inherently more meaningful than a sample mean. You might also consider whether dummy coding (0,1), as recommended by Donald, would be best or perhaps effect coding (-1, 1). Also a possibility, of course. Note that the interpretations of the several coefficients (b0, b2, and b3 in particular) change with changes in coding of the dichotomy A. -- DFB. Donald F. Burrill [EMAIL PROTECTED] 348 Hyde Hall, Plymouth State College, [EMAIL PROTECTED] MSC #29, Plymouth, NH 03264 603-535-2597 184 Nashua Road, Bedford, NH 03110 603-472-3742 = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
