Fits a multiple linear regression model using the Lp norm criterion.

Routine RLLP computes estimates of the regression coefficients in a multiple linear regression model y = X β + ɛ under the criterion of minimizing the Lp norm of the deviations for i = 1, …, n of the observed response yi from the fitted response

for a set on n observations and for p ≥ 1. For the case IWT = 0 and IFRQ = 0 the estimated regression coefficient vector,

(output in B) minimizes the Lp norm

The choice p = 1 yields the maximum likelihood estimate for β when the errors have a Laplace distribution. The choice p = 2 is best for errors that are normally distributed. Sposito (1989, pages 36‑40) discusses other reasonable alternatives for p based on the sample kurtosis of the errors.

Weights are useful if the errors in the model have known unequal variances

In this case, the weights should be taken as

Frequencies are useful if there are repetitions of some observations in the data set. If a single row of data corresponds to ni observations, set the frequency fi = ni. In general, RLLP minimizes the Lp norm

The asymptotic variance-covariance matrix of the estimated regression coefficients is given by

where R is from the QR decomposition of the matrix of regressors (output in R) and where an estimate of λ2 is output in SCALE2. An estimated asymptotic variance-covariance matrix of the estimated regression coefficients can be computed following the call to RLLP by invoking routine RCOVB with R and SCALE2.

In the discussion that follows, we will first present the algorithm with frequencies and weights all taken to be one. Later, we will present the modifications to handle frequencies and weights different from one.

Routine RLLP uses Newton’s method with a line search for p > 1.25 and, for p ≤ 1.25, uses a modification due to Ekblom (1973, 1987) in which a series of perturbed problems are solved in order to guarantee convergence and increase the convergence rate. The cutoff value of 1.25 as well as some of the other implementation details given in the remaining discussion were investigated by Sallas (1990) for their effect on CPU times.

In each case, for the first iteration a least-squares solution for the regression coefficients is computed using routine RGIVN. If p = 2, the computations are finished. Otherwise, the residuals from the k-th iteration,

are used to compute the gradient and Hessian for the Newton step for the (k + 1)-st iteration for minimizing the p-th power of the Lp norm. (The exponent 1/p in the Lp norm can be omitted during the iterations.)

For subsequent iterations, we first discuss the p > 1.25 case. For p > 1.25, the gradient and Hessian at the (k + 1)-st iteration depend upon

and

In the case 1.25 < p < 2 and

and the Hessian are undefined; and we follow the recommendation of Merle and Spath (1974). Specifically, we modify the definition of

to the following:

Let V(k+1) be a diagonal matrix with diagonal entries

and let z(k+1) be a vector with elements

In order to compute the step on the (k + 1)-st iteration, the R from the QR decomposition of [V(k+1)]1∕2X is computed using fast Givens transformations. Let R(k+1) denote the upper triangular matrix from the QR decomposition. Routine GIRTS (see Chapter 20, Mathematical Support) is used to solve the linear system [R(k+1)]TR(k+1)d(k+1) = XT z(k+1) for d(k+1) where R(k+1) is from the QR decomposition of [V(k+1)]1∕2X. The step taken on the (k + 1)-st iteration is

The first attempted step on the (k + 1)-st iteration is with α (k+1) = 1. If all of the

are nonzero, this is exactly the Newton step. See Kennedy and Gentle (1980, pages 528‑529) for further discussion.

If the first attempted step does not lead to a decrease of at least one-tenth of the predicted decrease in the p-th power of the Lp norm of the residuals, a backtracking linesearch procedure is used. The backtracking procedure uses a one-dimensional quadratic model to estimate the backtrack constant p. The value of ρ is constrained to be no less that 0.1. An approximate upper bound for p is 0.5. If after 10 successive backtrack attempts, α(k) = ρ1ρ2…ρ10 does not produce a step with a sufficient decrease, then RLLP issues a message with error code 5. For further details on the backtrack line-search procedure, see Dennis and Schnabel (1983, pages 126‑127).

Convergence is declared when the maximum relative change in the residuals from one iteration to the next is less than or equal to EPS. The relative change

in the i-th residual from iteration k to iteration k + 1 is computed as follows:

where s is the square root of the residual mean square from the least-squares fit on the first iteration.

For the case 1 ≤ p ≤ 1.25, we describe the modifications to the previous procedure that incorporate Ekblom’s (1973) results. A sequence of perturbed problems are solved with a successively smaller perturbation constant c. On the first iteration, the least-squares problem is solved. This corresponds to an infinite c. For the second problem, c is taken equal to s, the square root of the residual mean square from the least-squares fit. Then, for the (j + 1)-st problem, the value of c is computed from the previous value of c according to

Each problem is stated as

For each problem, the gradient and Hessian on the (k + 1)-st iteration depend upon

and

where

The linear system [R(k+1)]TR(k+1)d(k+1) = XTz(k+1) is solved for d(k+1) where R(k+1) is from the QR decomposition of [V (k+1)]1∕2X. The step taken on the (k + 1)-st iteration is

where the first attempted step is with α (k+1) = 1. If necessary, the backtracking line-search procedure discussed earlier is used.

Convergence for each problem is relaxed somewhat by using a convergence epsilon equal to max(EPS, 10−j) where j = 1, 2, 3, … indexes the problems (j = 0 corresponds to the least-squares problem).

and let t be the vector with elements ti. The initial estimate of β for the new problem with perturbation constant 0.01c is

where Δc = (0.01c ‑ c) = ‑0.99c, and where d is the solution of the linear system [R(k)]ΤR(k)d = XTt.

Convergence of the sequence of problems is declared when the maximum relative difference in residuals from the solution of successive problems is less than EPS.

The preceding discussion was limited to the case for which IWT = 0 and IFRQ = 0, i.e., the weights and frequencies are all taken equal to one. The necessary modifications to the preceding algorithm to handle weights and frequencies not all equal to one are as follows:

These replacements have the same effect as multiplying the i-th row of X and y by

and repeating the row fi times except for the fact that the residuals returned by RLLP are in terms of the original y and X.

Finally, R and an estimate of λ2 are computed. Actually, R is recomputed because on output it corresponds to the R from the initial QR decomposition for least squares. The formula for the estimate of λ− depends on p.

For p = 1, the estimator for λ2 is given by (McKean and Schrader 1987)

with

where z0.975 is the 97.5 percentile of the standard normal distribution, and where

are the ordered residuals where IRANK zero residuals are excluded. (Note that

For p = 2, the estimator of λ2 is the customary least-squares estimator given by

For 1 < p < 2 and for p > 2, the estimator for λ2 is given by (Gonin and Money 1989)

with

Different straight line fits to a data set are computed under the criterion of minimizing the Lp norm by using p equal to 1, 1.5, 2.0 and 2.5.