uppose β^1,β^2,⋯ ,β^pβ^1,β^2,⋯,β^pbeta, with, hat, on top, start subscript, 1, end subscript, comma, beta, with, hat, on top, start subscript, 2, end subscript, comma, \@cdots, comma, beta, with, hat, on top, start subscript, p, end subscript denote the estimators of regression coefficients in a linear regression obtained by minimizing
∑i=1n(yi−∑j=1pβjxij)2∑i=1n(yi−∑j=1pβjxij)2sum, start subscript, i, equals, 1, end subscript, start superscript, n, end superscript, left parenthesis, y, start subscript, i, end subscript, minus, sum, start subscript, j, equals, 1, end subscript, start superscript, p, end superscript, beta, start subscript, j, end subscript, x, start subscript, i, j, end subscript, right parenthesis, squared subject to ∑j=1p∣βj∣≤t∑j=1p∣βj∣≤tsum, start subscript, j, equals, 1, end subscript, start superscript, p, end superscript, \vert, beta, start subscript, j, end subscript, \vert, is less than or equal to, t
for some t≥0t≥0t, is greater than or equal to, 0. As we increase ttt from 0, the residual sum of square ∑i=1n(yi−∑j=1pβ^jxij)2∑i=1n(yi−∑j=1pβ^jxij)2sum, start subscript, i, equals, 1, end subscript, start superscript, n, end superscript, left parenthesis, y, start subscript, i, end subscript, minus, sum, start subscript, j, equals, 1, end subscript, start superscript, p, end superscript, beta, with, hat, on top, start subscript, j, end subscript, x, start subscript, i, j, end subscript, right parenthesis, squared
increases initially and then eventually decreases
decreases initially and then eventually increases
increases steadily and then remains constant
decreases steadily and then remains constant