UPHESC Assistant Professor Agricultural Statistics PYQs

Concept: Murthy (1957) obtained his unordered estimator by applying the Rao–Blackwell idea to an ordered estimator (Des Raj's), i.e. averaging over all orderings of the same set of units.

1. The unordered sample is a sufficient statistic; conditioning on it cannot increase variance.
2. Hence $V(M)\le V(O)$, with equality only in trivial cases.

Short trick: Unordering = Rao–Blackwellisation ⇒ variance never larger.

Concept: With σ known: $\overline{x}\pm 1.96\frac{\sigma }{\sqrt{n}}$.

1. σ = √25 = 5, n = 100 ⇒ SE = 5/10 = 0.5.
2. 1.96 × 0.5 = 0.98 ⇒ 15 ± 0.98 = (14.02, 15.98).

Short trick: The interval must be symmetric about 15 — only (D) is.

Concept: Genetic advance $GA=k{\sigma }_P{h}^2$ (k = 2.06 at 5% selection intensity).

1. High h² with high GA (as % of mean) indicates additive gene action ⇒ simple selection is effective — statement 1 true (Panse, 1957).
2. Johnson, Robinson and Comstock (1955): heritability is more useful when used together with genetic advance to predict response — statement 2 true.

Short trick: High h² + high GA ⇒ additive genes ⇒ select.

Concept: A full diallel includes all p² crosses — parents, F₁s and their reciprocals (Griffing's method 1).

1. Only a full diallel lets reciprocal (maternal/cytoplasmic) effects be estimated and removed.
2. Half diallel (p(p − 1)/2 or p(p + 1)/2 crosses) assumes no reciprocal differences; partial diallel samples only some crosses.

Short trick: Reciprocals matter ⇒ make reciprocals ⇒ full diallel.

Concept: In an idealised population of size N: $F_t=\frac{1}{2N}+(1-\frac{1}{2N})F_{t-1}$, so $1-F_t=(1-\frac{1}{2N}{)}^t(1-F_0)$.

1. $\mathrm{\Delta }F=\frac{1}{2N}>0$ every generation, so F rises steadily towards 1.
2. Smaller N ⇒ faster rise (random drift + relatedness).

Short trick: Finite N ⇒ F increases by 1/(2N) per generation.

Concept: Comstock and Robinson's NC designs differ in how many parents are sampled for a given number of crosses.

1. NC I (nested): each male is mated to a different set of females ⇒ m males + mf females — the largest number of parents, widest sample of the population.
2. NC III: n random F₂ plants backcrossed to the two parental lines ⇒ n + 2 parents for 2n crosses — intermediate.
3. NC II (factorial): every male × every female ⇒ only m + f parents for mf crosses — the narrowest coverage.
4. Order of coverage: NC I > NC III > NC II.

Short trick: Nested design uses fresh females for every male ⇒ most parents ⇒ best coverage.

Concept: Equal parent index assumes the daughter is the average of sire and dam: $\overline{D}=\frac{S+\overline{M}}{2}$ ⇒ $S=2\overline{D}-\overline{M}$.

1. So the correct equal-parent index is $I=2\overline{D}-\overline{M}$; the printed $2\overline{M}-\overline{D}$ has D and M interchanged — statement 1 false.
2. Krishnan's modified index corrects the daughters' average for the dams' deviation from the herd average using the intra-sire regression: $I=\overline{D}-b(\overline{M}-A)$ — statement 2 true.

Short trick: Sire = 2 × daughters − dams (not the other way).

Concept: Genetic correlation is a ratio involving the genetic covariance and two genetic variances, each estimated with error.

1. $r_G=\frac{{\mathrm{cov}}_G(x,y)}{\sqrt{{\sigma }_{G_x}^2{\sigma }_{G_y}^2}}$ combines three sampling errors, while $h^2$ involves one variance ratio.
2. Hence, from the same data, genetic correlations are estimated far less precisely: SE(h²) < SE(r_G).
3. This is why large samples are needed for reliable genetic correlations (Falconer).

Short trick: More components in the ratio ⇒ larger standard error ⇒ r_G is the least precise.

Concept: The three observations are the labels 1, 2 and 1, so $L(\theta )=p(1)p(2)p(1)={\theta }^2\cdot 2\theta (1-\theta )\cdot {\theta }^2=2{\theta }^5(1-\theta )$.

1. $\log L=\log 2+5\log \theta +\log (1-\theta )$.
2. $\frac{d}{d\theta }\log L=\frac{5}{\theta }-\frac{1}{1-\theta }=0\Rightarrow 5-5\theta =\theta \Rightarrow \hat{\theta }=\frac{5}{6}$.
3. Check by gene counting: allele '1' appears 2 + 1 + 2 = 5 times out of 6 genes ⇒ 5/6.

Short trick: HW model: MLE of θ = (2·n₁ + n₂)/(2n) — simple allele counting.

Concept: $d=\frac{\sum (e_t-e_{t-1}{)}^2}{\sum e_t^2}\approx 2(1-\hat{\rho })$, so 0 ≤ d ≤ 4.

1. DW statistic tests first-order autocorrelation of regression residuals: (a)-(2).
2. d = 2 ⇔ ρ̂ = 0, no autocorrelation: (b)-(4).
3. d below 2 (towards 0) ⇔ ρ̂ > 0, positive autocorrelation: (c)-(1).
4. d above 2 (towards 4) ⇔ ρ̂ < 0, negative autocorrelation: (d)-(3).

Short trick: d ≈ 2(1 − ρ): 0 = perfect positive, 2 = none, 4 = perfect negative.

Concept: (Multi)collinearity = strong linear relationship among the regressors (X variables).

1. It inflates the variances of OLS coefficients (VIF = 1/(1 − R²ⱼ)) and makes signs unstable.
2. Autocorrelation/serial correlation is correlation among the error terms over time, not among regressors.

Short trick: Errors correlated ⇒ autocorrelation; X's correlated ⇒ collinearity.

Concept: A time series = trend (T) + seasonal (S) + cyclical (C) + irregular (I) components.

1. Steady long-run rise in sugar demand → trend: (a)-(3).
2. Price rise every sowing season → seasonal (within-year, recurring): (b)-(1).
3. Prosperity phase of the business cycle → cyclical: (c)-(4).
4. Strike → unpredictable, irregular: (d)-(2).

Short trick: Trend = years, Seasonal = within a year, Cycle = boom/depression, Irregular = accidents/strikes.

Concept: Heteroscedasticity = error variance changing across observations.

1. In cross-section data units differ greatly in size (small vs large farms, poor vs rich households), so the spread of errors grows with scale.
2. Time-series data more typically suffer from autocorrelation.

Short trick: Cross-section ⇒ heteroscedasticity; time series ⇒ autocorrelation.

Concept: Standard results: $R_{1.23}^2=\frac{r_{12}^2+r_{13}^2-2r_{12}{r}_{13}{r}_{23}}{1-r_{23}^2}$ and the tie-corrected Spearman formula adds $\frac{p^3-p}{12}$ to $\sum d^2$ for every group of p tied ranks.

1. In (1), $r_{12}{r}_{23}{r}_{31}$ is the same product as $r_{12}{r}_{13}{r}_{23}$ (since $r_{31}=r_{13}$), and $0\le R_{1.23}\le 1$ — correct.
2. (2) is exactly the tie-corrected Spearman coefficient — correct.

Short trick: Multiple R is never negative; ties: add (p³ − p)/12 per tie group.

Concept: Remedies for heteroscedasticity work either on the scale of the data, the standard errors, or the estimator.

1. (1) Log transformation compresses scale and often stabilises variance ✓.
2. (2) Heteroscedasticity-consistent (White's robust) standard errors ✓.
3. (3) GLS/weighted least squares re-weights observations ✓.
4. (4) Adding lagged values is a remedy for dynamic misspecification/autocorrelation, not heteroscedasticity ✗.

Short trick: Log, robust SE, GLS/WLS — the three cures.

Concept: $a=b_{yx}=r\frac{{\sigma }_y}{{\sigma }_x}$ and $c=b_{xy}=r\frac{{\sigma }_x}{{\sigma }_y}$; both lines pass through $(\overline{x},\overline{y})$.

1. $\frac{c}{a}=\frac{{\sigma }_x^2}{{\sigma }_y^2}$ — (1) true.
2. $ac=r^2$ ⇒ $r=\sqrt{ac}$ (taking the common sign of a and c) — (2) true.
3. Solve $\overline{y}=a\overline{x}+b$, $\overline{x}=c\overline{y}+d$: $\overline{x}=c(a\overline{x}+b)+d$ ⇒ $\overline{x}=\frac{cb+d}{1-ac}$, and $\overline{y}=\frac{ad+b}{1-ac}$ — (3) true.

Short trick: $b_{xy}/b_{yx}={\sigma }_x^2/{\sigma }_y^2$; $b_{xy}{b}_{yx}=r^2$; lines meet at the means.

Concept: Degeneracy = a basic variable takes the value zero in a basic feasible solution.

1. Stage 1: in the initial basic feasible solution (some bᵢ = 0).
2. Stage 2: at a subsequent iteration, when there is a tie in the minimum-ratio test for the leaving variable.
3. Degeneracy can cause cycling; it is resolved by perturbation or Bland's rule.

Short trick: Degeneracy: at the start, or during iterations — two stages.

Concept: For any real t, $E[(X-tY{)}^2]\ge 0$, i.e. $E(X^2)-2tE(XY)+t^{2}E(Y^2)\ge 0$.

1. A quadratic in t that is never negative has discriminant ≤ 0: $4[E(XY){]}^2-4E(X^2)E(Y^2)\le 0$.
2. So $[E(XY){]}^2\le E(X^2)E(Y^2)$, with equality when Y is a multiple of X (hence ≤, not strict <).
3. Applied to deviations from means it gives |r| ≤ 1.

Short trick: Cauchy–Schwarz ⇒ |r| ≤ 1.

Concept: Gradient-based optimisation uses the first derivative to locate an optimum and the second derivative to classify it.

1. (1) Direct (derivative-free) search suits non-differentiable functions ✓.
2. (2) ∇f points in the direction of steepest ascent ✓.
3. (3) The Hessian = matrix of second-order partials = Jacobian of the gradient ✓.
4. (4) Reaching an optimal point is detected by the gradient (first derivative) being zero; the second derivative only tells whether it is a maximum, minimum or saddle ✗.

Short trick: First derivative = 0 → stationary point; second derivative → what kind of point.

Concept: Standard definitions (Kanti Swarup, Taha): solution → feasible solution → basic feasible solution → optimal solution.

1. Solution: any set of values of the decision variables satisfying the constraints (2) — statement 1 true.
2. Feasible solution: a solution that also satisfies the non-negativity restrictions (3) — statement 2 true.
3. Optimal solution: a feasible solution that optimises Z.

Short trick: Constraints ⇒ solution; + non-negativity ⇒ feasible; + best Z ⇒ optimal.